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Register a new userA local Torelli theorem for log symplectic manifolds
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We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing diagrams, and amenable to algorithmic classification. Applying the theory to four-dimensional projective space, we obtain a total of 40 irreducible components of the moduli space, most of which are new. Our main technique is a detailed analysis of the relevant deformation complex (the Poisson cohomology) as an object of the constructible derived category.
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For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauvilles. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandts theory of subnormal subgroups.
A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. In the published version of Mapping class group and a global Torelli theorem for hyperkahler manifolds I made an error based on a wrong quotation of Dennis Sullivans famous paper Infinitesimal computations in topology. I claimed that the natural homomorphism from the mapping class group to the group of automorphims of cohomology of a simply connected Kahler manifold has finite kernel. In a recent preprint arXiv:1907.05693, Matthias Kreck and Yang Su produced counterexamples to this statement. Here I correct this error and other related errors, observing that the results of Mapping class group and a global Torelli theorem remain true after an appropriate change of terminology.
Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $Bto C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $tilde{A}_{n-1}$ to that of $tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.
A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities $tilde{E}_6,tilde{E}_7$ and $tilde{E}_8$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskiis Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic.
Let $text{M}_C( 2, mathcal{O}_C) cong mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $mathbb{C}$. Let $beta_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( beta_{C}) = beta_C$ for some birational map $f$ from $text{M}_C( 2, mathcal{O}_C)$ to $text{M}_{C}( 2, mathcal{O}_{C})$, then the Jacobians of $C$ and of $C$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C$ are isomorphic. Similar statements hold for Kummer surfaces in $mathbb{P}^3$ and for quadratic line complexes.
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