Recent zbMATH articles in MSC 14Mhttps://www.zbmath.org/atom/cc/14M2021-04-16T16:22:00+00:00WerkzeugBook review of: S. Sullivant, Algebraic statistics.https://www.zbmath.org/1456.000152021-04-16T16:22:00+00:00"Kahle, Thomas"https://www.zbmath.org/authors/?q=ai:kahle.thomasReview of [Zbl 1408.62004].Virtual classes of parabolic \(\operatorname{SL}_2(\mathbb{C})\)-character varieties.https://www.zbmath.org/1456.140652021-04-16T16:22:00+00:00"González-Prieto, Ángel"https://www.zbmath.org/authors/?q=ai:gonzalez-prieto.angelLet \(\Sigma_g\) be the closed orientable surface of genus \(g\) and \(Q\) a parabolic structure on \(\Sigma_g\). In this paper, the author completes his study of the virtual classes of the \(\operatorname{SL}_2({\mathbb C})\)-character varieties of \((\Sigma_g,Q)\) by considering the case where there are parabolic points of semi-simple type. More precisely, let \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) denote the representation variety of \((\Sigma_g,Q)\) and \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q):={\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)//\operatorname{SL}_2({\mathbb C})\) the corresponding character variety. Now let \(\operatorname{K\mathbf{Var}}_{\mathbb C}\) be the Grothendieck ring of complex algebraic varieties and \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\) the localisation of this ring with respect to the multiplicative set generated by \(q\), \(q+1\) and \(q-1\), where \(q\) is the class of \({\mathbb C}\) in \(\operatorname{K\mathbf{Var}}_{\mathbb C}\). Then the author computes explicitly the virtual class of \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) in \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\) when there is at least one parabolic point with semi-simple holonomy and possibly some additional parabolic points with holonomy of Jordan type \(J_+\) (Theorem 5.6). From this, he deduces a formula for the virtual class of \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\), valid for all holonomies (Theorem 6.1).
Character varieties have been much studied in recent years by both arithmetic and geometric methods. Both methods have limitations when there are parabolic points. In his thesis, the author developed a method involving TQFTs to avoid these limitations and used this method to compute the classes of \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) and \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) in \(\operatorname{K\mathbf{MHS}}\) where the punctures are of Jordan type or type \(-\operatorname{Id}\). Here, \(\operatorname{K\mathbf{MHS}}\) is the Grothendieck ring of the category of mixed Hodge structures. (The relevant part of the author's et al. [Bull. Sci. Math. 161, Article ID 102871, 33 p. (2020; Zbl 1441.57031)]). However, there are new complications when parabolic points of semi-simple type are involved. In particular, the ``core submodule'' constructed by the author is no longer invariant under the TQFT. Moreover, if the punctures are non-generic, a new interaction phenomenon arises. These problems are addressed in the current paper.
In section 2, the author sketches the construction of the TQFT mentioned above together with a modification which allows computations in \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\). The key section 3 is concerned with \(\operatorname{SL}_2({\mathbb C})\)-representation varieties and is preliminary to the computation of the geometric TQFT in section 4. The interaction phenomenon is described in section 5, culminating in Theorem 5.6. Section 6 is directed towards proving Theorem 6.1.
The author comments that there is much work still to be done in extending his results to groups other than \(\operatorname{SL}_2({\mathbb C})\) and to more general spaces, for example singular and non-orientable surfaces or knot complements.
Reviewer: P. E. Newstead (Liverpool)Categorical localization for the coherent-constructible correspondence.https://www.zbmath.org/1456.140472021-04-16T16:22:00+00:00"Ike, Yuichi"https://www.zbmath.org/authors/?q=ai:ike.yuichi"Kuwagaki, Tatsuki"https://www.zbmath.org/authors/?q=ai:kuwagaki.tatsukiKontsevich's homological mirror symmetry(HMS) conjecture states that two categories associated to
a mirror pair are equivalent. For a Calabi-Yau(CY) variety, a mirror is also Calabi-Yau and the conjecture is a
quasi-equivalence between the dg category of coherent sheaves over one and the derived Fukaya category of
the other. For non-CY's, mirrors do not need to be varieties. For a Fano toric variety, its mirror is a
Landau-Ginzburg (LG) model, which is a holomorphic function on \((\mathbb C^\times)^n\) which can be read from
the defining fan of the toric variety which is in fact the specialization of Lagrangian potential
function of the toric \(A\)-model that is the generating function of open Gromov-Witten invariants of
a toric fiber [\textit{C.-H. Cho} and \textit{Y.-G. Oh}, Asian J. Math. 10, No. 4, 773--814 (2006; Zbl 1130.53055); \textit{K. Fukaya} et al., Duke Math. J. 151, No. 1, 23--175 (2010; Zbl 1190.53078)].
