On the flatness of local models for the symplectic group

Topological flatness of orthogonal local models in the split, even case. I

Local models are schemes, defined in terms of linear algebra, that were introduced by Rapoport and Zink to study the étale-local structure of integral models of certain PEL Shimura varieties over p-adic fields. A basic requirement for the integral models, or equivalently for the local models, is that they be flat. In the case of local models for even orthogonal groups, Genestier observed that the original definition of the local model does not yield a flat scheme. In a recent article, Pappas and Rapoport introduced a new condition to the moduli problem defining the local model, the so-called s…

Extension of 2-forms and symplectic varieties

This paper deals with symplectic varieties which do not have symplectic resolutions. Some moduli spaces of semi-stable torsion-free sheaves on a K3 surface, and symplectic V-manifolds are such varieties. We shall prove local Torelli theorem for symplectic varieties. Some results on symplectic singularities are also included.

The cluster symplectic double and moduli spaces of local systems

We prove a conjecture of Fock and Goncharov which provides a birational equivalence of a cluster variety called the cluster symplectic double and a certain moduli space of local systems associated to a surface.

Local symplectic algebra of quasi-homogeneous curves

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of an analytic curve is a finite dimensional vector space. We also show that the action of local diffeomorphisms preserving the curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain the complete symplectic classification of curves with the semig…

Central charges, symplectic forms, and hypergeometric series in local mirror symmetry

We study a cohomology-valued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich's homological mirror symmetry. In the case of local mirror symmetry, we will identify a symplectic form, and will conjecture an integral and symplectic monodromy property of a relevant hypergeometric series of Gel'fand-Kapranov-Zelevinski type.

A new family of isolated symplectic singularities with trivial local fundamental group

We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of $\mathbb{C}^4$ by the dihedral group of order $2d$, (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order $2d$ at equal parameters, (3) as singularities of a certain Slodowy slice in the $d$-fold cover of the nilpotent cone in ${\mathfrak{sl}}_d$.

Symplectic double for moduli spaces of G-local systems on surfaces

Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and define a collection of special rational coordinate systems on it. The moduli space D(G,S) is the symplectic double of the Poisson moduli space of framed G-local systems on S. Its symplectic form is upgraded to a K2-symplectic structure for which the special coordinates are K2-Darboux coordinates.

Symplectic Sμ singularities

We study the local symplectic algebra of the 1-dimensional isolated complete intersection singularity of type Sμ. We use the method of algebraic restrictions to classify symplectic Sμ singularities. We distinguish these symplectic singularities by discrete symplectic invariants. We also give the geometric description of them.

Frobenius 相交平坦性的变体及其在 Tate 代数中的应用

Variants on Frobenius Intersection Flatness and Applications to Tate Algebras

针对 Tate 代数等非局部优秀环不适用于现有 Frobenius 奇点理论的问题，本文提出 FOR、Frobenius 相交平坦与 FORT 三类框架，证明 Tate 代数具有 Frobenius 相交平坦性，并推出 affinoid 代数测试元及 $F$-纯轨迹开性的新结果。

A global Torelli theorem for singular symplectic varieties

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geomet…

A local Torelli theorem for log symplectic manifolds

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 di…

Topological recursion relations in the symplectic field theory of mapping tori and local symplectic field theory

Based on the localization result for descendants in rational SFT moduli spaces from our last joint paper, we prove topological recursion relations for the Hamiltonian in SFT of symplectic mapping tori and in local SFT. Combined with the dilaton equation in SFT, we use them to prove a reconstruction theorem for descendants from primaries. While it turns out that (in contrast to Gromov-Witten theory and non-equivariant cylindrical contact homology) the descendant Hamiltonian cannot be computed from the Hamiltonian without descendants alone, we get that the only additional piece of information ne…

Volume and symplectic structure for l-adic local systems

We introduce a notion of volume for an l-adic local system over an algebraic curve and, under some conditions, give a symplectic form on the rigid analytic deformation space of the corresponding geometric local system. These constructions can be viewed as arithmetic analogues of the volume and the Chern-Simons invariants of a representation of the fundamental group of a 3-manifold which fibers over the circle and of the symplectic form on the character varieties of a Riemann surface. We show that the absolute Galois group acts on the deformation space by conformal symplectomorphisms which exte…

Symplectic $W_8, W_9$ singularities

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $W_8$ and $W_9$ singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes.

Zero-dimensional symplectic isolated complete intersection singularities

We study the local symplectic algebra of the 0-dimensional isolated complete intersection singularities. We use the method of algebraic restrictions to classify these symplectic singularities. We show that there are non-trivial symplectic invariants in this classification.