Recent zbMATH articles in MSC 32https://www.zbmath.org/atom/cc/322021-02-12T15:23:00+00:00WerkzeugThe Taub-NUT ambitoric structure.https://www.zbmath.org/1452.320312021-02-12T15:23:00+00:00"Gauduchon, Paul"https://www.zbmath.org/authors/?q=ai:gauduchon.paulAn \textit{ambitoric structure} on an orientable \(4\)-manifold \(M\) is a pair \(((g_+, J_+), (g_-, J_-))\) formed by two Kähler structures satisfying the following conditions: (1) the Riemannian metric \(g_-\) is conformal to \(g_+\); (2) the complex structure \(J_-\) determines an orientation which is opposite to the one of \(J_+\); (3) both structures are toric with respect to same action of a \(2\)-torus. The main result of this paper consists in the proof of the existence of a one-parameter
family of ambitoric structures on \(\mathbb R^4\), which are naturally associated with the one-parameter family of the euclidean Taub-NUT metrics of \(\mathbb R^4\). More precisely, let \(T\) be the vector field on \(\mathbb R^4\) that generates the one-parameter group of diffeomorphisms, given by the right multiplications of quaternions by the complex numbers of \(e^{i t}\). Consider the (locally defined) \(T\)-invariant hyperkähler structures \((g^{(V)}, I_1^{(V)}, I_2^{(V)}, I_3^{(V)})\) on \(\mathbb R^4\) that are determined by some positive harmonic function \(V\) by means of the Gibbons-Hawking construction. The author shows that each such potential \(V\) determines not only a (positively oriented) \(T\)-invariant hyperkähler structure, but also a \(T\)-invariant almost-Kähler structure \((\widetilde g^{(V)}, J^{(V)})\) with opposite orientation. He then proves that the only \(V\), whose associated negatively oriented almost-Kähler structure \((\widetilde g^{(V)}, \widetilde J^{(V)})\) is actually Kähler, are either the affine functions of the coordinates \(x^1, x^2, x^3\) or have the form \(V = A + \frac{B}{\sqrt{\sum_{i = 1}^3 (x^i - x^i_o)}}\) for some constants \(A\), \(B\). From this the author infers that the only \(V\)'s which give a (positively oriented) hyperkähler structure and a negatively oriented Kähler structure, both defined over the whole \(\mathbb R^4\), are those of the euclidean Taub-NUT metrics, i.e., the \(V\) of the form \(V_a = a^2 + \frac{1}{\rho^2}\), \(a\geq 0\). He finally shows that, for any \(a \geq 0\), the pair of Kähler structures
\[
\left( (g^a_+ := g^{(V_a)}, J_+ := I_1^{(V_a)}); (g^a_- := \widetilde g^{(V_a)}, J_-:= \widetilde J^{(V_a)})\right)
\]
is an ambitoric structure on \(\mathbb R^4\). Both such Kähler structures can be identified with complete Kähler structures on \(\mathbb C^2\).
For the entire collection see [Zbl 1408.14005].
Reviewer: Andrea Spiro (Camerino)An approximation theorem for Hardy functions on \(\mathbb{C} \)-linearly convex domains of infinite type in \(\mathbb{C}^2\).https://www.zbmath.org/1452.320172021-02-12T15:23:00+00:00"Ha, Ly Kim"https://www.zbmath.org/authors/?q=ai:ha.ly-kimSummary: Let \(\varOmega\subset\mathbb{C}^2\) be a smoothly bounded, \(\mathbb{C}\)-linearly convex domain. We prove that if \(\varOmega\) is of \(F\)-type at all boundary points (for some type function \(F)\), then for all \(1\le p < \infty \), every Hardy function \(f\in H^p(\varOmega)\) is approximated by a sequence of holomorphic functions on \(\overline{\Omega }\) in the \(H^p\)-norm.Extensions of bounded holomorphic functions on the tridisk.https://www.zbmath.org/1452.320062021-02-12T15:23:00+00:00"Kosiński, Łukasz"https://www.zbmath.org/authors/?q=ai:kosinski.lukasz"Mccarthy, John E."https://www.zbmath.org/authors/?q=ai:mccarthy.john-eLet \(\Omega\subset\mathbb C^d\) be a bounded domain and let \(\emptyset\neq\mathcal V\subset\Omega\). Assume that \(\mathcal V\) is a relatively polynomial convex subset of \(\Omega\) (i.e., \(\overline{\mathcal V}\) is polynomially convex and \(\overline{\mathcal V}\cap\Omega=\mathcal V\)) and has the polynomial extension property with respect to \(\Omega\) (i.e., for every polynomial \(f\in\mathbb C[z_1,\dots,z_n]\) there exists a \(\varphi\in H^\infty(\Omega)\) such that \(\varphi=f\) on \(\mathcal V\) and \(\sup_\Omega|\varphi|=\sup_{\mathcal V}|f|\)). The authors study the problem whether \(\mathcal V\) is a holomorphic retract of \(\Omega\). It is known that the answer is positive in the case where e.g. \(\Omega=\mathbb D^2\) is the bidisk or \(\Omega\) is the Euclidean ball. The authors discuss the case where \(\Omega=\mathbb D^3\) and they prove the following two theorems:
\begin{itemize}
\item Assume that \(\dim\mathcal V=1\). If \(\mathcal V\) is algebraic or has polynomially convex projections, then \(\mathcal V\) is a retract of \(\mathbb D^3\).
\item Assume that \(\dim\mathcal V=2\). Then either \(\mathcal V\) is a retract or there exist domains \(U_r\subset\mathbb D^2\) and a holomorphic functions \(h_r:U_r\longrightarrow\mathbb D\), \(j=1,2,3\), such that \(\mathcal V =\{(z_1, z_2, h_3(z_1, z_2)) : (z_1, z_2)\in U_3\}= \{(z_1, h_2(z_1, z_3), z_3) : (z_1 , z_3)\in U_2\}=\{(h_1(z_2, z_3), z_2, z_3) : (z_2, z_3)\in U_1\}\).
\end{itemize}
Reviewer: Marek Jarnicki (Kraków)Homogeneous models for Levi degenerate CR manifolds.https://www.zbmath.org/1452.320422021-02-12T15:23:00+00:00"Santi, Andrea"https://www.zbmath.org/authors/?q=ai:santi.andreaLet \((M, \mathcal D, J)\) be a CR manifold of hypersurface type, i.e., a \((2n+1)\)-dimensional manifold \(M\), equipped with a contact distribution \(\mathcal D\subset TM\) and a smooth field of complex structures \(J_x: \mathcal D_x \to \mathcal D_x\), for which the corresponding \((+i)\)-eigendistribution \(\mathcal D^{10}\subset\mathcal D^{\mathbb C}\) is involutive. For any complex subbundle \(\mathcal {\mathcal K} \subset T^{\mathbb C} M\), let us denote by \(\underline{\mathcal K}\) the space of the sections of \(\mathcal K\). Consider the nested sequence of spaces of complex vector fields \[\underline{\mathcal F_{-1}} \supset \underline {\mathcal F_0} \supset \underline {\mathcal F_1} \supset \ldots \supset \underline {\mathcal F_{p-1}} \supset \underline {\mathcal F_{p}} \supset \ldots \] inductively defined by \[\underline {\mathcal F_{-1}}:= \underline{\mathcal D^{\mathbb C}_{-1}}\ ,\ \ \underline {\mathcal F_{p}}:= \underline{ \mathcal F^{10}_{p}} + \overline{\underline{\mathcal F_{p}^{01}}}\ \ \text{where}\ \underline {\mathcal F^{10}_{-1}}:= \underline {\mathcal D^{10}_{-1}}\] \[\text{and}\ \ \underline{\mathcal F^{10}_p} := \big\{ X \in \underline{\mathcal F^{10}_{p-1}}:[X, \underline{\mathcal D^{10}}] = 0{\hskip -0.3 cm}\mod {\hskip -0.1 cm}\underline{\mathcal F^{10}_{p-1}}{\oplus} \underline{\mathcal D^{01}}\big\}\ ,\] and assume that each space \(\underline {\mathcal F_{p}}\) consists of the sections of some regular complex subbundle \(\mathcal F_p \subset \mathcal D^{\mathbb C}\). The CR manifold \((M, \mathcal D, J)\) is called \textit{(Levi) \(k\)-nondegenerate} if there is an integer \(p\) such that \(\mathcal F_p = 0\) and \(p = k-1\) is the smallest integer for which this occurs. Note that a CR manifold of hypersurface type is Levi nondegenerate in the usual sense if and only if it is \(1\)-non\-degenerate.
The main results of this paper establish new CR invariants for this class of CR manifolds of hypersurface type. In fact, the author at first shows that, at all points \(x\) of a \(k\)-nondegenetate CR manifold \(M\), there are naturally associated graded vector spaces, called \textit{cores}, \[\mathfrak m_x = \mathfrak m^{-2}_x + \mathfrak m^{-1}_x + \mathfrak m^{0}_x + \ldots + \mathfrak m^{k-2}_x\ ,\] which are generalisations of the classical Tanaka symbols \(\mathfrak m_x = \mathfrak m_x^{-2} + \mathfrak m_x^{-1}\) associated with the points of any Levi non-degenerate hypersurface. The \textit{cores} are preserved under local CR equivalences and, if \((M, \mathcal D, J)\) is locally homogenous, all of its cores are isomorphic one to the other. The author then introduces two definitions, the notion of \textit{abstract core} and the notion of \textit{model for an abstract core \(\mathfrak m\)}. The first concept captures all of the algebraic properties of the cores of the \(k\)-nondegenerate CR manifolds. The second characterises the Lie algebras of the infinitesimal CR automorphisms of the locally homogeneous CR manifolds with cores isomorphic to a given abstract core \(\mathfrak m\). Finally the author classifies the abstract cores, which might be associated with the \(7\)-dimensional \(k\)-nondegenerate CR manifolds (note that in dimension \(7\) only the values \(k=2\) or \(k = 3\) are allowed) and determines all models for such cores that satisfy appropriate simplifying assumptions. The locally homogeneous CR manifolds corresponding to such models are proven to be CR inequivalent one to the other and provide an interesting new class of examples of \(k\)-nondegenerate CR manifolds.
Reviewer: Andrea Spiro (Camerino)An analogue of the squeezing function for projective maps.https://www.zbmath.org/1452.320162021-02-12T15:23:00+00:00"Nikolov, Nikolai"https://www.zbmath.org/authors/?q=ai:nikolov.nikolai-marinov"Thomas, Pascal J."https://www.zbmath.org/authors/?q=ai:thomas.pascal-jAuthors' abstract: ``In the spirit of Kobayashi's applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel's work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong-Rosay theorem.''
As a matter of fact, the authors' detailed results are:
Theorem~1.
\begin{itemize}
\item[1.] For every \(d\in {\mathbb{N}}^*\), there is \(r_d>0\) such that for any proper convex domain \(D\subset {\mathbb{R}}^d\), for any \(z\in D\), \(s_D(z)\geq r_d\).
\item[2.] If \(D\subset {\mathbb{R}}^d\) is a domain such that \(\inf_{z\in D}s_D(z)>0\), then \(D\) is convex and proper.
\end{itemize}
Theorem~2. Let \(D\subset {\mathbb{R}}^n\) be a convex domain, and \(p\in\partial D\). If \(p\) is a strictly convex boundary point, then \(\lim_{x \rightarrow p}s_D(x)=1\).
Moreover, using the squeezing function and their Theorem~2, the authors give a new and short proof of a result, which is related to the Wong-Rosay theorem:
Theorem~5. Let \(D\) be a domain in \({\mathbb{R}}^d\). Assume that there exist points \(p\in\partial D\), \(q\in D\) and a sequence \((\varphi_j)\) of projective automorphisms of \(D\) such that \(q_j:=\varphi_j(q)\rightarrow p\). If \(p\) is strictly convex, then \(D\) is projectively equivalent to the unit ball.
Reviewer: Juhani Riihentaus (Joensuu)A vector bundle version of the Monge-Ampère equation.https://www.zbmath.org/1452.320472021-02-12T15:23:00+00:00"Pingali, Vamsi Pritham"https://www.zbmath.org/authors/?q=ai:pingali.vamsi-prithamThe aim of this paper is to introduce a vector bundle version of the complex Monge-Ampère equation motivated by the study of stability conditions involving higher Chern forms. Let \(M\) be a compact complex \(n\)-dimensional manifold, \(h\) be a metric on a holomorphic vector bundle \(E\) over \(M\). The vector bundle Monge-Ampère equation is
\[
\left(\frac {i\Theta_h}{2\pi}\right)^n=\eta \mathrm{Id},
\]
where \(\Theta_h\) is the curvature of the Chern connection of \((E,h)\) and \(\eta\) is a given volume form. Motivated by the Riemann surface case, the existence of a solution of the vector bundle complex Monge-Ampère equation would require some positivity condition on \(i\Theta_h\).
The first main result of the paper provides some consequences of the existence of a positively curved solution to this equation: stability (involving the second Chern character) and a Kobayashi-Lübke-Bogomolov-Miyaoka-Yau-type inequality. Let \(E\) be a holomorphic rank-\(2\) vector bundle on a smooth projective surface \(M\), \(\eta>0\) is a given volume form. If there exists a smooth metric \(h\) such that \((i\Theta_h)^2=\eta \mathrm{Id}\) and \(i\Theta_h\) is Griffiths positive, then the following hold:
\begin{itemize}
\item[(1)] Stability: If \(E\) is indecomposable then \(E\) is \(MA\)-stable.
\item[(2)] Chern class inequality: \(c_1^2(E)-4c_2(E)\leq 0\) with equality holding if and only if \(\Theta\) is projectively flat.