For a smooth Fano, it has been proven for many special cases that the dg category of coherent sheaves
\(\mathbf{coh}\, X_\Sigma\) over the toric variety \(X_\Sigma\) associated to the fan \(\Sigma\) is quasi-equivalent to
the Fukaya-Seidel category \(\mathfrak{Fuk}(W_\Sigma)\) of the associated Laurent polynomial \(W_\Sigma\).
When a variety is not complete, \(\mathbf{coh}\, X_\Sigma\) is of infinite dimensional nature and its Fukaya-type
category also should have an infinite-dimensional nature. Such a construction is known to be
(partially) wrapped Fukaya categories. In this regard, the main theme of the present paper in review is
to establish a quasi-isomorphism \(\mathbf{coh}(X\setminus D)\cong \mathbf{coh} X/\mathbf{coh}_D X\) in some special cases
in the microlocal world: Here
\(X\setminus D\) is the complement of a divisor \(D\) and \(\mathbf{coh} X/\mathbf{coh}_D X\) is the dg category of
sheaves supported in \(D\) by relating the isomorphism to a similar isomorphism
\[
W_{\mathbf{s}\setminus\mathbf{r}}(M) \cong W_{\mathbf{s}}(M)/\mathfrak B_{\mathbf{r}}
\]
of \textit{Z. Sylvan} [J. Topol. 12, No. 2, 372--441 (2019; Zbl 1430.53097)] in the Fukaya-Seidel side: Here \(\mathbf{s}\) is a collection
of symplectic stops and \(\mathbf{r} \subset\mathbf{s}\) is a sub-collection thereof, and
\(\mathfrak B_{\mathbf{r}}\) is the full subcategory spanned by Lagrangians near the sub-stops \(\mathbf{r}\).
The paper extends a version of coherent-constructible correspondence [\textit{B. Fang} et al., Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011); \textit{K. Bongartz} et al., Adv. Math. 226, No. 2, 1875--1910 (2011; Zbl 1223.16004)] to the dg category of
\emph{quasi-coherent shaves} over \(X_\Sigma\) in dimension 2.
Reviewer: Yong-Geun Oh (Pohang)GLSMs for exotic Grassmannians.https://www.zbmath.org/1456.814342021-04-16T16:22:00+00:00"Gu, Wei"https://www.zbmath.org/authors/?q=ai:gu.wei"Sharpe, Eric"https://www.zbmath.org/authors/?q=ai:sharpe.eric-r"Zou, Hao"https://www.zbmath.org/authors/?q=ai:zou.haoSummary: In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.Immaculate line bundles on toric varieties.https://www.zbmath.org/1456.140612021-04-16T16:22:00+00:00"Altmann, Klaus"https://www.zbmath.org/authors/?q=ai:altmann.klaus"Buczyński, Jarosław"https://www.zbmath.org/authors/?q=ai:buczynski.jaroslaw"Kastner, Lars"https://www.zbmath.org/authors/?q=ai:kastner.lars"Winz, Anna-Lena"https://www.zbmath.org/authors/?q=ai:winz.anna-lenaFor an algebraic variety \(X\) over an algebraically closed field \({\mathbb K}\) of arbitrary characteristic, a sheaf \({\mathcal F}\) on \(X\) is called \textit{immaculate} if all cohomology groups \(H^p(X,{\mathcal F})=0\) for all \(p\in{\mathbb Z}\). The main focus of the paper under review is the structure of the family of all immaculate line bundles on \(X\) as a subset of the group \(\text{Pic}(X)\).
