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A period map for generalized deformations. (English) Zbl 1189.14018

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Let \(X\) be a compact Kähler manifold and let \(H^\ast(X,\mathbb C)\) be the graded vector space of its de Rham cohomology. The goal of this article is to define a natural transformation \(\Phi:\text{Def}_X\rightarrow\text{Aut}_{H^\ast(X,\mathbb C)}\). For every local Artinian \(\mathbb C\)-algebra \(A\) and every deformation of \(X\) over \(\text{Spec}(A)\) the authors define in a functorial way a canonical morphism of schemes \(\text{Spec}(A)\rightarrow\text{GL}(H^\ast(X,\mathbb C))=\prod_n\text{GL}(H^n(X,\mathbb C)).\) The construction of this morphism is carried out by using the interplay of Cartan homotopies and \(L_\infty\)-morphisms and is compatible with classical constructions of the theory of infinitesimal variations of Hodge structures:
(1)
Via the natural isomorphism \(H_X\simeq\bigoplus_{p,q}H^q(\Omega^p_X)\) induced by Dolbeault’s theorem and the \(\partial\overline\partial\)-lemma, the differential of \(\Phi\), \[ d\Phi:H^1(T_X)\rightarrow\text{Hom}^0(H^\ast(X,\mathbb C),H^\ast(X,\mathbb C)), \] is identified with the contraction operator \(d\Phi(\xi)=\mathbf{i}_{\xi}\) where \(\mathbf{i}_{\xi} =\xi\lrcorner\omega\).
(2)
The contraction \(\mathbf{i}:H^2(T_X)\rightarrow\text{Hom}^1(H^\ast(X,\mathbb C),H^\ast(X,\mathbb C))\) is a morphism of obstruction theories. In particular every obstruction to deformation of \(X\) is contained in the kernel of \(\mathbf{i}\).
(3)
For every \(m\) let \(H^\ast(F^m)\subseteq H^\ast(X,\mathbb C)\) be the subspace of cohomology classes of closed \((p,q)\)-forms, with \(p\geq m\). Then the composition of \(\Phi\) with the natural projection \(\text{GL}(H^\ast(X,\mathbb C))\rightarrow\text{Grass}(H^\ast(X,\mathbb C))=\prod_n\text{Grass}(H^n(X,\mathbb C)),\) sending \(f\) to \(f(H^\ast(F^m)),\) is the classical period map.
Besides of using the framework of \(L_\infty\)-algebras, the authors also give a more geometric definition of the morphism \(\Phi\): Denote by \(A_X=\bigoplus_i A^i_X\) the space of complex valued differential forms on \(X\), by \(d=\partial+\overline{\partial}:A^i_X\rightarrow A^{i+1}_X\) the de Rham differential and by \(\partial A_X\subseteq A_X\) the subspace of \(\partial\)-exact forms. A small variation of the almost complex structure is determined by a form \(\xi\in A^{0,1}_X(T_X)\). The integrability condition on \(\xi\) is equivalent to \((d+\mathbf{l}_\xi)^2=0,\) where \(\mathbf{l}_\xi:A^i_X\rightarrow A^{i+1}_X\) is the holomorphic Lie derivative. Assuming therefore that \((d+\mathbf{l}_\xi)^2=0,\) according to the \(\partial\overline\partial\)-lemma, the complex \((\partial A_X,d)\) is acyclic so that id \(\xi\) is sufficiently small, the complex \((\partial A_X,d+\mathbf{l}_xi)\) is still acyclic. Let \([\omega]\in H^\ast(X,\mathbb C)\) and choose a \(d\)-closed form \(\omega_0\in A_X\) representing \([\omega]\) and such that \(\partial\omega_0=0.\) Because \((d+\mathbf{l}_\xi)\omega_0=\partial(\xi\lrcorner\omega_0)\in\partial A_X\) and \((d+\mathbf{l}_\xi)^2\omega_0=0\), there exists \(\beta\in A_X\) such that \((d+\mathbf{l}_\xi)\omega_0=(d+\mathbf{l}_\xi)\partial\beta\). If \(\mathbf{i}_\xi\) is the contraction, then \(d(e^{\mathbf{i}_\xi}(\omega_0-\partial\beta))=0,\) and the cohomology class of \(e^{\mathbf{i}_\xi}(\omega_0-\partial\beta)\) does not depend on the choice of \(\beta\) and \(\omega_0\). So the authors may define \(\Phi_\xi([\omega])\) as the cohomology class of \(e^{\mathbf{i}_\xi}(\omega_0-\partial\beta)\).
As a direct consequence of the \(L_\infty\)-approach, the authors find that \(\Phi\) is invariant under the gauge action, where two integrable small variations of the almost complex structure \(\xi_1\), \(\xi_2\) are gauge equivalent if they give isomorphic deformations of \(X\).
The authors construction generalizes directly to generalized deformations of \(X\), defined as the solution, up to gauge equivalence, of the Maurer-Cartan equation in the differential graded Lie algebra \[ \text{Poly}_X=\bigoplus_i\text{Poly}_X^i,\;\text{Poly}_X^i=\bigoplus_{b-a=i-1} A_X^{0,b}(\bigwedge^a T_X), \] endowed with the opposite Dolbeault differential and the Schouten-Nijenhuys bracket.
At the end, the authors get for every \(m\) a commutative diagram of morphisms of functors of Artin rings \[ \begin{tikzcd} \widetilde{\mathrm{Def}}_X \rar["\Phi"] & \mathrm{Aut}_{H^\ast(X,\mathbb C)}\dar["\pi"] \\ \mathrm{Def}_X \uar["i"] \rar["p" '] & \mathrm{Grass}_{H^\ast(F^m),H^\ast(X,\mathbb C)} \end{tikzcd} \] where \(\widetilde{\text{Def}}_X\) is the functor of generalized deformations of \(X\), \(i\) is the natural inclusion, \(\text{Grass}_{H^\ast(F^m),H^\ast(X,\mathbb C)}\) is the Grassmann functor with base point \(H^\ast(F^m)\), \(\pi\) is the smooth morphism defined as \(\pi(f)=f(H^\ast(F^m))\) and \(p\) is the classical \(m\)th period map. Thus \(\pi\Phi\) is a natural candidate for a period map for generalized deformations.
The authors studies deformation functors associated with DGLA morphisms in general, and they give an example from Kähler geometry. The authors also formalize, under the notion of Cartan homotopy, a set of standard identities that arise in deformation theory of algebraic geometric objects. These identities are used in the study of polyvector fields and generalized periods, and end all in all in the following nice theorem:
For any \(m\geq 0\), the map \(\Phi:\text{Def}\rightarrow\text{Aut}_{H^\ast(X,\mathbb C)}\) lifts the \(m\)-th period map \(\text{Def}_X\rightarrow\text{Grass}_{H^\ast(F^m),H^\ast(X,\mathbb C)}.\)
The article is very good written and highly nontrivial. With the correct prerequisites, it gives nice results and good examples.

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
17B70 Graded Lie (super)algebras
13D10 Deformations and infinitesimal methods in commutative ring theory
32G20 Period matrices, variation of Hodge structure; degenerations
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