\end{itemize}
The second main result of this paper is the following theorem. Let \((L,h_0)\) be a holomorphic line bundle over a compact Riemann surface \(X\) such that its curvature \(\Theta_0\) defines a Kähler form \(\omega_{\Sigma}=i\Theta_0\) over \(M\). Assume that the degree of \(L\) is \(1\). Let \(r_1,r_2\geq 2\) be two integers, and \(\phi\in H^0(X,L)\) which is not identically \(0\). Then the following are equivalent.
\begin{itemize}
\item[(1)] Stability: \(r_1>r_2\).
\item[(2)] Existence: There exists a smooth metric \(h\) on \(L\) such that the curvature \(\Theta_h\) of its Chern connection \(\nabla_h\) satisfies the Monge-Ampère Vortex equation
\[
i\Theta_h=(1-|\phi|_h^2)\frac {\mu\omega_{\Sigma}+i\nabla_h^{1,0}\phi\wedge\nabla^{0,1}\phi^{*_h}}{(2r_2+|\phi|_h^2)(2+2r_2-|\phi|_h^2)},
\]
where \(\mu=2(r_2(r_1+1)+r_1(r_2+1))\) and \(\phi^{*_h}\) is the adjoint of \(\phi\) with respect to \(h\) when \(\phi\) is considered as an endomorphism of the trivial line bundle to \(L\).
\end{itemize}
Moreover, if a solution \(h\) of the above equation satisfying \(|\phi|_h^2\leq 1\) exists, then it is unique.
Reviewer: Rafał Czyz (Krakow)Iitaka conjecture. An introduction.https://www.zbmath.org/1452.140022021-02-12T15:23:00+00:00"Fujino, Osamu"https://www.zbmath.org/authors/?q=ai:fujino.osamuLet \(X\) be a complex projective manifold, and denote by \(K_X := \det \Omega_X\) its canonical bundle. The Kodaira dimension of \(X\) measures the positivity of the canonical bundle by looking at the asymptotic growth of sections:
if the linear system \(|mK_X|\) is non-empty for some \(m \in \mathbb N\), one defines
\[
\kappa (X) := \limsup_{m \to \infty} \frac{
\log \dim H^0(X, m K_X)
}
{
\log m
}.
\]
If no positive multiple of \(K_X\) has a non-zero global section, one simply sets \(\kappa(X)=-\infty\). More geometrically speaking, see
[\textit{S. Iitaka}, Algebraic geometry. An introduction to birational geometry of algebraic varieties. New York - Heidelberg - Berlin: Springer-Verlag (1982; Zbl 0491.14006)], the Kodaira dimension is the dimension of the image \(Y\) of the Iitaka fibration
\[
\varphi_{|m K_X|} : X \dashrightarrow Y
\]
for \(m \in \mathbb N\) sufficiently divisible. Since a general fibre \(F\) of the Iitaka fibration has \(\kappa(F)=0\), a rather special property, one can aim to understand the geometry of \(X\) in terms of the base \(Y\) and the general fibre \(F\). Note however that this strategy can only succeed if the Kodaira dimension behaves well with respect to fibrations. The famous Iitaka conjecture \(C_{n,m}\) claims that one has indeed a simple subadditivity property: let \(f: X \rightarrow Y\) be a surjective morphism with connected fibres from a projective manifold \(X\) of dimension \(n\) onto a projective manifold \(Y\) of dimension \(m\). Then one has
\[
\kappa(X) \geq \kappa(X_y) + \kappa(Y)
\]
where \(X_y\) is a sufficiently general fibre. This conjecture is a central tool in Iitaka's program aiming for a birational classification of projective varieties in terms of irregularity and Kodaira dimension.
After forty years of investigation Iitaka's conjecture is still open, although many famous mathematicians have proven special cases and the statement is known to be a consequence of the minimal model program (including the abundance conjecture). An important by-product of the work on Iitaka's conjecture, initiated by
foundational papers of \textit{T. Fujita} [J. Math. Soc. Japan 30, 779--794 (1978; Zbl 0393.14006)], \textit{Y. Kawamata} [Invent. Math. 66, 57--71 (1982; Zbl 0461.14004)], \textit{J. Kollár} [Adv. Stud. Pure Math. 10, 361--398 (1987; Zbl 0659.14024)] and \textit{E. Viehweg} [Compos. Math. 35, 197--223 (1977; Zbl 0357.14014); Adv. Stud. Pure Math. 1, 329--353 (1983; Zbl 0513.14019)], is the theory of positivity of direct image sheaves
that has found many applications in classification theory of projective varieties. It is also the starting point of the book under review. Let me give a short, rather incomplete, summary of its contents:
In the first two chapters, the author explains the basic technique and the numerous tricks involved in proving statements about direct image sheaves. He gives a review of classical notions like Viehweg's weak positivity and statements like Kollár torsion-freeness, and combines them with more recent results like the weak semistable reduction of Abramovich-Karu or the effective freeness of Popa-Schnell.
After this technical preparation, Section 4 contains complete proofs of important special cases, most notably when the base \(Y\) is of general type or when the general fibre \(X_y\) is of general type. While the proof of the latter case traditionally involves a fair amount of Hodge theory,
the author uses the existence of minimal models proven by \textit{C. Birkar} et al. [J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)] to obtain a simplified proof of a slightly sharper result.
Finally in Section 5 the author goes back to another central aspect of Iitaka's program: a classification theory that includes also open, i.e. quasi-projective manifolds. Given a quasi-projective manifold \(V\) one chooses a compactification \(V \subset X\) by a projective manifold \(X\) such that the complement \(D:= X \setminus V\) is a simple normal crossing divisor. The Kodaira dimension \(\kappa(V)\) is then defined as the Kodaira dimension of the log-canonical divisor \(K_X+D\) which is in fact independent of the choices made in the construction. Given
a fibration \(f: X \rightarrow Y\) and boundary divisors \(D_X \subset X\) (resp. \(D_Y \subset Y\)) such that \(f^* D_Y \subset D_X\) one can now formulate the even more difficult log-version of Iitaka's conjecture \(\bar C_{n,m}\). The author gives a mostly self-contained proof of
\(\bar C_{n,n-1}\), initially proven by \textit{Y. Kawamata} [in: Proc. int. Symp. on algebraic geometry, Kyoto 207--217 (1977; Zbl 0437.14018)].
The importance of Iitaka's conjecture is obvious to anyone who is aware of the notion of Kodaira dimension and its rôle in classification theory.
It is also notoriously difficult. Yet even if one does not hope to prove Iitaka's conjecture, one might want to take a closer look at this book:
the literature on positivity of direct image sheaves is vast and highly technical, this text gathers the most fundamental tools and makes Viehweg's theory more accessible. The use of the more recent contributions to the minimal model program simplifies some arguments, exposed in the author's precise style of writing.
Reviewer: Andreas Höring (Nice)Equality of norms for a class of Bloch and symmetrically weighted Lipschitz spaces of vector valued functions and derivation inequalities for Pick functions.https://www.zbmath.org/1452.320082021-02-12T15:23:00+00:00"Jocić, Danko R."https://www.zbmath.org/authors/?q=ai:jocic.danko-rSummary: If \(\mathcal{X}\) and \(\mathcal{Y}\) are Banach spaces and \(f : \mathbb{B}_{\mathcal{X}} \rightarrow \mathcal{Y}\) is Fréchet differentiable on the open unit ball \(\mathbb{B}_{\mathcal{X}}\) of \(\mathcal{X}\), then for every operator monotone function \(\varphi :(- 1, 1) \rightarrow \mathbb{R}\), which satisfies \(\varphi^{\prime \prime} \geqslant 0\) on \([a, b)\), \[\sup_{a, b \in \mathbb{B}_{\mathcal{X}}, a \neq b} \frac{\| f(a) - f(b) \|}{\sqrt{\varphi^\prime(\| a \|)} \| a - b \| \sqrt{\varphi^\prime(\| b \|)}} = \sup_{a \in \mathbb{B}_{\mathcal{X}}} \frac{\| D f(a) \|}{\varphi^\prime(\| a \|)} . \tag{1}\] This generalizes Holland-Walsh-Pavlović criterium for the membership in Bloch type spaces for functions defined in the unit ball of a Banach space and taking values in another Banach space. We also established relations of the induced Bloch and Lipschitz spaces with other spaces of vector valued functions.General Schwarz lemmata and their applications.https://www.zbmath.org/1452.320262021-02-12T15:23:00+00:00"Ni, Lei"https://www.zbmath.org/authors/?q=ai:ni.leiThe main purpose of this paper is to prove a generalization of Schwarz Lemma for holomorphic maps between Kähler manifolds. In particular the following theorem generalises a previous result of the author [``Liouville theorems and a Schwarz lemma for holomorphic mappings between Kähler manifolds'', Preprint, \url{arXiv:1807.02674}]).
Theorem. Let \(f:M^m \rightarrow N^n\) be a holomorphic map, \(M\) is a complete manifold. If \(M\) is noncompact assume either the bisectional curvature is bounded from below or for some exhaustion function \(\rho\) the following holds: \({{\lim \sup}_{\rho \rightarrow \infty}} (\frac {|\partial \rho|+[\sqrt {-1} \partial {\bar \partial \rho}]_+}{\rho})=0\). Let \(l\leq \dim(M)\) be a positive integer. Assume that the holomorphic sectional curvature of \(N\) satisfies \(H^N (Y) \leq -k|Y|^4\) and \(M\) has Ricci tensor restricted to \(l\)-dimensional subspaces satisfying \(\mathrm{Ric}_l^M\geq -K\), for some \(K\geq0\), \(k>0\). Then \(\sigma_l(x)\leq \frac {2l'}{l'+1} \frac {K}{k}\), where \(\sigma _l (x)={\sum} _{\alpha =1}^l |\lambda _\alpha|^2(x)\) and \(l'=\min\{l,\dim(f(M))\}\), where in normal coordinates \(df(\frac{\partial}{\partial z^\alpha})=\lambda_\alpha \delta _{i\alpha}\frac{\partial}{\partial w_i}\). In particular if \(K=0\) then \(f\) is constant.
Methods are similar to those in [loc. cit.], namely extensions of \(\partial \bar \partial\)-Bochner formulae and viscosity consideration. Moreover applications are discussed.
Reviewer: Antonella Nannicini (Firenze)Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree.https://www.zbmath.org/1452.320342021-02-12T15:23:00+00:00"Brotbek, Damian"https://www.zbmath.org/authors/?q=ai:brotbek.damian"Deng, Ya"https://www.zbmath.org/authors/?q=ai:deng.yaIn [Hyperbolic manifolds and holomorphic mappings. New York, NY: Marcel Dekker, Inc. (1970; Zbl 0207.37902)] \textit{S. Kobayashi} made the following famous conjecture, which is often called the logarithmic Kobayashi conjecture in the literature.
Conjecture (Kobayashi). The complement \(\mathbb{P}^n \backslash D\) of a general hypersurface \(D \subset \mathbb{P}^n\) of sufficiently large degree \(d \geq d_n\) is Kobayashi hyperbolic.
In this paper, the authors prove that in any projective manifold, the complements of general hypersurfaces of sufficiently
large degree, are Kobayashi hyperbolic. The proof also provides an effective lower bound on the degree. The naturality of the results allows us to enunciate the following quite clear version.
Main Theorem. Let \(Y\) be a smooth complex projective variety having dimension at least 2. Fix any very ample line bundle \(A\) on \(Y\). Then for a general smooth hypersurface \(D \in |A^d |\) with
\[
d \geq (n + 2)^{n+3} (n + 1)^{n+3}\sim_{ n \to \infty}e^3 n^{2n+6},
\]
the following assertions hold.
(i) The complement \(Y \backslash D\) is hyperbolically embedded into \(Y\). In particular, \(Y \backslash D\) is Kobayashi hyperbolic.
(ii) For any holomorphic entire curve (possibly algebraically degenerate) \(f : \mathbb{C} \longrightarrow Y\) which is not contained in \(D\), one has
\[
T_f (r, A)\leq N_f^{(1)} (r, D) + C (\log T_f (r, A) + \log r) \vert \vert
.\]
Here \(T_f (r, A)\) is the Nevanlinna order function, \(N_f^{(1)} (r, D)\) is the truncated counting function, and the symbol \(\vert \vert\) means that the inequality holds outside a Borel subset of \((1, + \infty)\) of finite Lebesgue measure.
(iii) The (Campana) orbifold \((Y, (1 - \frac{1}{d})D)\) is orbifold hyperbolic, i.e., there exists no entire curve \(f : \mathbb{C} \longrightarrow Y\) so that
\[
f(\mathbb{C})\not\subset D\quad\text{with}\quad\text{mult}_t (f^* D) \ge d\quad \text{for all}\quad t \in f^{- 1} (D).
\]
(iv) Let \(\pi : X \longrightarrow Y\) be the cyclic cover of \(Y\) obtained by taking the \(d\)-th root along \(D\). Then \(X\) is Kobayashi hyperbolic.
The proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies on the construction of higher order logarithmic connections,
allowing the construction logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials, that the authors are able to construct. See also [\textit{D. Brotbek}, Publ. Math., Inst. Hautes Étud. Sci. 126, 1--34 (2017; Zbl 06827883)] and [\textit{Y. Deng}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 3, 787--814
(2020; Zbl 1447.32042)].
Compare the result (iii), about the orbifold hyperbolicity for generic geometric orbifolds, with [\textit{F. Campana} et al., ``Orbifold hyperbolicity'', Preprint, \url{arXiv:1803.10716}].
A result related with (iv) is due to [\textit{X. Roulleau} et al., J. Lond. Math. Soc., II. Ser. 87, No. 2, 453--477 (2013; Zbl 1276.14053)].