Classically, the cohomology of a Weil divisor on a toric variety is calculated using polyhedra complexes contained in \(N_{\mathbb R}\), where \(N\) is the lattice of \(1\)-parameter subgroups of the torus acting on \(X\). The first contribution of the paper under review is a shift of the classical approach, viewing now the cohomology of a toric \({\mathbb Q}\)-Cartier Weil divisor using polytopes in the space \(M_{\mathbb R}\), where \(M\) is the dual lattice of \(N\). For projective toric varieties, Theorem 3.6 describes the \(M\)-graded cohomology groups \(H^i(X,{\mathcal O}(D))\) in terms of the polyhedra associated to a decomposition of the divisor \(D\) as the difference \(D^+-D^-\) of two nef divisors.
For the main objective, a description of the locus of all immaculate line bundles in the class group \(\text{Pic}(X)\) of a toric variety \(X\), the first results establish some general invariance properties of immaculacy (or a relative version of it) of locally free sheaves under various types of morphisms between toric varieties. Next, to describe the immaculate locus the authors use the map \(\pi:{\mathbb Z}^{\Sigma(1)}\to \text{Pic}(X)\) that assigns to a \(T\)-invariant divisor its class. Using this map, the first task is to identify the \(T\)-invariant divisors whose images carry some cohomology by using an approach similar to the one used for acyclic line bundles as in [\textit{L. Borisov} and \textit{Z. Hua}, Adv. Math. 221, No. 1, 277--301 (2009; Zbl 1210.14006)] and [\textit{A. I. Efimov}, J. Lond. Math. Soc., II. Ser. 90, No. 2, 350--372 (2014; Zbl 1318.14047)]. In Section 5 of the paper under review the authors identify some subsets of \(\Sigma(1)\) whose images under \(\pi\) either carry some cohomology or not. One of the main results, Theorem 5.24, essentially describes the locus of immaculate line bundles for a complete simplicial toric variety. Moreover, in some concrete instances the conditions on the subsets of \(\Sigma(1)\) can be used to describe the locus of immaculate bundles, for example for smooth projective toric varieties of Picard rank \(2\) in Theorem 6.2 . Using the classification of smooth projective toric varieties of Picard rank \(3\) of \textit{V. L. Batyrev} [Tôhoku Math. J., II. Ser. 43, No. 4, 569--585 (1991; Zbl 0792.14026)] in Section 8 the authors consider this situation in two cases, depending on the splitting of the fan of the toric variety.
Reviewer: Felipe Zaldívar (Ciudad de México)Polynomials from combinatorial \(K\)-theory.https://www.zbmath.org/1456.051712021-04-16T16:22:00+00:00"Monical, Cara"https://www.zbmath.org/authors/?q=ai:monical.cara"Pechenik, Oliver"https://www.zbmath.org/authors/?q=ai:pechenik.oliver"Searles, Dominic"https://www.zbmath.org/authors/?q=ai:searles.dominicSummary: We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a \(K\)-theoretic deformation of the quasi-key basis and also a lift of the \(K\)-analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the \(K\)-analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)].
The second new basis is the kaon basis, a \(K\)-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis.
Throughout, we explore how the relationships among these \(K\)-analogues mirror the relationships among their cohomological counterparts. We make several ``alternating sum'' conjectures that are suggestive of Euler characteristic calculations.Brasselet number and Newton polygons.https://www.zbmath.org/1456.140622021-04-16T16:22:00+00:00"Dalbelo, Thaís M."https://www.zbmath.org/authors/?q=ai:dalbelo.thais-maria"Hartmann, Luiz"https://www.zbmath.org/authors/?q=ai:hartmann.luizAuthors' abstract: We present a formula to compute the Brasselet number of \( f:(Y,0)\rightarrow (\mathbb{C},0)\) where \(Y\subset X\) is a non-degenerate complete intersection in a toric variety \(X\). As applications we establish several results concerning invariance of the Brasselet number for families of non-degenerate complete intersections. Moreover, when \((X,0)=(\mathbb{C} ^{n},0)\) we derive sufficient conditions to obtain the invariance of the Euler obstruction for families of complete intersections with an isolated singularity which are contained in \(X\).
