[Paper Review] Extension of 2-forms and symplectic varieties

Yoshinori Namikawa|arXiv (Cornell University)|Oct 12, 2000

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes foundational theorems on symplectic varieties by proving the Stability Theorem, which ensures that symplectic 2-forms extend in flat families of symplectic varieties, and the Local Torelli Theorem, which shows that the period map is a local isomorphism under certain conditions. These results generalize deformation theory and Hodge theory to singular symplectic spaces with terminal singularities, particularly when the singular locus has codimension at least 4.

ABSTRACT

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.

Motivation & Objective

• To establish the stability of symplectic structures under flat deformations of symplectic varieties.
• To generalize the Local Torelli theorem to singular symplectic varieties with terminal singularities.
• To clarify the behavior of cohomology and holomorphic 2-forms on resolutions of symplectic varieties.
• To provide conditions under which the period map for symplectic varieties is a local isomorphism.
• To extend Hodge-theoretic and deformation-theoretic results to non-smooth symplectic varieties with controlled singularities.

Proposed method

• Uses the Kuranishi family of a symplectic variety to study its local deformations, assuming the singular locus has codimension ≥4.
• Applies the Stability Theorem to show that the symplectic 2-form extends in flat families, ensuring each fiber remains symplectic.
• Constructs a quadratic form q on H^2(U, ℂ) using the holomorphic 2-form ω and its lifts to a resolution ν: ~Z → Z.
• Defines a period map p: S → ℙ(H) by mapping each deformation parameter s to the class of the extended 2-form ω_s in cohomology.
• Relies on Hodge decomposition and mixed Hodge theory to analyze cohomology of fibers and show that H^2(U_s, ℂ) is isomorphic to H^2(Z, ℂ).
• Uses the condition that ∫~Z (ωω̄)^l = 1 to normalize the 2-form and ensure well-definedness of the quadratic form and period map.

Experimental results

Research questions

• RQ1 Can a symplectic 2-form on the smooth locus of a symplectic variety be extended to all fibers in a flat family?
• RQ2 Under what conditions is the period map for a symplectic variety a local isomorphism?
• RQ3 How does the cohomology of the smooth locus relate to the cohomology of the total space in a deformation of a symplectic variety?
• RQ4 What is the role of the singular locus's codimension in the deformation theory of symplectic varieties?
• RQ5 Can the Local Torelli theorem be extended to non-projective symplectic varieties with terminal singularities?

Key findings

• The symplectic 2-form ω on the smooth locus U of a projective symplectic variety Z extends to a symplectic form on all fibers Z_t in a sufficiently small neighborhood of the central fiber in a flat family.
• The restriction map H^2(Z, ℂ) → H^2(U, ℂ) is an isomorphism, implying that cohomology of the singular variety captures the cohomology of its smooth locus.
• The quadratic form q on H^2(U, ℂ) is independent of the choice of resolution ν: ~Z → Z, and is defined via integration of ω and its conjugate against lifts of cohomology classes.
• The period map p: S → ℙ(H) is a local isomorphism, where H = H^2(U, ℂ), and the image lies in the locus ℒ = {x ∈ ℙ(H) | q(x) = 0, q(x + x̄) > 0}.
• The local system R^2π_*(π^{-1}𝒪_S) is trivialized over the base S, and its fiber is isomorphic to H^2(U_s, ℂ) for each s ∈ S.
• The condition (*) on the image of cohomology classes from the resolution ~Z to the pushforward sheaf R^2ν_*ℚ ensures the validity of the Local Torelli Theorem in the non-projective case.