As far as we known, the optimatility of the bound in (i) remains as an open problem. Certainly, the main theorem in the present article is a genuine constribution to the study of the Kobayashi hyperbolicity property.
Reviewer: Jesus Muciño Raymundo (Morelia)Explicit absolute parallelism for 2-nondegenerate real hypersurfaces \(M^5 \subset \mathbb{C}^3\) of constant Levi rank 1.https://www.zbmath.org/1452.320442021-02-12T15:23:00+00:00"Merker, Joël"https://www.zbmath.org/authors/?q=ai:merker.joel"Pocchiola, Samuel"https://www.zbmath.org/authors/?q=ai:pocchiola.samuelIn this paper, the authors solve \textit{in an explicit manner} the biholomorphic equivalence problem between \(5\)-dimensional \(2\)-nondegenerate real hypersurfaces in \(\mathbb C^3\) of constant Levi rank \(1\). Despite the complexity of the appearing computations and motivated by \textit{S. M. Webster}'s remarks in [Duke Math. J. 104, No. 3, 463--475 (2000; Zbl 0971.32019)], their devised \textit{differential-algebraic} approach enables them to eventually discover two fundamental differential invariants \(J\) and \(W\) of the equivalence problem in terms of derivations of the single defining equation of the CR manifolds, under consideration.
In particular, it turns out that when the appearing two differential invariants vanish identically on the CR manifold, then it is holomorphically equivalent to the known tube over the light cone:
\[
(\Re z_1)^2+(\Re z_2)^2+(\Re z_3)^2=0, \quad \Re z_1>0
\]
defined in the complex space \(\mathbb C^3_{\{z_1, z_2, z_3\}}\).
It is also worth to notice that the equivalence problem studied in this work has been also considered before by \textit{A. Isaev} and \textit{D. Zaitsev} [J. Geom. Anal. 23, No. 3, 1571--1605 (2013; Zbl 1281.32030)] and by \textit{C. Medori} and \textit{A. Spiro} [Int. Math. Res. Not. 2014, No. 20, 5602--5647 (2014; Zbl 1305.32021)]. Nevertheless, none of these two works succeeded to achieve the two fundamental differential invariant \(J\) and \(W\), explicitly in terms of the defining function of the CR manifolds in question.
Reviewer: Masoud Sabzevari (Shahr-e Kord)Boundary of the pyramidal equisymmetric locus of \(\mathcal{M}_n\).https://www.zbmath.org/1452.320182021-02-12T15:23:00+00:00"Díaz, Raquel"https://www.zbmath.org/authors/?q=ai:diaz.raquel-gomez"González-Aguilera, Víctor"https://www.zbmath.org/authors/?q=ai:gonzalez-aguilera.victorSummary: The augmented moduli space \(\widehat{\mathcal M}_n\) is a compactification of moduli space \(\mathcal M_n\) obtained by adding stable hyperbolic surfaces. The different topological types of the added stable surfaces produce a stratification of \(\partial \widehat{\mathcal M}_n\). Let \(\mathcal{P}_n \subset \mathcal{M}_n\) be the pyramidal locus in moduli space, i.e., the set of hyperbolic surfaces of genus \(n\) such that the topological action of its preserving-orientation isometry group is the pyramidal action of the dihedral group \(D_n\). The purpose of this paper is to state the complete list of strata in the boundary of \(\mathcal{P}_n\).Lectures on the Ax-Schanuel conjecture.https://www.zbmath.org/1452.140072021-02-12T15:23:00+00:00"Bakker, Benjamin"https://www.zbmath.org/authors/?q=ai:bakker.benjamin"Tsimerman, Jacob"https://www.zbmath.org/authors/?q=ai:tsimerman.jacobSummary: Functional transcendence results have in the last decade found a number of important applications to the algebraic and arithmetic geometry of varieties Xadmitting flat or hyperbolic uniformizations: Pila and Zannier's new proof of the Manin-Mumford conjecture, the proof of the André-Oort conjecture for \(A_g\), and the generic Shafarevich conjecture for hypersurfaces of Lawrence-Venkatesh, to name a few. The key insight (originally stemming from work of Pila and Zannier) is the use of o-minimality to pass between the geometry of Xand that of its uniformizing space. The goal of these lectures is to give a tour through the main elements of the proof of the Ax-Schanuel conjecture for variations of Hodge structures intended for non-experts. We start by introducing the basic notions of o-minimal geometry with a view towards the two algebraization theorems of Pila-Wilkie and Peterzil-Starchenko. We then show how these results are combined with local volume bounds in the style of Hwang-To to prove the Ax-Schanuel conjecture.
These notes originated from the lecture series by the authors at the workshop Shimura varieties and hyperbolicity of moduli spaces, UQAM (Montreal), May 28-June 1, 2018. The authors are grateful to the organizers for the invitation and for the wonderful conference.
For the entire collection see [Zbl 07235518].Meromorphic extensions of \((\cdot,W)\)-meromorphic functions.https://www.zbmath.org/1452.320072021-02-12T15:23:00+00:00"Quang, Thai Thuan"https://www.zbmath.org/authors/?q=ai:quang.thai-thuan"Lam, Lien Vuong"https://www.zbmath.org/authors/?q=ai:lam.lien-vuongSummary: In this paper we establish some classes of subspaces \(W\) of the dual \(F'\) of a locally convex space \(F\) such that every \(F\)-valued \((F, W)\)-meromorphic function (with/without local boundedness) on a domain \(D\) in \(\mathbb{C}^n\), in the sense \(u \circ f\) is meromorphic for all \(u \in W,\) is meromorphic. Further, combining those results with studing on (\textit{BB})-Zorn property we give conditions for Fréchet spaces \(E, F\) and subspaces \(W\) of \(F'\) under which \((F, W)\)-meromorphic functions can be meromorphically extended to a domain \(D\) of \(E\) from a subset \(D \cap E_B\) where \(E_B\) is the linear hull of some balanced convex compact subset \(B\) of \(E\). Using these results we get the answers of the following questions: (1) When does the domain of meromorphy of a \((\cdot,W)\)-meromorphic function on a Riemann domain \(D\) over a Fréchet space coincide with the envelope of holomorphy of \(D\)? (2) When will \((\cdot,W)\)-meromorphic functions be able to extend meromorphically through an analytic subset of codimension \(\geq 2\) of a domain in a Fréchet space?Criterion for existence of a logarithmic connection on a principal bundle over a smooth complex projective variety.https://www.zbmath.org/1452.320222021-02-12T15:23:00+00:00"Gurjar, Sudarshan"https://www.zbmath.org/authors/?q=ai:gurjar.sudarshan-rajendra"Paul, Arjun"https://www.zbmath.org/authors/?q=ai:paul.arjunSummary: Let \(X\) be a connected smooth complex projective variety of dimension \(n \ge 1\). Let \(D\) be a simple normal crossing divisor on \(X\). Let \(G\) be a connected complex Lie group, and \(E_G\) a holomorphic principal \(G\)-bundle on \(X\). In this article, we give criterion for existence of a logarithmic connection on \(E_G\) singular along \(D\).Algebraically hyperbolic manifolds have finite automorphism groups.https://www.zbmath.org/1452.320332021-02-12T15:23:00+00:00"Bogomolov, Fedor A."https://www.zbmath.org/authors/?q=ai:bogomolov.fedor-alekseivich"Kamenova, Ljudmila"https://www.zbmath.org/authors/?q=ai:kamenova.ljudmila"Verbitsky, Misha"https://www.zbmath.org/authors/?q=ai:verbitsky.mishaA complex projective manifold \(M\) is said to be algebraically hyperbolic if there exists a constant \(A>0\) such that the degree of any curve of genus \(g\) in \(M\) is bounded from above by \(A(g-1)\). This notion has its origins in the lecture notes [Proc. Symp. Pure Math. 62, 285--360 (1997; Zbl 0919.32014)] by \textit{J.-P. Demailly}. Among other results in these notes, Demailly showed that every projective (compact) Kobayashi hyperbolic manifold is algebraically hyperbolic. It is not known whether the converse is true. It is well known that the group of holomorphic automorphisms of a compact Kobayashi hyperbolic manifold is finite. The last three statements are the motivations for the main result of this paper: the group of holomorphic automorphisms of an algebraically hyperbolic manifold is finite.
For an algebraically hyperbolic manifold \(M\), let \(\mathrm{Aut}(M)\) denote the group of holomorphic automorphisms of \(M\). The authors' proof involves considering the image \(\mathfrak{G}\) of \(\mathrm{Aut}(M)\) in \(H^{1,1}(M; \mathbb{R})\) and arriving at various contradictions that the study of \(\mathfrak{G}\) leads to if one assumes that \(\mathrm{Aut}(M)\) is infinite. The authors also give a slightly different proof of Demailly's result that every projective Kobayashi hyperbolic manifold is algebraically hyperbolic.
Reviewer: Gautam Bharali (Bangalore)Fermat's cubic, Klein's quartic and rigid complex manifolds of Kodaira dimension one.https://www.zbmath.org/1452.140352021-02-12T15:23:00+00:00"Bauer, Ingrid"https://www.zbmath.org/authors/?q=ai:bauer.ingrid-c"Gleissner, Christian"https://www.zbmath.org/authors/?q=ai:gleissner.christianThe following are shown in [\textit{I. Bauer} and \textit{F. Catanese}, Adv. Math. 333, 620--669 (2018; Zbl 1407.14003)]: (\(1\)) A rigid compact complex surface has Kodaira dimension either \(-\infty\) or \(2\). (\(2\)) For each \(n \geq 3\) and \(2 \leq k \leq n\), there exist rigid compact complex manifolds \(X_{n,k}\) of dimension \(n\) and Kodaira dimension \(k\). (\(3\)) For each \(n \geq 4\), there exist rigid compact complex manifolds of dimension \(n\) and Kodaira dimension \(0\). In [\textit{A. Beauville}, Prog. Math. 39, 1--26 (1983; Zbl 0537.53057)] it has been shown that there exist rigid threefolds of Kodaira dimension \(0\).
This article shows that for each \(n \geq 3\) there exist infinitesimally rigid compact complex manifolds \(\hat{X_n}\) of Kodaira dimension \(1\). As a corollary one obtains the existence of rigid but not infinitesimally rigid compact complex manifolds of dimension \(n \geq 5\) and Kodaira dimension \(3\). The authors first construct infinitesimally rigid normal projective varieties with isolated canonical cyclic quotient singularities with Kodaira dimension \(1\) by taking the quotient of a product of \((n-1)\) Fermat Cubics and the Klein Quartic. Then they compare the deformations of the singular variety with those of a suitable resolution of singularities obtained by techniques of toric geometry to show that the resolutions are rigid compact complex manifolds of Kodaira dimension \(1\).
Reviewer: Jayan Mukherjee (Lawrence)A tour through Mirzakhani's work on moduli spaces of Riemann surfaces.https://www.zbmath.org/1452.320032021-02-12T15:23:00+00:00"Wright, Alex"https://www.zbmath.org/authors/?q=ai:wright.alexSummary: We survey Mirzakhani's work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of nonexperts.Jumping numbers of analytic multiplier ideals (with an appendix by Sébastien Boucksom).https://www.zbmath.org/1452.320372021-02-12T15:23:00+00:00"Kim, Dano"https://www.zbmath.org/authors/?q=ai:kim.dano"Seo, Hoseob"https://www.zbmath.org/authors/?q=ai:seo.hoseobLet \(X\) be a complex manifold, let \(U\subset X\) be an open set, and let \(\varphi\) be a plurisubharmonic function on \(U\). The multiplier ideal sheaf \(\mathcal J(\varphi)\) of \(\varphi\) is the coherent ideal sheaf on \(U\) consisting of holomorphic germs \(u\) such that \(|u|^2e^{-2\varphi}\) is locally Lebesgue integrable. A number \(\alpha>0\) is a jumping number of \(\varphi\) at \(x\), \(\alpha\in\operatorname{Jump}(\varphi)_x\), if \(\mathcal J(c\varphi)_x=\mathcal J(\alpha\varphi)_x\) for \(c\in[\alpha, \alpha+\delta)\) (for some \(\delta>0\)) and \(\mathcal J ((\alpha +\delta)\varphi)_x\varsubsetneq \mathcal J (\alpha\varphi)_x\). The authors study various properties of jumping numbers. Assume additionally that \(\varphi\) is toric in the unit polydisk \(\mathbb D^n\), i.e., \(\varphi(z_1,\dots,z_n)=\varphi(|z_1|,\dots,|z_n|)\), \(z_1,\dots,z_n\in\mathbb D\). It is well known that \(\varphi(z_1,\dots,z_n)=\widehat\varphi(\log|z_1|,\dots,\log|z_n|)\), where \(\widehat\varphi\) is convex and non-decreasing in each variable. Let \(P(\varphi)\) be the Newton convex body of \(\varphi\) defined by \(\widehat\varphi\). The main result of the paper states that if \(n=2\), then the set \(\operatorname{Jump}(\varphi)_0\) has at least one cluster point if and only if at least one of the following conditions holds:
(1) \(x_0:=\inf\operatorname{pr}_1(P(\varphi))> 0\) and \(\{(x_0 , t) : t\in\mathbb R\}\cap P(\varphi) = \emptyset\),
(2) \(y_0:=\inf\operatorname{pr}_2(P(\varphi))> 0\) and \(\{(t, y_0 ) : t\in\mathbb R\}\cap P(\varphi) = \emptyset\).