Reviewer: Tadeusz Krasiński (Łódź)Chiral algebra, localization, modularity, surface defects, and all that.https://www.zbmath.org/1456.813682021-04-16T16:22:00+00:00"Dedushenko, Mykola"https://www.zbmath.org/authors/?q=ai:dedushenko.mykola"Fluder, Martin"https://www.zbmath.org/authors/?q=ai:fluder.martinThe authors study Lagrangian \(\mathcal{N} = 2\) superconformal field theories in four dimensions.
By employing supersymmetric localization on a rigid background of the form \(S^3 \times S^1_y\) they explicitly localize a given Lagrangian superconformal field theory and obtain the corresponding two-dimensional vertex operator algebra VOA (chiral algebra) on the torus \(S^1\times S^1_y\subset S^3\times S^1_y\). To derive the VOA the authors define the appropriate rigid supersymmetric \(S^3 \times S^1_y\) background reproducing the superconformal index. They analyze the supersymmetry algebra and classify the possible fugacities and their preserved subalgebras. Although the minimal amount of supersymmetry needed to retain the VOA construction is \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(1|1)_r\) it appears that it is possible to turn on fugacities preserving an \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(2|1)_r\) subalgebra which can be further broken to the minimal one by defects. Specifically, discrete fugacities \(M,N \in \mathbb{Z}\) can be turned on. The authors argue that these deformations do not affect the VOA construction but change the complex structure of the
torus and affect the boundary conditions (spin structure) upon going around one of the cycles, \(S^1_y\)
The authors address the two-dimensional theory corresponding to the localization of the \(\mathcal{N} = 2\) vector multiplets and hypermultiplets. In the latter case they show that the remnant classical piece in the localization precisely reduces to the two-dimensional symplectic boson theory on the boundary torus \(S^1\times S^1_y\). The authors show that in the presence of flavor holonomies, which appear as mass-like central charges in the supersymmetry algebra, vertex operators charged under the flavor symmetries fail to remain holomorphic while the sector that remains holomorphic is formed by flavor-neutral operators.
The authors study the modular properties of the four-dimensional Schur index. They introduce formal partition functions \(Z^{(\nu_1,\nu_2)}_{(m,n)}\), which are defined as the partition function in the given spin structure \((\nu_1,\nu_2)\), but with the modified contour of the holonomy integral in the localization formula, labeled by two integers \(m\) and \(n\). The authors suggest that the objects \(Z^{(\nu_1,\nu_2)}_{(m,n)}\) furnish an infinite-dimensional projective representation of \(\mathrm{SL}(2,\mathbb{Z})\).
Finally the authors comment on the flat \(\Omega\)-background underlying the chiral algebra.
Reviewer: Farhang Loran (Isfahan)Correspondence scrolls.https://www.zbmath.org/1456.140482021-04-16T16:22:00+00:00"Eisenbud, David"https://www.zbmath.org/authors/?q=ai:eisenbud.david"Sammartano, Alessio"https://www.zbmath.org/authors/?q=ai:sammartano.alessioThis paper introduces schemes called ``correspondence scrolls'' and gives a general study of them. For a closed subscheme \(Z\) in \(\Pi _{i=1}^n {\mathbb A}^{a_i+1}\), defined by a multigraded ideal \(I\subset A:={\Bbbk}[x_{i,j} : 1\le i \le n, 0 \le j \le a_i]\), and for \(\mathbf{b}=(b_1, \ldots , b_n) \in \mathbb{N}_+^n\), consider the homomorphism \({\Bbbk}[z_{i,\alpha }] \rightarrow A/I\) which sends a variable \(z_{i, \alpha }\) to the monomial \(x_i^\alpha\), of degree \(b_i\). Here \(x_i^ {\alpha }\) denotes \(x_{i,0}^{\alpha _0} \cdots x_{i, a_i}^{\alpha _{a_i}}\). The kernel of the above map defines a closed projective subscheme \(C(Z, {\mathbf b}) \subset { \mathbb P}^N\) (\(N= \sum \binom{a_i+b_i}{a_i}-1)\), called \textit{correspondence scroll}. This definition includes classical correspondences as well as interesting non-classical ones: rational normal scrolls, double structures which are degenerate \(K3\) surfaces, degenerate Calabi-Yau threefolds, etc. Many invariants or properties of correspondence scrolls are studied: dimension, degree, nonsingularity, Cohen-Macaulay and Gorenstein property and others. The paper is very well written and invites to further research.