Moreover, the set of cluster points of \(\operatorname{Jump}(\varphi)_0\) equals \(\{k/m : k\in\mathbb Z_{>0},\; m\in S\}\), where \(S\subset\{x_0, y_0\}\) is such that (\(x_0\in S\) if (1) holds) and (\(y_0\in S\) if (2) holds).
Reviewer: Marek Jarnicki (Kraków)The Miyaoka-Yau inequality and uniformisation of canonical models.https://www.zbmath.org/1452.320322021-02-12T15:23:00+00:00"Greb, Daniel"https://www.zbmath.org/authors/?q=ai:greb.daniel"Kebekus, Stefan"https://www.zbmath.org/authors/?q=ai:kebekus.stefan"Peternell, Thomas"https://www.zbmath.org/authors/?q=ai:peternell.thomas-martin"Taji, Behrouz"https://www.zbmath.org/authors/?q=ai:taji.behrouzAs a result of his proof of the Calabi conjecture, Yau established an inequality known since as the Miyaoka-Yau inequality, concerning Chern classes of holomorphic vector bundles of compact Kähler manifolds, generalizing an inequality known as the Bogomolov-Gieseker inequality. In the paper under review, the authors establish the Miyaoka-Yau inequality in terms of orbifold Chern classes for the tangent sheaf of any complex projective variety of general type with Kawamata log-terminal singularities and nef canonical divisor. In the case of equality for a variety with at worst terminal singularities, they prove that the associated canonical model is the quotient of the unit ball by a discrete group action. The proof involves the introduction of an appropriate definition of Higgs sheaves on singular spaces and an associated notion of stability.
Reviewer: Athanase Papadopoulos (Strasbourg)Gradient estimate of positive eigenfunctions of sub-Laplacian on complete pseudo-Hermitian manifolds.https://www.zbmath.org/1452.350702021-02-12T15:23:00+00:00"Ren, Yibin"https://www.zbmath.org/authors/?q=ai:ren.yibinThe paper's main theme is a classical Cheng-Yau gradient estimate for positive harmonic functions in the setting of complete noncompact pseudo-Hermitian manifolds with bounded geometric conditions. The operator considered is a sub-Laplacian, therefore standard Riemannian techniques are not applicable. The results include an estimate of the greatest lower bound for the \(L^{2}\)-spectrum of the sub-Laplacian. As a typical application of the Cheng-Yau estimate, the author proves the Liouville theorem of positive pseudo-harmonic functions on complete noncompact Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.
Reviewer: Maria Gordina (Storrs)Towards the topological recursion for double Hurwitz numbers.https://www.zbmath.org/1452.140512021-02-12T15:23:00+00:00"Do, Norman"https://www.zbmath.org/authors/?q=ai:do.norman"Karev, Maksim"https://www.zbmath.org/authors/?q=ai:karev.maksim-vSummary: Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers are defined analogously, but with prescribed ramification over both zero and infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz numbers also arise as intersection numbers on moduli spaces.
In this paper, we repackage double Hurwitz numbers as enumerations of branched covers weighted by certain monomials and conjecture that they are governed by the topological recursion. Evidence is provided in the form of the associated quantum curve and low genus calculations. We furthermore reduce the conjecture to a weaker one, concerning a certain polynomial structure of double Hurwitz numbers. Via the topological recursion framework, our main conjecture should lead to a direct connection to enumerative geometry, thus shedding light on the aforementioned conjecture of Goulden, Jackson and Vakil.
For the entire collection see [Zbl 1404.14006].\(L^2\) extension of \(\overline{\partial}\)-closed forms from a hypersurface.https://www.zbmath.org/1452.320462021-02-12T15:23:00+00:00"McNeal, Jeffery D."https://www.zbmath.org/authors/?q=ai:mcneal.jeffery-d"Varolin, Dror"https://www.zbmath.org/authors/?q=ai:varolin.drorIn this paper, the authors study the extension problems of Ohsawa-Takegoshi type in complex analytic geometry. To be precise, let \(X\) be a Stein manifold with Kähler form \(\omega\) and let \(\iota:Z\hookrightarrow X\) be a smooth hypersurface. Let \(L\to X\) be a holomorphic line bundle with smooth Hermitian metric \(e^{-\varphi}\). Assume the line bundle \(E_{Z}\to X\) associated to the smooth divisor \(Z\) has a section \(f_Z\in H^0(X,E_Z)\) and a smooth Hermitian metric \(e^{-\lambda_Z}\) such that \(Z\) is the divisor of \(f_Z\) and
\[
\sup_X |f_Z|^2e^{-\lambda_Z}\le 1.
\]
The authors determine whether all \(L\)-valued \(\bar\partial\)-closed \((0,q)\)-forms on \(Z\) can be extended to \(L\)-valued \(\bar\partial\)-closed forms on \(X\) with certain \(L^2\) estimates. When \(q\ge1\), the authors introduce two different notions of extension. Under certain curvature assumptions, the authors are able to obtain affirmative results for the two types of extension. By comparing Berndtsson's result on compact Kähler manifold, the authors also obtain extension results with cohomology bounds in the Stein case. But they give an example in which there are \(\bar\partial\)-closed \((0,q)\)-forms that do not have \(\bar\partial\)-closed extensions with arbitrarily small \(L^2\) norm at the end of the paper.
The proof of the main result is technical. To obtain \(L^2\) extension, the authors extend the form in an \(\varepsilon\)-neighborhood of the hypersurface \(Z\), then employ the twisted \(\bar\partial\)-equation to obtain a globally \(\bar\partial\)-closed form. To obtain the smooth extension, instead of minimizing over all possible extensions, the authors minimize over a subset of extensions that still have a uniform norm bound, but also carry a fixed amount of mass along the hypersurface.
Reviewer: Liwei Chen (Columbus)Maximality of plurifinely plurisubharmonic functions.https://www.zbmath.org/1452.320392021-02-12T15:23:00+00:00"Nguyen Xuan Hong"https://www.zbmath.org/authors/?q=ai:nguyen-xuan-hong.Let \(\Omega\subset\mathbb C^n\) be open in the plurifine topology, i.e., the weakest topology in which all plurisubharmonic functions on \(\Omega\) are continuous. A function \(u:\Omega\longrightarrow[-\infty,+\infty)\) is called \(\mathcal F\)-plurisubharmonic if \(u\) is \(\mathcal F\)-upper semicontinuous and finely subharmonic on every complex affine line.
We say that \(u\) is \(\mathcal F\)-maximal in \(\Omega\) if for every bounded \(\mathcal F\)-open set \(G\) with \(\overline G\subset\Omega\), and for every \(\mathcal F\)-plurisubharmonic function \(v\) on \(G\) that is bounded from above and extends \(\mathcal F\)-upper semicontinuously to \(\overline G^{\mathcal F}\), if \(v\leq u\) on \(\partial_{\mathcal F}G\), then \(v\leq u\) on \(G\).
We say that \(u\) belongs to \(\mathcal M(\Omega)\) if for every \(\mathcal F\)-open sets \(G\) with \(\overline G\subset\Omega\) there exists a negative plurisubharmonic function \(\varphi\) defined on a Euclidean neighborhood of \(G\) such that \(\forall \varepsilon>0\exists M>1/\varepsilon: \varepsilon\varphi\leq u\) on \(G\cap\{u=-M\}\).
The main result of the paper states that if \(u\in\mathcal M(\varOmega)\), \(u<0\), then \(u\) is \(\mathcal F\)-locally \(\mathcal F\)-maximal in \(\Omega\) if and only if \(u\) is \(\mathcal F\)-maximal in \(\Omega\).
It is a generalization of a theorem from the paper [the author et al., Potential Anal. 48, No. 1, 115--123 (2018; Zbl 1383.32009)].
Reviewer: Marek Jarnicki (Kraków)A brief introduction to Berezin-Toeplitz operators on compact Kähler manifolds.https://www.zbmath.org/1452.320022021-02-12T15:23:00+00:00"Le Floch, Yohann"https://www.zbmath.org/authors/?q=ai:le-floch.yohannThe central theme of this book is to give a short and simple overview on Berezin-Toeplitz operators on compact Kähler manifolds. It contains 9 chapters and an appendix. The first part contains a review of relevant material from complex geometry. Examples are presented with explicit detail and computation, prerequisites have been kept to a minimum. The chapters are devoted to give a proof of the main properties of the Berezin-Toeplitz operators on compact Kähler manifolds: almost complex structures, tangent bundles, complex line bundles with connections, geometric quantization of compact Kähler manifolds, Berezin-Toeplitz operators, Schwartz kernels, \dots The book is carefully designed to supply graduate students. Much of the material, covered in the book still represents an essential prerequisite for anyone who
wants to work in the field. The author have managed to make it readable by non-specialists.
Reviewer: Béchir Dali (Bizerte)On the boundedness of invariant hyperbolic domains.https://www.zbmath.org/1452.320232021-02-12T15:23:00+00:00"Ning, Jiafu"https://www.zbmath.org/authors/?q=ai:ning.jiafu"Zhou, Xiangyu"https://www.zbmath.org/authors/?q=ai:zhou.xiangyuThe main result is a generalization of a result of \textit{A. Kodama} [Proc. Japan Acad., Ser. A 58, 227--230 (1982; Zbl 0515.32011)]. Specifically, let \(K\) be a compact Lie group with a Lie group homomorphism \(\rho\colon K\to\mathrm{GL}({\mathbb C}^n)\), which naturally defines a representation of \(K\) in the space of entire functions \(\mathcal{O}({\mathbb C}^n)\). The authors prove that if every \(K\)-invariant entire function is constant and \(\Omega\subseteq {\mathbb C}^n\) is a \(K\)-invariant orbit convex domain with \(0\in\Omega\), then \(\Omega\) is bounded if and only if it is Kobayashi hyperbolic.
Reviewer: Daniel Beltiţă (Bucureşti)Topology and complex structures of leaves of foliations by Riemann surfaces.https://www.zbmath.org/1452.320252021-02-12T15:23:00+00:00"Sibony, Nessim"https://www.zbmath.org/authors/?q=ai:sibony.nessim"Wold, Erlend Fornæss"https://www.zbmath.org/authors/?q=ai:wold.erlend-fornaessThe authors study Brody hyperbolic holomorphic foliations on compact complex surfaces and address the problem of finding conditions for a generic leaf to be a holomorphic disk. This is motivated by Anosov's conjecture: for a generic holomorphic foliation on \(\mathbb{P}^2\), all but countable many leaves are disks. The main result is the following.
Theorem 1.1. Let \((X,\mathcal{L}, E)\) be a Brody hyperbolic holomorphic foliation on a compact complex manifold of dimension \(2\), where the singular set \(E\) is finite. Assume that there is no compact weakly directed by \(\mathcal{L}^1\), and that all singularities are hyperbolic. Suppose that there exists a sequence \(\lbrace z_j\rbrace\subset X\setminus E\) of points such that \(\rho(z_j)\rightarrow 1\) and \(z_j\rightarrow z\in X\setminus E\). Then there is a non-trivial minimal closed saturated set \(Y\subset X\) such that all but countably many leaves in \(Y\) are disks.
Here \(\rho(z) := k(z,v)/ k_i(z,v)\), where \(k\) denotes the leafwise Kobayashi metric and \(k_i\) the injective Kobayashi metric.
In particular a generic holomorphic foliation on \(\mathbb{P}^2\) satisfies the assumptions of Theorem 1.1. As a consequence of Theorem 1.1, either all leaves are far from resembling the disk or all but countably many leaves are disks (Theorem 1.2).
The authors also provide various examples of foliations with leaves having different topological type.
Reviewer: Judith Brinkschulte (Leipzig)The strata do not contain complete varieties.https://www.zbmath.org/1452.140232021-02-12T15:23:00+00:00"Gendron, Quentin"https://www.zbmath.org/authors/?q=ai:gendron.quentinThe moduli space of holomorphic differentials on Riemann surfaces with prescribed numbers of zeros and multiplicities is called a stratum. In this paper the author shows that any stratum of holomorphic differentials does not contain complete algebraic curves (i.e. no compact Riemann surfaces can be embedded in the stratum algebraically). The proof applies the maximum modulus principle in a cute way to the shortest saddle connections joining zeros of the differentials under the induced flat metric. It remains an open question to determine whether the projectivized strata (i.e. parameterizing the underlying effective canonical divisors) can contain a complete algebraic curve or not. Note that if one considers the strata of strictly meromorphic differentials or non-effective canonical divisors, then they do not contain any complete algebraic curve [\textit{D. Chen}, J. Inst. Math. Jussieu 18, No. 6, 1331--1340 (2019; Zbl 1423.14184)].
Reviewer: Dawei Chen (Chestnut Hill)Deformation theory of scalar-flat Kähler ALE surfaces.https://www.zbmath.org/1452.320282021-02-12T15:23:00+00:00"Han, Jiyuan"https://www.zbmath.org/authors/?q=ai:han.jiyuan"Viaclovsky, Jeff A."https://www.zbmath.org/authors/?q=ai:viaclovsky.jeff-aAn ALE Kähler surface \((X,g,J)\) is a Kähler manifold of complex dimension 2 with an extra property. These are interesting since they are extremal in the sense of Calabi, and they arise as ``bubbles'' in gluing constructions for extremal Kähler metrics. The main result in this paper shows that for any scalar-flat Kähler ALE surface, all small deformations of the complex structure admit scalar-flat Kähler ALE metrics. The proof is analytic in nature. The paper begins with the definition of weighted Hölder spaces which will be used throughout the paper. Then the authors give some analysis of the complex analytic compactifications of Kähler ALE spaces. After that, they study the deformation of complex structures using an adaptation of Kuranishi's theory to ALE spaces. Next a refined gauging procedure is carried out, to construct the Kuranishi family of ``essential'' deformations. After that previous results are generalized or adapted and the main results are obtained.
Reviewer: Gabriela Paola Ovando (Rosario)The Monge-Ampère equation for non-integrable almost complex structures.https://www.zbmath.org/1452.320352021-02-12T15:23:00+00:00"Chu, Jianchun"https://www.zbmath.org/authors/?q=ai:chu.jianchun"Tosatti, Valentino"https://www.zbmath.org/authors/?q=ai:tosatti.valentino"Weinkove, Ben"https://www.zbmath.org/authors/?q=ai:weinkove.benLet \((M^{2n}, J, g)\) be a compact almost Hermitian \(2n\)-manifold and denote by \(\omega = g(J \cdot, \cdot)\) its associated Kähler form. By Yau's Theorem, if \(J\) is integrable and \((M, J, g)\) is Kähler, for any smooth volume form \(\text{vol}^{(F)} = e^F \omega^n\) satisfying the normalisation condition \(\int_M \text{vol}^{(F)} = \int_M \omega^n\), there exists a unique smooth function \(\varphi\) satisfying the conditions
\[ (\omega + \sqrt{-1} \partial \bar \partial \varphi)^n = \text{vol}^{(F)},\qquad \omega + \sqrt{-1} \partial \bar \partial \varphi > 0,\qquad \sup_M \varphi = 0\tag{\(\ast\)}.\]
The result has been generalised by the second and third author in [J. Amer. Math. Soc. 23, 19--40 (2010; Zbl 1208.53075)] to arbitrary Hermitian manifolds up to addition to \(F\) of a (uniquely determined) constant \(b\). In this paper this result is proved to be true on arbitrary compact almost Hermitian manifolds, provided that the operator \(\varphi \mapsto \sqrt{-1} \partial \bar \partial \varphi\) is replaced by the operator \(\varphi \mapsto \frac{1}{2}(d (J d \varphi))^{(1,1)}\), where
\((\cdot)^{(1,1)}\) denotes the natural pointwise projection onto the \(2\)-forms of bidegree \((1,1)\) with respect to the complex structure \(J_x\), \(x \in M\). Note that if \(J\) is integrable, this reduces to the classical definition of the \(\partial \bar \partial\) operator.
The result is crucially based on the following theorem, whose proof represents the hardest part of the whole paper: ``Given a compact almost Hermitian manifold \((M, J, g)\), for any pair of smooth real functions \(F\) and \(\varphi\) satisfying \((\ast)\), there exist a priori \(\mathcal C^\infty\) estimates on \(\varphi\) depending only on \((M, J, g)\) and bounds for \(F\).''
The main result of this paper provides a positive solution to a problem posed by Gromov, provided that \(\partial \bar \partial \varphi\) is understood as \((\frac{1}{2} d (J d \varphi))^{(1,1)}\). The original statement of the problem, where \(\partial \bar \partial \varphi\) is understood as \(\frac{1}{2} d (J d \varphi)\), is known to have a negative answer by the results of \textit{P. Delanoë} [Osaka J. Math. 33, No. 4, 829--846 (1996; Zbl 0878.53030)] and \textit{M. Warren} and \textit{Y. Yuan} [Commun. Pure Appl. Math. 62, No. 3, 305--321 (2009; Zbl 1173.35388)].
Reviewer: Andrea Spiro (Camerino)The optimal jet \(L^2\) extension of Ohsawa-Takegoshi type.https://www.zbmath.org/1452.320152021-02-12T15:23:00+00:00"Hosono, Genki"https://www.zbmath.org/authors/?q=ai:hosono.genkiBasing on the Berndtsson-Lempert method, the author proves the following optimal version of the Ohsawa-Takegoshi extension theorem. Let \(D\subset\mathbb C^n\) be a bounded pseudoconvex domain and \(S\subset D\) be a closed submanifold of codimension \(k\). Let \(G\) be negative plurisubharmonic function on \(D\), continuous on \(D\setminus S\), such that near each point of \(S\) the function \(G\) may be represented in local coordinates as \(\log(|z_1|^2+\dots+|z_k|^2)\) \(+\) a continuous function. Let \(\varphi\) be a continuous plurisubharmonic function on \(D\). Fix an integer \(p\geq2\). Let \(\mathcal I_S\) stand for the ideal sheaf associated to \(S\) and let \(J^{(p-1)}\) be the vector bundle \(\mathcal I_S^{p-1}/\mathcal I_S^p\). Define a Hermitian metric
\[\langle f_x, g_x\rangle_{J^{(p-1)}}:=\lim_{t\to-\infty}\int_{\{U_x\cap\{t<G<t+1\}\}} f_x\overline g_x e^{-\varphi-(k+p-1)G}dV_{U_x},\] where \(f_x, g_x\in J_x^{(p-1)}\), \((z_1,\dots,z_n)\) are local variables in a neighborhood \(U\) of \(x\) such that \(S\cap U=\{z_1,\dots,z_k=0\}\), \(G=\log(|z_1|^2+\dots+|z_k|^2)+\) a continuous term, \(U_x:=\{q\in U: z_{k+1}(q)=z_{k+1}(x), \dots, z_n(q)=z_n(x)\}\), and \(dV_{U_x}\) is the volume form on \(U_x\) such that \(dV_{U_x}(z_1,\dots, z_k) \cdot dV_S (z_{k+1}, \dots, z_n) = dV_{\mathbb C^n}\). Let \(A^2(S,J^{(p-1)})\) be the space of all \(L^2\) holomorphic sections of \(J^{(p-1)}\) on \(S\) with respect to the metric \(|\;|_{J^{(p-1)}}\). Then \(f\) has an extension \(F_0\in H^0(D,\mathcal I_S^{p-1)})\) such that \(\int_D|F_0|^2e^{-\varphi-(k+p-2)G}dV_{\mathbb C^n}\leq\|f\|^2_{J^{(p-1)}}\).
The author also presents a generalization of the above result to the case where \(D\) is unbounded.
Reviewer: Marek Jarnicki (Kraków)An inequality between complex Hessian measures of Hölder continuous \(m\)-subharmonic functions and capacity.https://www.zbmath.org/1452.320382021-02-12T15:23:00+00:00"Kołodziej, Sławomir"https://www.zbmath.org/authors/?q=ai:kolodziej.slawomir"Nguyen, Ngoc Cuong"https://www.zbmath.org/authors/?q=ai:nguyen.ngoc-cuongLet \(\Omega\subset\mathbb C^n\) be open. The authors study the complex \(m\)-Hessian operator in \(\Omega\). Let \(\mathcal{SH}_m(\Omega)\) stand for the set of all \(m\)-subharmonic functions on \(\Omega\) and let \(\beta:=dd^c\|z\|^2\). Fix a \(\varphi\in\mathcal{SH}(\Omega)\cap\mathcal C^\alpha(\overline\Omega)\) with \(0<\alpha\leq1\). The main results of the paper are the following two theorems.
-- There exist constants \(C, \alpha_0>0\) such that for every positive Borel measure \(\mu\) with compact support in \(\Omega\) such that \(\mu\leq(dd^c\varphi)^m\wedge\beta^{n-m}\) we have (*) \(\mu(K)\leq C\big(\operatorname{cap}_m(K)\big)^{1+\alpha_0}\) for every compact \(K\subset\Omega\), where \(\operatorname{cap}_m(K):=\sup\big\{\int_K(dd^c w)^m\wedge\beta^{n-m}: w\in\mathcal{SH}_m(\Omega),\; 0\leq w\leq1\big\}\).
-- Let \(\Omega\) be a bounded strictly \(m\)-pseudoconvex domain (i.e., there exists a strictly \(m\)-subharmonic \(\mathcal C^2(\overline\Omega)\) defining function for \(\Omega\)), let \(\mu\) be a positive Borel measure with compact support in \(\Omega\) such that (*) is satisfied, and let \(\psi\in\mathcal C(\partial\Omega)\). Then there exists a unique solution to the Dirichlet problem \((dd^c u)^m\wedge\beta^{n-m}=\mu\), \(u\in\mathcal{SH}_m(\Omega)\cap\mathcal C(\overline\Omega)\), \(u|_{\partial\Omega)}=\psi\). Moreover, if \(\psi\) is Hölder continuous on \(\partial\Omega\), then the solution \(u\) is Hölder continuous on \(\overline\Omega\).
For the entire collection see [Zbl 1446.58001].
Reviewer: Marek Jarnicki (Kraków)A brief survey of FJRW theory.https://www.zbmath.org/1452.140562021-02-12T15:23:00+00:00"Francis, Amanda E."https://www.zbmath.org/authors/?q=ai:francis.amanda-e"Jarvis, Tyler J."https://www.zbmath.org/authors/?q=ai:jarvis.tyler-j"Priddis, Nathan"https://www.zbmath.org/authors/?q=ai:priddis.nathanSummary: In this paper we describe some of the constructions of FJRW theory. We also briefly describe its relation to Saito-Givental theory via Landau-Ginzburg mirror symmetry and its relation to Gromov-Witten theory via the Landau-Ginzburg/Calabi-Yau correspondence. We conclude with a discussion of some of the recent results in the field, including the gauged linear sigma model, which is expected to provide a geometric framework for unifying many of these ideas.
For the entire collection see [Zbl 1446.53004].Reflection maps.https://www.zbmath.org/1452.320192021-02-12T15:23:00+00:00"Peñafort-Sanchis, Guillermo"https://www.zbmath.org/authors/?q=ai:penafort-sanchis.guillermoIn a nutshell, the main results are on the obstructions and criteria for stability and \(\mathcal{A}\)-finiteness of reflection maps and families of \(\mathcal{A}\)-finite reflection maps of rank zero.
Reviewer: Mohammed El Aïdi (Bogotá)On multipliers of some new analytic \(M_\alpha^{p,q}\), \(M^{p,\infty,\alpha}\), and \(M^{\infty,p,\alpha}\) type spaces and related spaces on the unit polydisc.https://www.zbmath.org/1452.320092021-02-12T15:23:00+00:00"Shamoyan, Romi F."https://www.zbmath.org/authors/?q=ai:shamoyan.romi-f"Mihić, Olivera R."https://www.zbmath.org/authors/?q=ai:mihic.olivera-rSummary: We study certain new spaces of coefficient multipliers of new analytic Lizorkin-Triebel type spaces \(M_\alpha^{p,q}\), \(M^{p,\infty,\alpha}\), and \(M^{\infty,p,\alpha}\) and related analytic spaces in the unit polydisk with some restriction on parameters. Our results extend some previously known assertions on coefficient multipliers of classical analytic Bergman \(A^p_\alpha\) and analytic weighted Hardy \(H^p_\alpha\) type spaces in the unit disk. Many results are new even in one-dimensional case of unit disk. We define and study also spaces of multipliers of some new analytic Besov type spaces in polydisk.Supplement to the Khinchin-Ostrovskiĭ multidimensional theorem and applications.https://www.zbmath.org/1452.320102021-02-12T15:23:00+00:00"Gavrilov, V. I."https://www.zbmath.org/authors/?q=ai:gavrilov.valerian-ivanovich"Subbotin, A. V."https://www.zbmath.org/authors/?q=ai:subbotin.a-v(no abstract)The geometry of the space of BPS vortex-antivortex pairs.https://www.zbmath.org/1452.300062021-02-12T15:23:00+00:00"Romão, N. M."https://www.zbmath.org/authors/?q=ai:romao.nuno-m"Speight, J. M."https://www.zbmath.org/authors/?q=ai:speight.james-martinSummary: The gauged sigma model with target \({\mathbb{P}}^1\), defined on a Riemann surface \(\Sigma \), supports static solutions in which \(k_+\) vortices coexist in stable equilibrium with \(k_-\) antivortices. Their moduli space is a noncompact complex manifold \({\mathsf{M}}_{(k_+,k_-)}(\Sigma)\) of dimension \(k_++k_-\) which inherits a natural Kähler metric \(g_{L^2}\) governing the model's low energy dynamics. This paper presents the first detailed study of \(g_{L^2}\), focussing on the geometry close to the boundary divisor \(D=\partial \, {\mathsf{M}}_{(k_+,k_-)}(\Sigma)\). On \(\Sigma =S^2\), rigorous estimates of \(g_{L^2}\) close to \(D\) are obtained which imply that \({\mathsf{M}}_{(1,1)}(S^2)\) has finite volume and is geodesically incomplete. On \(\Sigma ={\mathbb{R}}^2\), careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for \(g_{L^2}\) in the limits of small and large separation. All these results make use of a localization formula, expressing \(g_{L^2}\) in terms of data at the (anti)vortex positions, which is established for general \({\mathsf{M}}_{(k_+,k_-)}(\Sigma)\). For arbitrary compact \(\Sigma \), a natural compactification of the space \({{\mathsf{M}}}_{(k_+,k_-)}(\Sigma)\) is proposed in terms of a certain limit of gauged linear sigma models, leading to formulae for its volume and total scalar curvature. The volume formula agrees with the result established for \(\text{Vol}(\mathsf{M}_{(1,1)}(S^2))\), and allows for a detailed study of the thermodynamics of vortex-antivortex gas mixtures. It is found that the equation of state is independent of the genus of \(\Sigma \), and that the entropy of mixing is always positive.Whitney theorem for complex polynomial mappings.https://www.zbmath.org/1452.320052021-02-12T15:23:00+00:00"Farnik, M."https://www.zbmath.org/authors/?q=ai:farnik.michal"Jelonek, Z."https://www.zbmath.org/authors/?q=ai:jelonek.zbigniew"Ruas, M. A. S."https://www.zbmath.org/authors/?q=ai:ruas.maria-aparecida-soaresFor a smooth algebraic affine variety \(X^{n}\) the topology of generic polynomial mappings \(F:X\rightarrow\mathbb{C}^{m}\) with bounded degrees of its components (by \(d_{1},\ldots,d_{m}\)) is studied. The space of all such mappings is denoted by \(\Omega_{X}(d_{1},\ldots,d_{m})\). Each element \(F\in\Omega_{X}(d_{1},\ldots,d_{m})\) generates the mapping \(j^{q}(F)\) (defined by \(x\rightarrow(x,F(x),(\partial^{\alpha}F(x))_{1\leq|\alpha|\leq q})\) in local coordinates) of \(X\) into the space of \(q\)-jets \(J^{q}(X,\mathbb{C}^{m})\), and similarly the mapping \(j^{q_{1},\ldots,q_{r}}(F)\) of \(X\) into the space of multi-jets \(J^{q_{1},\ldots,q_{r}}(X,\mathbb{C}^{m})\). One of general theorems (for arbitrary \(n\) and \(m\)) is that a generic element of \(\Omega_{X}(d_{1},\ldots,d_{m})\), treated as the above mapping, is transversal to any algebraic modular submanifold of the space of multi-jets \(J^{q_{1},\ldots,q_{r}}(X,\mathbb{C}^{m})\). An effective result is that a generic element of \(\Omega_{X}(d_{1},\ldots,d_{m})\) is transversal to any smooth (locally closed) algebraic subvariety if \(d_{i}\geq\sum_{j=1}^{r}q_{j}+r-1\), \(i=1,\ldots,m\). More specific results are given in particular cases \(X=\mathbb{C}^{2}\) or \(X=\{x^{2}+y^{2}+z^{2}=1\}\subset\mathbb{C}^{3}\) and \(m=2\). The authors prove that a generic element \(\Omega_{X}(d_{1},d_{2})\) has only cusps, folds and double folds as singularities and compute the numbers of these singularities. Moreover, they describe the topology of the set of critical points \(C(F)\) of generic \(F\) and the topology of the discriminant \(\Delta(F)\) of \(F\).