Reviewer: Nicolae Manolache (Bucureşti)Linearly dependent powers of binary quadratic forms.https://www.zbmath.org/1456.110442021-04-16T16:22:00+00:00"Reznick, Bruce"https://www.zbmath.org/authors/?q=ai:reznick.bruceSummary: Given an integer \(d \ge 2\), what is the smallest \(r\) so that there is a set of binary quadratic forms \(\{f_1,\dots,f_r\}\) for which \(\{f_j^d\}\) is nontrivially linearly dependent? We show that if \(r \le 4\), then \(d \le 5\), and for \(d \ge 4\), construct such a set with \(r = \lfloor d/2\rfloor + 2\). Many explicit examples are given, along with techniques for producing others.On the eigenpoints of cubic surfaces.https://www.zbmath.org/1456.140432021-04-16T16:22:00+00:00"Celik, Turku Ozlum"https://www.zbmath.org/authors/?q=ai:celik.turku-ozlum"Galuppi, Francesco"https://www.zbmath.org/authors/?q=ai:galuppi.francesco"Kulkarni, Avinash"https://www.zbmath.org/authors/?q=ai:kulkarni.avinash"Sorea, Miruna-Ştefana"https://www.zbmath.org/authors/?q=ai:sorea.miruna-stefanaThe aim of the paper is to study the eigenscheme of order three partially symmetric and symmetric tensors. They also show that a subvariety of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenscheme of \(4 \times 4 \times 4\) symmetric tensors.
The spectral theory of tensors is a multi-linear generalization of the study of eigenvalues and eigenvectors in the case of matrices. The eigenscheme \(E(\mathcal{T})\) of a tensor \(\mathcal{T}\) can be roughly thought as the set of eigenpoints of the tensor, i.e. eigenvectors of \(\mathbb{C}^{n+1}\) of a particular contraction of the tensor. In the case of partial symmetric or symmetric tensors of order \(3\) a contraction may be the following. A partial symmetric tensor \(\mathcal{T} \in \operatorname{Sym}^2 \mathbb{C}^{n+1} \otimes \mathbb{C}^{n+1}\) can be seen as an \((n+1)\)-tuple of quadratic forms \((q_0,\dots,q_n)\) in the variables \(x_i\). Analogously, given a a symmetric tensor \(f \in \operatorname{Sym}^3 \mathbb{C}^{n+1}\), i.e. a homogeneous cubic polynomial, one can associate to it an \((n+1)\)-tuple of quadratic forms given by its derivatives \(\frac{\partial f}{\partial x_i}\). The authors investigates the eigenscheme and some its particular subschemes of the aforementioned tensors with those contractions.
At first they recall some basic notions regarding the theory. In particular they introduce the irregular eigenscheme \(\operatorname{Irr}(\mathcal{T})\) and the regular eigenscheme \(\operatorname{Reg}(\mathcal{T})\). The first can be thought as the subscheme of \(E(\mathcal{T})\) given by points with zero eigenvalue, while the second is the residue of \(E(\mathcal{T})\) with respect to \(\operatorname{Irr}(\mathcal{T})\). After that they focus on the case of order \(3\) symmetric tensors providing bounds on the dimensions and geometric properties of the irregular and regular eigenschemes. Numerous examples of symmetric tensors satisfying all the described properties are provided. As they observe, if the regular eigenscheme of a cubic polynomial is \(0\) dimensional, then it consists of at most \(2^{n+1}-1\) points. Therefore they investigate in the ternary and quaternary case whether there exists a cubic polynomial with a prescribed number of regular eigenpoints. Eventually they show that a open subvariety of a linear subspace of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenschemes of order \(3\) quaternary symmetric tensors.
Reviewer: Reynaldo Staffolani (Trento)Deformations of smooth complete toric varieties: obstructions and the cup product.https://www.zbmath.org/1456.140632021-04-16T16:22:00+00:00"Ilten, Nathan"https://www.zbmath.org/authors/?q=ai:ilten.nathan-owen"Turo, Charles"https://www.zbmath.org/authors/?q=ai:turo.charlesLet \(X\) be a complete \(\mathbb Q\)-factorial toric variety with tangent sheaf \(\mathcal T_X\).
The main result of the paper gives a combinatorical description of \(H^2(X,\mathcal T_X)\) and the cup product map
\[H^1(X,\mathcal T_X)\times H^1(X,\mathcal T_X)\longrightarrow H^2(X,\mathcal T_X).\]
As an application an example of a smooth toric threefold is given for which the cup product map does not vanish, i.e. smooth complete
toric varieties may have obstructed deformations.
Reviewer: Gerhard Pfister (Kaiserslautern)Generalized moment graphs and the equivariant intersection cohomology of BXB-orbit closures in the wonderful compactification of a group.https://www.zbmath.org/1456.140642021-04-16T16:22:00+00:00"Oloo, Stephen"https://www.zbmath.org/authors/?q=ai:oloo.stephenBraden and MacPherson provided a combinatorial approach to computing equivariant intersection cohomology of (certain) varieties equipped with a torus action using the notion of moment graph [\textit{T. Braden} and \textit{R. MacPherson}, Math. Ann. 321, No. 3, 533--551 (2001; Zbl 1077.14522)]. The moment graph encodes the data of zero and one-dimensional torus orbits, and Braden and MacPherson's approach relies on the fact that it is possible to compute equivariant intersection cohomology using data from these orbits only, proved by \textit{M. Goresky} et al. [Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)]. This approach applies for example to Schubert varieties in flag varieties, which are Borel orbit closures.
In the paper under review, the author proposes a generalization of moment graphs and applies it to provide a combinatorial description of the torus equivariant intersection cohomology of the Borel orbit closures in the wonderful compactification of a semisimple adjoint complex algebraic group. The author expects applications of his approach to the representation theory of Hecke algebras.
The main difference with the previous situation is that instead of torus fixed points, the vertices of the generalized moment graph encode minimal torus orbits inside Borel orbits, which may be of arbitrary dimension. Some of the main arguments in the paper then involve working on transverse slices to such orbits, which in particular allows to describes edges of the generalized moment graph by using results of \textit{M. Brion} [Transform. Groups 4, No. 2--3, 127--156 (1999; Zbl 0953.14004)].
Following the ideas of Brendan and MacPherson, the author defines a so-called Brendan-MacPherson sheaf, of combinatorial nature, on the generalized moment graph. There is as well an intersection cohomology sheaf, directly related to intersection cohomology of the Borel orbit closures. The main result of the paper is that these two sheaves are isomorphic in an essentially canonical way, that is, the isomorphism is unique up to scalar multiplication.