Reviewer: Tadeusz Krasiński (Łódź)Conjectural large genus asymptotics of Masur-Veech volumes and of area Siegel-Veech constants of strata of quadratic differentials.https://www.zbmath.org/1452.140262021-02-12T15:23:00+00:00"Aggarwal, Amol"https://www.zbmath.org/authors/?q=ai:aggarwal.amol"Delecroix, Vincent"https://www.zbmath.org/authors/?q=ai:delecroix.vincent"Goujard, Élise"https://www.zbmath.org/authors/?q=ai:goujard.elise"Zograf, Peter"https://www.zbmath.org/authors/?q=ai:zograf.peter"Zorich, Anton"https://www.zbmath.org/authors/?q=ai:zorich.antonThe moduli space of quadratic differentials (with at worst simple poles) on Riemann surfaces can be stratified according to the number and multiplicities of singularities of the differentials. For each stratum of quadratic differentials, period coordinates with respect to the induced flat metric give a natural volume form, leading to a finite volume on the hyperboloid of unit-area differentials, called the Masur-Veech volume. Computing Masur-Veech volumes and analyzing their asymptotic behavior has motivated a number of interesting works in the fields of Teichmüller dynamics, intersection theory, representation theory, and combinatorics. For the case of abelian differentials (whose global squares give quadratic differentials), large genus asymptotics of Masur-Veech volumes were conjectured by \textit{A. Eskin} and \textit{A. Zorich} [Arnold Math. J. 1, No. 4, 481--488 (2015; Zbl 1342.32012)], and were settled respectively by \textit{A. Aggarwal} [J. Am. Math. Soc. 33, No. 4, 941--989 (2020; Zbl 1452.14025)] via a combinatorial method and by \textit{D. Chen} et al. [Invent. Math. 222, No. 1, 283--373 (2020; Zbl 1446.14015)] via intersection theory. In this paper the authors make conjectural descriptions on the large genus asymptotic behavior of Masur-Veech volumes of the strata of (primitive) quadratic differentials as well as their (area) Siegel-Veech constants. Numerical evidence is also provided for these conjectures.
Reviewer: Dawei Chen (Chestnut Hill)Left-invariant Grauert tubes on \(\operatorname{SU}(2)\).https://www.zbmath.org/1452.320112021-02-12T15:23:00+00:00"Aslam, Vaqaas"https://www.zbmath.org/authors/?q=ai:aslam.vaqaas"Burns, Daniel M. jr."https://www.zbmath.org/authors/?q=ai:burns.daniel-m-jun"Irvine, Daniel"https://www.zbmath.org/authors/?q=ai:irvine.danielLet \((M,g)\) be a real analytic Riemannian manifold. An adapted complex structure on \(TM\) is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation, that is the tangent foliation to \(g\)-geodesics, are complex submanifolds. Such \(M_\mathbb{C}\) is called a Grauert tube. If this structure can be extended to the whole of \(TM\), it is called an entire tube. Even for left-invariant metrics on a Lie group \(G\) it is not easy to decide if \(TG\) is entire, except for bi-invariant metrics \(g\) in which case it is.
It was known that some left-invariant metrics may not give entire Grauert tubes \(G_\mathbb{C}\), for instance tubes of left-invariant metrics on \(\operatorname{SU}(2)\) with negative scalar curvature cannot be entire by a previous result of Lempert and Szöke. Other necessary conditions have been known for Grauert tubes.
This paper studies in detail the case \(G=\operatorname{SU}(2)\) with a left-invariant metric \(g\). The approach is based on integrability of the geodesic flow of \(g\) and investigations of complex geodsics on \(G_\mathbb{C}\). I turns out that for generic metric \(g\), when the isometry group is just \(G\), the map \(\exp_\mathbb{C}\) is not well defined on all of \(iTG\) because most complexified geodesics are not given by entire functions
into \(G_\mathbb{C}\).
The remaining cases, when there is a non-trivial isotropy but the metric is non bi-invariant, namely when \((G,g)\) is the quotient of \(\operatorname{SU}(2)\times \operatorname{U}(1)\) by the diagonal action of \(S^1\), \(g\) is given by one parameter \(\lambda\). Actually the inertia operator is \(\mathrm{diag}(1,1,\lambda)\). It is proven that \(\lambda<1\) (and indeed \(\lambda=1\)) correspond to entire tubes, while for \(\lambda>1\) the metric \(g\) has tube of finite radius, which is described.
Along the way the authors find a new obstruction to tubes being entire, exploring complete integrability of the geodesic flow of \(g\).
Reviewer: Boris S. Kruglikov (Tromsø)Vojta's conjecture on rational surfaces and the \(abc\) conjecture.https://www.zbmath.org/1452.110902021-02-12T15:23:00+00:00"Yasufuku, Yu"https://www.zbmath.org/authors/?q=ai:yasufuku.yuIn the first of the three main theorems in the paper under review, the author considers three lines \(L_1 , L_2 , L_3\) of \({\mathbb{P}}^2\) defined over \(\overline{\mathbb{Q}}\) in general position. Let \(X_1\) be the blowup of \({\mathbb{P}}^2\) at a point defined over \(\overline{\mathbb{Q}}\) in \(L_1 \setminus (L_2 \cup L_3)\), with \(E_1\) as the exceptional divisor. For \(n \ge 2\), construct \(X_n\)
inductively by blowing up \(X_{n-1}\) at (the unique) point of \(E_{n-1} \cap \widetilde{L}_1\), obtaining the exceptional divisor \(E_n\). Then
Vojta's conjecture holds for \(X_n\) with respect to the divisor
\[
\widetilde{L}_1 +\widetilde{L}_2 +\widetilde{L}_3 +\widetilde{E}_1 +\cdots+\widetilde{E}_{n-1} +E_n.
\]
The author remarks that the special case of \(X_1\) had been treated in his earlier work [Monatsh. Math. 163, No. 2, 237--247 (2011; Zbl 1282.11086)].
The proof of this first result is based on Ridout's Theorem.
For his second main result, the author considers the case of multiple blowups, where he starts from the same \(X_1\), but blows up at a point not in \( \widetilde{L}_1\) at least once. He shows that Vojta's Conjecture for this situation implies a special case of the \(abc\) conjecture. In the third theorem, he discusses the implication in the other direction. One key ingredient in his proof is a new auxiliary lemma on Farey series organized in the Stern-Brocot tree.
The author points out that his arguments for proving the first and third results carry over to Nevanlinna theory; this enables him to obtain new cases of Griffiths's conjecture.
Reviewer: Michel Waldschmidt (Paris)The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties.https://www.zbmath.org/1452.140172021-02-12T15:23:00+00:00"Javanpeykar, Ariyan"https://www.zbmath.org/authors/?q=ai:javanpeykar.ariyanSummary: These notes grew out of a mini-course given from May 13th to May 17th at UQÀM in Montréal during a workshop on Diophantine Approximation and Value Distribution Theory.
For the entire collection see [Zbl 07235518].Masur-Veech volumes of quadratic differentials and their asymptotics.https://www.zbmath.org/1452.140342021-02-12T15:23:00+00:00"Yang, Di"https://www.zbmath.org/authors/?q=ai:yang.di"Zagier, Don"https://www.zbmath.org/authors/?q=ai:zagier.don-bernard"Zhang, Youjin"https://www.zbmath.org/authors/?q=ai:zhang.youjinLet \(\mathcal Q_{g,n}\) be the moduli space of quadratic differentials on genus \(g\) Riemann surfaces with at worst simple poles at the \(n\) marked points. It carries a natural volume form given by period coordinates with respect to the induced flat metric, and the associate volume \(\mathrm{Vol}~\mathcal Q_{g,n}\) (after suitable normalization) is called the Masur-Veech volume of the principal strata of quadratic differentials. In [``Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials'', Preprint, \url{arXiv:1912.02267}] \textit{D. Chen} et al. gave an expression of \(\mathrm{Vol}~\mathcal Q_{g,n}\) via certain linear Hodge integrals on the Deligen-Mumford moduli space of curves. Based on this formula, the authors apply the theory of integrable systems to derive a number of relations for the generating series of \(\mathrm{Vol}~\mathcal Q_{g,n}\). Moreover, they provide refinements of the conjectural formulas given in [\textit{A. Aggarwal} et al., Arnold Math. J. 6, No. 2, 149--161 (2020; Zbl 1452.14026)] for the large genus asymptotics of the \(\mathrm{Vol}~\mathcal Q_{g,n}\) as well as the associated area Siegel-Veech constants. As a remark, the original version of the large genus asymptotic conjecture for \(\mathrm{Vol}~\mathcal Q_{g,n}\) has been recently setted by \textit{A. Aggarwal} [``Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials'', Preprint, \url{arXiv:2004.05042}].
Reviewer: Dawei Chen (Chestnut Hill)A uniqueness theorem for the two-dimensional sigma function.https://www.zbmath.org/1452.300152021-02-12T15:23:00+00:00"Domrin, A. V."https://www.zbmath.org/authors/?q=ai:domrin.andrei-victorovichSummary: We prove that the sigma functions of Weierstrass \((g = 1)\) and Klein \((g = 2)\) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of \(2g\) linear differential heat equations in a nonholonomic frame (for a function of \(3g\) variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic solutions of these systems can be extended analytically to entire functions of angular variables. For \(g =1\), we give a complete description of the envelopes of holomorphy of such solutions.A Kollár-type vanishing theorem.https://www.zbmath.org/1452.320202021-02-12T15:23:00+00:00"Wu, Jingcao"https://www.zbmath.org/authors/?q=ai:wu.jingcaoIn [Ann. Math. (2) 123, 11--42 (1986; Zbl 0598.14015)] \textit{J. Kollár} established the following result:
Theorem. Let \(f: X \to Y\) be a surjective map between a projective manifold \(X\) and a projective variety \(Y\). If \(A\) is an ample divisor on \(Y\), then for any \(i > 0\) and \(q \ge 0\),
\[H^i (Y, R^qf_*(K_X) \otimes \mathcal O_Y(A)) = 0.\]
In this paper the author considers similar properties for the adjoint bundle \(K_X\otimes L\) where \(L\) is a line bundle on \(X\). When \(L\) is endowed with a smooth semi-positive Hermitian metric, in [\textit{C. Mourougane} and \textit{S. Takayama}, Ann. Sci. Éc. Norm. Supér. (4) 41, No. 6, 905--924 (2008; Zbl 1167.14027); \textit{K. Takegoshi}, Math. Ann. 303, No. 3, 389--416 (1995; Zbl 0843.32018)] it was shown that also holds a Kollar-type vanishing result. Here the author is interested in the singular case. In fact such a result was first developed by Kawamata (see Theorem 2.86 in [\textit{T. de Fernex} et al., Math. Res. Lett. 10, No. 2--3, 219--236 (2003; Zbl 1067.14013)]). Here the author notices that in this situation, there are 2 key ingredients:
a) the injectivity theorem and
b) the torsion freeness of the higher direct images.
Now there is an analytic version of the injectivity theorem in [\textit{Y. Gongyo} and \textit{S.-I. Matsumura}, Ann. Sci. Éc. Norm. Supér. (4) 50, No. 2, 479--502 (2017; Zbl 1401.14083)] which is the starting point of this paper. As far as whether \(R^qf_*(K_{X/Y}\otimes L)\) is torsion free when \(L\) is pseudoeffective, it turns out not to be the case in general. However when the singularity of \(L\) is mild, the author shows that (Theorem 1.4) \[R^qf_*(K_{X/Y} \otimes L)\] is reflexive.
By combining those results, the author establishes the following result.
Theorem 1.5. Let \(f:X \to Y\) be a surjective fibration between 2 projective manifolds \(X\) and \(Y\). Let \((L, h)\) be a \(\mathbb{Q}\)-effective line bundle on \(X\) so that \(\mathcal{I}\)(h) (:= the multiplier ideal associated with a singular metric) is equal to \(\mathcal O_X\).