Reviewer: Thibaut Delcroix (Montpellier)Dynamics near an idempotent.https://www.zbmath.org/1456.370232021-04-16T16:22:00+00:00"Shaikh, Md. Moid"https://www.zbmath.org/authors/?q=ai:shaikh.md-moid"Patra, Sourav Kanti"https://www.zbmath.org/authors/?q=ai:patra.sourav-kanti"Ram, Mahesh Kumar"https://www.zbmath.org/authors/?q=ai:ram.mahesh-kumarSummary: \textit{N. Hindman} and \textit{I. Leader} [Semigroup Forum 59, No. 1, 33--55 (1999; Zbl 0942.22003)]
first introduced the notion of the semigroup of ultrafilters converging to zero for a dense subsemigroup of \(((0, \infty), +)\). Using the algebraic structure of the Stone-Čech compactification, \textit{M. A. Tootkaboni} and \textit{T. Vahed} [Topology Appl. 159, No. 16, 3494--3503 (2012; Zbl 1285.54017)]
generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent \(e\) for a dense subsemigroup of a semitopological semigroup \((R, +)\) and they gave the combinatorial proof of the Central Sets Theorem near \(e\). Algebraically one can define quasi-central sets near \(e\) for dense subsemigroups of \((R, +)\). In a dense subsemigroup of \((R, +)\), C-sets near \(e\) are the sets, which satisfy the conclusions of the Central Sets Theorem near \(e\). \textit{S. K. Patra} [Topology Appl. 240, 173--182 (2018; Zbl 1392.37008)]
gave dynamical characterizations of these combinatorially rich sets near zero. In this paper, we shall establish these dynamical characterizations for these combinatorially rich sets near \(e\). We also study minimal systems near \(e\) in the last section of this paper.A direct proof that toric rank \(2\) bundles on projective space split.https://www.zbmath.org/1456.140542021-04-16T16:22:00+00:00"Stapleton, David"https://www.zbmath.org/authors/?q=ai:stapleton.david-pSummary: The point of this paper is to give a short, direct proof that rank \(2\) toric vector bundles on \(n\)-dimensional projective space split once \(n\) is at least \(3\). This result is originally due to \textit{J. Bertin} and \textit{G. Elencwajg} [Duke Math. J. 49, 807--831 (1982; Zbl 0512.14007)], and there is also related work by \textit{T. Kaneyama} [Nagoya Math. J. 111, 25--40 (1988; Zbl 0820.14010)], \textit{A. A. Klyachko} [Math. USSR, Izv. 35, No. 2, 337--375 (1990; Zbl 0706.14010); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1001--1039 (1989)], and \textit{N. Ilten} and \textit{H. Süss} [Transform. Groups 20, No. 4, 1043--1073 (2015; Zbl 1387.14125)]. The idea is that, after possibly twisting the vector bundle, there is a section which is a complete intersection.Non-elliptic webs and convex sets in the affine building.https://www.zbmath.org/1456.051732021-04-16T16:22:00+00:00"Akhmejanov, Tair"https://www.zbmath.org/authors/?q=ai:akhmejanov.tairSummary: We describe the \(\mathfrak{sl}_3\) non-elliptic webs in terms of convex sets in the affine building. Kuperberg defined the non-elliptic web basis in his work on rank-\(2\) spider categories. \textit{B. Fontaine} et al. [Compos. Math. 149, No. 11, 1871--1912 (2013; Zbl 1304.22016)] showed that the \(\mathfrak{s}l_3\) non-elliptic webs are dual to CAT(0) triangulated diskoids in the affine building. We show that each such triangulated diskoid is the intersection of the min-convex and max-convex hulls of a generic polygon in the building. Choosing a generic polygon from each of the components of the Satake fiber produces (the duals of) the non-elliptic web basis. The convex hulls in the affine building were first introduced by \textit{G. Faltings} [Prog. Math. 195, 157--184 (2001; Zbl 1028.14002)] and are related to tropical convexity, as discussed in work by \textit{M. Joswig} et al. [Albanian J. Math. 1, No. 4, 187--211 (2007; Zbl 1133.52003)] and by \textit{L. Zhang} [``Computing convex hulls in the affine building of \(\mathfrak{sl}_d\)'', Preprint, \url{arXiv:1811.08884}].Isotropic and coisotropic subvarieties of Grassmannians.https://www.zbmath.org/1456.140592021-04-16T16:22:00+00:00"Kohn, Kathlén"https://www.zbmath.org/authors/?q=ai:kohn.kathlen"Mathews, James C."https://www.zbmath.org/authors/?q=ai:mathews.james-cThe authors extend to subvarieties of any codimension in Grassmannians the notion of coisotropic hypersurfaces introduced by Gel'fand, Kapranov and Zelevinsky. A subvariety \(W\) is strongly coisotropic if every homomorphism in the conormal spaces of \(W\) has rank at most one. It turns out that the characterization of coisotropic hypersurfaces in terms of Chow hypersurfaces associated to projective varieties extends to strongly coisotropic subvarieties. Indeed the authors show that \(W\subset \mathrm{Gr}(a,\mathbb P^n)\) is strongly coisotropic if and only if there is an irreducible variety \(X\subset\mathbb P^n\) such that \(W\) is the Zariski closure of the set of all \(a\)-dimensional projective subspaces which intersect \(X\) at some smooth point non-transversely. A weaker notion of coisotropic subvarieties in Grassmannians is provided: \(W\) is coisotropic if each conormal space lies in the Zariski closure of the set of linear spaces that are spanned by homomorphisms of rank one. The authors study several properties of coisotropic subvarieties. They also prove that for a general hypersurface \(X\) of degree at least three in \(\mathbb P^n\) and for \(2\leq m\leq n,\deg(X)\), the closure of the variety of lines which intersect \(X\) at some smooth point with multiplicity \(m\) is coisotropic. Finally, the authors introduce the notion of isotropic subvarieties, for which the previous rank-one conditions hold on the tangent vectors, instead of on conormal vectors. The authors prove a full characterization of isotropic curves in Grassmannians in terms of preimages of obsculating subspaces to curves under suitable projections.