If \(A\) is an ample divisor on \(Y\), then for any \(i > 0\) and \(q\ge 0\)
\[H^i(Y, R^qf_*(K_X \otimes L) \otimes \mathcal O_Y(A)) = 0\]
Also the author remarks that Theorem 1.5 can be used to prove the positivity of the sheaves
\[R^qf_*(K_X \otimes L).\]
From this, the author establishes the following.
Theorem 1.6. Under same assumptions as in Theorem 1.5, assume that \(A\) is an ample and globally generated line bundle and \(A'\) is a nef line bundle on \(Y\). Then the sheaf
\[R^qf_*(K_X \otimes L) \otimes A^m \otimes A'\]
is globally generated for any \(q \ge 0\) and \(m \ge \dim Y +1\).
Reviewer: Tan VoVan (Suffolk)Spin from defects in two-dimensional quantum field theory.https://www.zbmath.org/1452.811582021-02-12T15:23:00+00:00"Novak, Sebastian"https://www.zbmath.org/authors/?q=ai:novak.sebastian"Runkel, Ingo"https://www.zbmath.org/authors/?q=ai:runkel.ingoSummary: We build two-dimensional quantum field theories on spin surfaces starting from theories on oriented surfaces with networks of topological defect lines and junctions. The construction uses a combinatorial description of the spin structure in terms of a triangulation equipped with extra data. The amplitude for the spin surfaces is defined to be the amplitude for the underlying oriented surface equipped with a defect network dual to the triangulation. Independence of the triangulation and of the other choices follows if the line defect and junctions are obtained from a \(\Delta \)-separable Frobenius algebra with involutive Nakayama automorphism in the monoidal category of topological defects. For rational conformal field theory, we can give a more explicit description of the defect category, and we work out two examples related to free fermions in detail: the Ising model and the \(so (n)\) WZW model at level 1.
{\copyright 2020 American Institute of Physics}Heat kernel asymptotics, local index theorem and trace integrals for Cauchy-Riemann manifolds with \(S^1\) action.https://www.zbmath.org/1452.320432021-02-12T15:23:00+00:00"Cheng, Jih-Hsin"https://www.zbmath.org/authors/?q=ai:cheng.jih-hsin"Hsiao, Chin-Yu"https://www.zbmath.org/authors/?q=ai:hsiao.chin-yu"Tsai, I-Hsun"https://www.zbmath.org/authors/?q=ai:tsai.i-hsunSummary: Among the transversally elliptic operators initiated by Atiyah and Singer, Kohn's \(\square_b\) operator on CR manifolds with \(S^1\) action is a natural one of geometric significance for complex analysts. Our first main result establishes an asymptotic expansion for the heat kernel of such an operator with values in its Fourier components, which involves a contribution in terms of a distance function from lower dimensional strata of the \(S^1\)-action. Our second main result computes a local index density, in terms of tangential characteristic forms, on such manifolds including Sasakian manifolds of interest in String Theory, by showing that certain non-trivial contributions from strata in the heat kernel expansion will eventually cancel out by applying Getzler's rescaling technique to off-diagonal estimates. This leads to a local result which can be thought of as a type of local index theorem on these CR manifolds. We give examples of these CR manifolds, some of which arise from Brieskorn manifolds. Moreover in some cases, we can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a complex orbifold equipped with an orbifold holomorphic line bundle, as an index theorem obtained by a single integral over a smooth CR manifold. We achieve this without use of equivariant cohomology method and our method can naturally drop the contributions arising from lower dimensional strata as done in previous works.Embedding Riemann surfaces with isolated punctures into the complex plane.https://www.zbmath.org/1452.320132021-02-12T15:23:00+00:00"Kutzschebauch, Frank"https://www.zbmath.org/authors/?q=ai:kutzschebauch.frank"Poloni, Pierre-Marie"https://www.zbmath.org/authors/?q=ai:poloni.pierre-marieThe authors obtain a new list of open Riemann surfaces that admit a proper holomorphic embedding in \(\mathbb{C}^2\). The results in this paper, as an addition to the known results, include surfaces that have additional punctures. In this sense, the list improves a list obtained by \textit{A. Sathaye} in the paper [``On planar curves'', Am. J. Math. 99, No. 5, 1105--1135 (1977)], who considered compact surfaces with only a finite set removed.
Reviewer: Athanase Papadopoulos (Strasbourg)Regularity of weak minimizers of the K-energy and applications to properness and K-stability.https://www.zbmath.org/1452.320272021-02-12T15:23:00+00:00"Berman, Robert J."https://www.zbmath.org/authors/?q=ai:berman.robert-j"Darvas, Tamás"https://www.zbmath.org/authors/?q=ai:darvas.tamas"Lu, Chinh H."https://www.zbmath.org/authors/?q=ai:lu.chinh-hIn this paper, \((X,\omega)\) is a compact Kähler manifold and \(\mathcal{H}\) the space of Kähler metrics cohomologous to \(\omega\). The authors show that if a constant scalar curvature Kähler (csck) metric exists in \(\mathcal{H}\), then all finite energy minimizers of the extended K-energy are smooth csck metrics. This partially confirms a conjecture of Y. A. Rubinstein and the second author, and a less general conjecture of X. X. Chen. As an immediate application, they obtain that existence of a csck metric in \(\mathcal{H}\) implies J-properness (a notion due to Tian) of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result the authors prove that an ample line bundle \((X,L)\) admitting
a csck metric in \(c_1(L)\) is K-polystable. When the automorphism group is finite, the properness result, combined with a result of Boucksom-Hisamoto-Jonsson, also implies that \((X,L)\) is uniformly K-stable. This K-stability results fit into a circle of ideas surrounding the Yau-Tian-Donaldson conjecture on a polarized manifold \((X, L)\), saying that the first Chern class of \(L\) contains a Kähler metric with constant scalar curvature if and only if \((X, L)\) is stable in an appropriate sense, inspired by geometric invariant theory.
Reviewer: Athanase Papadopoulos (Strasbourg)Isometries in spaces of Kähler potentials.https://www.zbmath.org/1452.320292021-02-12T15:23:00+00:00"Lempert, László"https://www.zbmath.org/authors/?q=ai:lempert.laszloGiven a Kähler manifold \((X, \omega)\), the space of (relative) Kähler potentials
\[
\mathcal{H}= \mathcal{H} (\omega)= \{ u \in {\mathcal C}^{\infty} (X) \mid \omega + \partial \overline \partial u >0 \},
\]
is a convex open subset of the space of real-valued smooth functions \({\mathcal C} ^{\infty} (X)\) on \(X\). The space \(\mathcal H\) inherits a Fréchet manifold structure from \({\mathcal C}^{\infty} (X)\) and each tangent space \(T_u \mathcal H\) can be canonically identified with \(\mathcal C^{\infty} (X)\) itself. Moreover, \(\mathcal H\), endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. The curvature of \(\mathcal H\) is covariantly constant, a property that for manifolds of finite dimension would imply the existence of local symmetries, self-isometries of neighborhoods of an arbitrary point, that act on tangent vectors issued from the point by \(- \mathrm{Id}\).
In the paper the author study the existence, uniqueness, and other properties of local isometries in spaces of Kähler potentials. In particular, he characterizes local isometries between spaces of Kähler potentials, and proves existence and uniqueness for such isometries.
Reviewer: Anna Fino (Torino)CR embedded submanifolds of CR manifolds.https://www.zbmath.org/1452.320412021-02-12T15:23:00+00:00"Curry, Sean N."https://www.zbmath.org/authors/?q=ai:curry.sean-n"Gover, A. Rod"https://www.zbmath.org/authors/?q=ai:gover.ashwin-rodIn the monograph under review the authors develop a tractor calculus technique for embeddings of a hypersurface type CR manifold into another hypersurface type CR manifold. This presents a new powerful approach to long-standing problems, such as the problem of obstructions for embedding CR manifolds into spheres and hyperquadrics. This problem is solved by a CR Bonnet theorem, which is inspired by Burstall and Calderbank's treatment of the conformal Bonnet theorem. According to classical results on pseudohermitian structures by Tanaka and Webster, CR manifolds can be studied via a Riemannian structure that is induced by the choice of a contact form. To obtain meaningful results in the CR context one needs to track the changes of the Riemannian objects under rescaling of the contact form. This is similar to the study of conformal geometry by choosing a metric. Tractors are geometric objects that incorporate that change of scale. They have been successfully used in conformal geometry and also in CR geometry. The monograph gives a gentle introduction to tractor calculus in the context of CR manifolds in Chapter 3. Chapter 2 is devoted to a detailed exposition of the Tanaka Webster pseudohermitian calculus. In Chapter 4 and 5 the authors investigate the relations between the tractors of a CR submanifold of codimension two and of its ambient CR manifold. A key ingredient is the synchronisation of the scales of the contact forms in sub- and ambient manifold. The authors introduce a CR second fundamental form and prove CR versions of the Gauss, Codazzi and Ricci equations. In Chapter 6 the results are extended to the case of higher-codimensional embeddings of hypersurface type CR manifolds. Chapter 7 is devoted to one of the applications of the developed theory, namely manufacturing of invariants of CR embeddings. The main result of the monograph is the CR Bonnet theorem in Chapter 8, which gives a condition for transversal embeddability of the hypersurface type CR manifold into a model hyperquadric.
Reviewer: Gerd Schmalz (Armidale)Bicomplex Bergman and Bloch spaces.https://www.zbmath.org/1452.300262021-02-12T15:23:00+00:00"Reséndis O., L. F."https://www.zbmath.org/authors/?q=ai:resendis-ocampo.lino-feliciano"Tovar S., L. M."https://www.zbmath.org/authors/?q=ai:tovar.luis-manuelSummary: In this article, we define the bicomplex weighted Bergman spaces on the bidisk and their associated weighted Bergman projections, where the respective Bergman kernels are determined. We study also the bicomplex Bergman projection onto the bicomplex Bloch space.Addendum to: ``Explicit absolute parallelism for 2-nondegenerate real hypersurfaces \(M^5 \subset {\mathbb{C}}^3\) of constant Levi rank 1''.https://www.zbmath.org/1452.320452021-02-12T15:23:00+00:00"Merker, Joël"https://www.zbmath.org/authors/?q=ai:merker.joel"Pocchiola, Samuel"https://www.zbmath.org/authors/?q=ai:pocchiola.samuelThis article is an addendum to the work [J. Geom. Anal. 30, No. 3, 2689--2730 (2020; Zbl 1452.32044)] of the authors, published in the same volume of the journal. In that original paper, it is studied the biholomorphic equivalence problem between 5-dimensional 2-nondegenerate hypersurfaces \(M^5\subset\mathbb C^3\) of the constant Levi rank 1. As the main result, the authors had found -- by applying a computational \textit{differential-algebraic} approach -- \textit{explicit} expressions of the two fundamental invariants \(J\) and \(W\) of the mentioned problem.
In particular, among the results of the original work, the authors proved that when the two differential invariants \(J\) and \(W\) vanish identically on \(M^5\), then the equivalence problem to this CR manifold extends to a certain equivalence problem of a 10-dimensional prolonged space with \textit{constant} structure equations precisely as those of the well-known model light cone. As a consequence, then in this case \(M^5\) is biholomorphically equivalent to the mentioned model.
But in the original work and due to the complexity of the computations and the length of the appearing intermediate expressions, the authors skipped to present a detailed proof for the explicit expressions of the constant structure equations associated with the mentioned prolonged space. In this supplementary addendum, they attempt to provide a detailed proof for this interesting result in CR geometry.
Reviewer: Masoud Sabzevari (Shahr-e Kord)On complex Legendre duality.https://www.zbmath.org/1452.320302021-02-12T15:23:00+00:00"Lempert, László"https://www.zbmath.org/authors/?q=ai:lempert.laszloLet \(X\) be a compact Kähler manifold with a smooth Kähler form \(\omega_0\). The space of its Kähler potentials is defined by
\[
\mathcal{H}=\left\{u\in C^\infty(X): \omega_u=\omega_0+i \partial\bar\partial u>0\right\}.
\]
With the Mabuchi metric, the space \(\mathcal{H}\) is naturally an infinite-dimensional Riemannian manifold. In [\textit{B. Berndtsson} et al., Am. J. Math. 142, No. 1, 323--339 (2020; Zbl 07176037)], the following generalization of the Legendre transformation from Euclidean spaces to Kähler manifolds is proved.
Theorem. Suppose \(u\in \mathcal{H}\) and \(\omega_u\) is real analytic. Then \(u\) has a neighborhood \(\mathcal{U}\subset \mathcal{H}\) and there is a \(C^\infty\) diffeomorphism \(F: \mathcal{U}\to \mathcal{U}\) that is an involution, an isometry of the Mabuchi metric, fixes \(u\), and its differential \(F_*\) acts on \(T_u\mathcal{U}\) as multiplication by \(-1\).
In this article, the author proves that the real analyticity condition for \(\omega_u\) is necessary for the existence of such an involution to exists.
Theorem. Let \(\mathcal{U}\) be an open neighborhood of \(u\) in \(\mathcal{H}\). Suppose that there exists a \(C^\infty\) map \(F: \mathcal{U}\to \mathcal{H}\) such that \(F: \mathcal{U}\to F(\mathcal{U})\) is a diffeomorphism, \(F\) preserves the Mabuchi metric, \(F(u)=u\) and \(F_*\xi=-\xi\) for all \(\xi\in T_u\mathcal{U}\). Then, \(\omega_u\) is real analytic.