Reviewer: Luca Chiantini (Siena)Wilson loop algebras and quantum K-theory for Grassmannians.https://www.zbmath.org/1456.814352021-04-16T16:22:00+00:00"Jockers, Hans"https://www.zbmath.org/authors/?q=ai:jockers.hans"Mayr, Peter"https://www.zbmath.org/authors/?q=ai:mayr.peter"Ninad, Urmi"https://www.zbmath.org/authors/?q=ai:ninad.urmi"Tabler, Alexander"https://www.zbmath.org/authors/?q=ai:tabler.alexanderSummary: We study the algebra of Wilson line operators in three-dimensional \(\mathcal{N} = 2\) supersymmetric \(\mathrm{U}(M)\) gauge theories with a Higgs phase related to a complex Grassmannian \(\mathrm{Gr}(M,N)\), and its connection to K-theoretic Gromov-Witten invariants for \(\mathrm{Gr}(M,N)\). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of \(\mathrm{Gr}(M,N)\), isomorphic to the Verlinde algebra for \(\mathrm{U}(M)\), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.Birational superrigidity and \(K\)-stability of Fano complete intersections of index \(1\).https://www.zbmath.org/1456.140502021-04-16T16:22:00+00:00"Zhuang, Ziquan"https://www.zbmath.org/authors/?q=ai:zhuang.ziquanA Fano variety \(X\) is said to be birationally superrigid if it has terminal singularities,
it is \({\mathbb Q}\)-factorial of Picard number 1, and every birational map \(X\)
to a Mori fiber space is an isomorphism. On the other hand, \(X\) is \(K\)-stable
with respect to its anticanonical bundle if, essentially, it admits a Kähler-Einstein
metric, and \(K\)-stability is encoded in the positivity of the invariants \(\beta(F)\)
for \(F\) any dreamy prime divisor \(F\) over \(X\) (see 2.2 for details). In the paper under
review the author shows (see Thm. 1.2 and 1.3) that for a \(n\)-dimensional smooth Fano complete
intersection \(X \subset {\mathbb P}^{n+r}\) of index one, if \(n \geq 10r\) then \(X\) is birationally
superrigid and \(K\)-stable. Moreover, the smooth complete intersection of a quadric and a cubic
in \({\mathbb P}^5\) is also \(K\)-stable. For a Fano manifold (see Def. A.1 in the appendix of
the paper under review) \(X\) is said to be conditionally birationally superrigid if
every birational map from \(X\) to a Mori fiber space whose undefined locus has
codimension at least \(1\) plus the index of \(X\) is an isomorphism.
In the Appendix, the authors show that Fano complete intersections of higher index in large dimension (see Cor. A.3
for details) are conditionally birationally superrigid.
Reviewer: Roberto Muñoz (Madrid)Elliptic classes of Schubert varieties.https://www.zbmath.org/1456.140602021-04-16T16:22:00+00:00"Kumar, Shrawan"https://www.zbmath.org/authors/?q=ai:kumar.shrawan"Rimányi, Richárd"https://www.zbmath.org/authors/?q=ai:rimanyi.richard"Weber, Andrzej"https://www.zbmath.org/authors/?q=ai:weber.andrzejSummary: We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces \(G/P\). While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi-Tarasov-Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.