Using the WZW equation, the author proves that the existence of the map \(F\) implies that the wave front set of \(u\) is empty, and then concludes that \(u\) is real analytic.
Reviewer: Botong Wang (Madison)Siu's lemma, optimal \(L^2\) extension and applications to twisted pluricanonical sheaves.https://www.zbmath.org/1452.320142021-02-12T15:23:00+00:00"Zhou, Xiangyu"https://www.zbmath.org/authors/?q=ai:zhou.xiangyu"Zhu, Langfeng"https://www.zbmath.org/authors/?q=ai:zhu.langfengThe starting point of this technical paper originated from an attempt to improve a technical Lemma by [\textit{D. H. Phong} and \textit{J. Sturm}, Ann. Math. (2) 152, No. 1, 277--329 (2000; Zbl 0995.11065)] for plurisubharmonic functions with non-trivial multiplier ideal sheaves which is the main content of Theorem 1.2. Then by using Theorem 1.2, the authors establish an improved version (Theorem 1.3) of the optimal \(L^2\) extension theorem.
By using Theorems 1.2 and 1.3, they obtained an optimal \(L^{2/m}\) extension result (Theorem 1.4) for twisted pluricanonical bundles.
As applications of the above results, they establish the positivity of twisted relative pluricanonical bundles and its direct images (Theorem 1.5) where \(\Pi: X \to Y\) is a surjective proper holomorphic map from a Kähler manifold \((X, \omega)\) of dimension \(n\) to a connected manifold \(Y\) of dimension k with \(1 \le k \le n\). \(X\) is equipped with a holomorphic line bundle \((L, h)\) with a singular hermitian metric \(h\) such that its associated curvature current is \(\ge 0\).
Now let \(\mathcal{E} :=\Pi_*(K_{X \backslash Y} + L)\) and \(\mathcal{E}_h :=\Pi_*(K_{X \backslash Y} + L \otimes I(h))\). Then by using Theorem 1.3, they prove the positivity of \(\mathcal{E}\) and \(\mathcal{E}_h\)
Furthermore, let \(\mathcal{E}_{m-1} : = \Pi_*(m K_{X \backslash Y} + L)\). Then they show (Corollary 1.8) the semi positivity of \(\mathcal{E}_{m-1}\) for \(m \ge 2\)
Reviewer: Tan VoVan (Suffolk)Mass equidistribution for random polynomials.https://www.zbmath.org/1452.320042021-02-12T15:23:00+00:00"Bayraktar, Turgay"https://www.zbmath.org/authors/?q=ai:bayraktar.turgaySummary: The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contains a wide range of random variables including standard Gaussian and all bounded random variables. We prove that for almost every sequence of random polynomials their normalized zero currents become equidistributed with respect to a deterministic extremal current. The main ingredients of the proof are Bergman kernel asymptotics, mass equidistribution of random polynomials and concentration inequalities for subgaussian quadratic forms.Zermelo deformation of Hermitian metrics by holomorphic vector fields.https://www.zbmath.org/1452.530182021-02-12T15:23:00+00:00"Aldea, Nicoleta"https://www.zbmath.org/authors/?q=ai:aldea.nicoletaSummary: In this paper we first present the real homogeneous complex Finsler metrics (\( \mathbb{R} \)-complex Finsler metrics) and make use of them to generalize the Zermelo navigation on Hermitian manifolds. \( \mathbb{R} \)-complex Randers metrics are obtained by the Zermelo deformation of Hermitian metrics, with variable space-dependent ship's relative speed under action of weak complex vector fields. Next, we indicate how some properties of a Hermitian metric, e.g. Kähler property, holomorphic sectional curvature behave by the Zermelo deformation in a special holomorphic wind. Lastly, the results are illustrated with some relevant examples.Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry.https://www.zbmath.org/1452.320402021-02-12T15:23:00+00:00"Berman, Robert J."https://www.zbmath.org/authors/?q=ai:berman.robert-jThe background for this paper is the author's probabilistic constructions of Kähler-Einstein metrics in complex algebraic geometry [Commun. Math. Phys. 354, No. 3, 1133--1172 (2017; Zbl 1394.32019); in: Algebraic geometry: Salt Lake City 2015. 2015 summer research institute in algebraic geometry, University of Utah, Salt Lake City, UT, USA, July 2015. Proceedings. Part 1. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. 29--73 (2018; Zbl 1446.32017)]. In these constructions, the author shows how the asymptotic behavior of certain (deformed) determinantal point processes can be described in terms of volume forms of Kähler-Einstein metrics or, more generally, the Monge-Ampère measures of solutions to complex Monge-Ampère equations. This paper provides a survey of the techniques used in this, emphasizing their roots in complex analysis (in particular: approximation and interpolation theory) and contains new results generalizing the constructions to the non-compact setting of \(\mathbb{C}^n\).
The main result of the paper is a large deviation principle for a class of (deformed) determinantal point processes in \(\mathbb{C}^n\). Both the point processes and the complex Monge-Ampère equations that describe their asymptotics are part of a thermodynamic framework and can be parametrized by a temperature variable. Two additional results in the paper, generalizing analogous results in complex geometry, describe the limit of solutions to these Monge-Ampère equations as the temperature tends to zero and (positive) infinity, respectively. In these results, the zero temperature limit is given by Siciak's weighted extremal function for a weighted set and the infinite temperature limit is given by a weak solution of a Calabi-Yau equation.
Reviewer: Jakob Hultgren (Washington)Complex algebraic foliations.https://www.zbmath.org/1452.320362021-02-12T15:23:00+00:00"Lins Neto, Alcides"https://www.zbmath.org/authors/?q=ai:lins-neto.alcides"Scárdua, Bruno"https://www.zbmath.org/authors/?q=ai:scardua.bruno-c-azevedoThis book provides a more or less self-contained introduction to the theory of holomorphic foliations, in particular complex algebraic foliations.
The first chapter introduces the basic notions related to foliations with or without singularities. The authors discuss elementary results related to the extension of holomorphic foliations and also explain ways of constructing foliations, in particular by suspension of a group of biholomorphisms.
In the second chapter, the authors pass to holomorphic foliations on complex projective spaces. In particular they explain the characterization of foliations on \(\mathbb{CP}^n\) of fixed degree. Special attention is given to holomorphic foliations of codimension one on \(\mathbb{CP}^n\), proving that they are obtained by foliations defined by polynomial vector fields on \(\mathbb{C}^n\).
In the third chapter, the authors discuss the existence or nonexistence of algebraic leaves of holomorphic foliations. The index theorem of Camacho-Sad and the Baum-Bott theorem for \(\mathbb{CP}^2\) are proved.
Chapter four discusses holomorphic foliations with algebraic limit sets. The central result in this chapter states that under certain conditions, a foliation on \(\mathbb{CP}^2\) with algebraic limit set is given by the inverse image of a foliation of degree one on \(\mathbb{CP}^2\) under a rational map.
Chapter five is devoted to prove the rigidity theorem of Ilyashenko, which assures topological rigidity for a large class of holomorphic foliations on \(\mathbb{CP}^2\).
Chapter six deals with transverse structures of foliations, in particular with affine or projective transverse structures. Several classification results for foliations admitting such structures are presented.
In the appendix, some basic results of the theory of holomorphic functions of several variables are recalled and applied to the extension of foliations.
The book provides a good and well-motivated introduction to the topic of holomorphic foliations. Every chapter contains several exercises. Unfortunately, the book was not proofread by an English speaker.
Reviewer: Judith Brinkschulte (Leipzig)Homogeneous principal bundles over manifolds with trivial logarithmic tangent bundle.https://www.zbmath.org/1452.320242021-02-12T15:23:00+00:00"Azad, Hassan"https://www.zbmath.org/authors/?q=ai:azad.hassan"Biswas, Indranil"https://www.zbmath.org/authors/?q=ai:biswas.indranil"Khadam, M. Azeem"https://www.zbmath.org/authors/?q=ai:khadam.m-azeemLet \(X\) be a compact connected complex manifold and \(D\) a reduced effective normal crossing divisor in \(X\). Suppose further that the logarithmic tangent bundle \(TX(-\log D)\) is holomorphically trivial. Let \({\mathbb G}\) be the connected component of the identity in the group of holomorphic automorphisms of \(X\) preserving \(D\). There is a known characterisation of the pairs \((X,D)\) due to \textit{J. Winkelmann} [Osaka J. Math. 41, No. 2, 473--484 (2004; Zbl 1058.32011)].
If \(H\) is a connected complex Lie group, a holomorphic principal \(H\)-bundle \(E_H\) over \(X\) is called homogeneous if the pullback \(g^*E_H\) is holomorphically isomorphic to \(E_H\) for all \(g\in{\mathbb G}\). The object of this paper is to obtain two further characterisations of homogeneity for holomorphic principal bundles. In the first place, \(E_H\) is homogeneous if and only if \(E_H\) admits a logarithmic connection singular over \(D\) (Corollary 4.5). Secondly, \(E_H\) is homogeneous if and only if the pullback of \(E_H\) by the tautological map \({\mathbb G}\times X\to X\), considered as a family parameterised by \({\mathbb G}\), is infinitesimally rigid at the identity of \({\mathbb G}\) (Proposition 5.1).
Reviewer: P. E. Newstead (Liverpool)Large genus asymptotics for volumes of strata of abelian differentials.https://www.zbmath.org/1452.140252021-02-12T15:23:00+00:00"Aggarwal, Amol"https://www.zbmath.org/authors/?q=ai:aggarwal.amolThe moduli space of abelian differentials on Riemann surfaces can be stratified according to the number and multiplicities of zeros of the differentials. For each stratum of abelian differentials, period coordinates with respect to the induced flat metric give a natural volume form, leading to a finite volume on the hyperboloid of unit-area differentials, called the Masur-Veech volume. One way to evaluate Masur-Veech volumes is via enumerating certain branch covers of tori and analyzing their large degree asymptotics, as such covers correspond to lattice points in the strata. \textit{A. Eskin} and \textit{A. Okounkov} used this idea to describe a general algorithm to determine the volumes of the strata [Invent. Math. 145, No. 1, 59--103 (2001; Zbl 1019.32014)]. The numerical data out of this algorithm further motivated \textit{A. Eskin} and \textit{A. Zorich} to make a conjectural description on the large genus asymptotics of Masur-Veech volumes [Arnold Math. J. 1, No. 4, 481--488 (2015; Zbl 1342.32012)]. In this paper the author establishes the volume asymptotic for all strata as predicted by Eskin and Zorich. The proof employs remarkable combinatorial techniques to analyze the original algorithm of Eskin and Okounkov. An upshot is to find dominate terms in the summation expression of the Masur-Veech volume and show that the non-dominant terms in the sum decay rapidly. An alternative solution to this problem via intersection theory on moduli spaces can be found in [\textit{D. Chen} et al., Invent. Math. 222, No. 1, 283--373 (2020; Zbl 1446.14015)].
Reviewer: Dawei Chen (Chestnut Hill)Analysis and applications: the mathematical work of Elias Stein.https://www.zbmath.org/1452.320012021-02-12T15:23:00+00:00"Fefferman, Charles"https://www.zbmath.org/authors/?q=ai:fefferman.charles-louis"Ionescu, Alex"https://www.zbmath.org/authors/?q=ai:ionescu.alex"Tao, Terence"https://www.zbmath.org/authors/?q=ai:tao.terence-c"Wainger, Stephen"https://www.zbmath.org/authors/?q=ai:wainger.stephenSummary: This article discusses some of Elias M. Stein's seminal contributions to analysis.Fractional exponential decay in the forbidden region for Toeplitz operators.https://www.zbmath.org/1452.320212021-02-12T15:23:00+00:00"Deleporte, Alix"https://www.zbmath.org/authors/?q=ai:deleporte.alixSummary: We prove several results of concentration for eigenfunctions in Toeplitz quantization. Under mild regularity assumptions, we prove that eigenfunctions are \(O(\exp(-cN^{\delta}))\) away from the corresponding level set of the symbol, where \(N\) is the inverse semiclassical parameter and \(0<\delta<1\) depends on the regularity. As an application, we prove a precise bound for the free energy of spin systems at high temperatures, sharpening a result of Lieb.Generalized fractional Cauchy-Riemann operator associated with the fractional Cauchy-Riemann operator.https://www.zbmath.org/1452.352352021-02-12T15:23:00+00:00"Ceballos, Johan"https://www.zbmath.org/authors/?q=ai:ceballos.johan"Coloma, Nicolás"https://www.zbmath.org/authors/?q=ai:coloma.nicolas"Di Teodoro, Antonio"https://www.zbmath.org/authors/?q=ai:di-teodoro.antonio-nicola|teodoro.antonio-di"Ochoa-Tocachi, Diego"https://www.zbmath.org/authors/?q=ai:ochoa-tocachi.diegoSummary: In this paper, we present a characterization of all linear fractional order partial differential operators with complex-valued coefficients that are associated to the generalized fractional Cauchy-Riemann operator in the Riemann-Liouville sense. To achieve our goal, we make use of the technique of an associated differential operator applied to the fractional case.A remark on the tautness modulo an analytic hypersurface of Hartogs-type domains.https://www.zbmath.org/1452.320122021-02-12T15:23:00+00:00"Pham, D. T."https://www.zbmath.org/authors/?q=ai:pham.duong-thanh|pham.duc-thoan|pham.duc-truong|pham.dinh-tuan-antoine|pham-dinh-tao.|pham.duong-trieuSummary: We present sufficient conditions for the tautness modulo an analytic hypersurface of Hartogs-type domains \(\Omega_H(X)\) and Hartogs-Laurent-type domains \(\Sigma_{ u,v }(X)\). We also propose a version of Eastwood's theorem for tautness modulo an analytic hypersurface.