[Paper Review] Period map for non-compact holomorphically symplectic manifolds

D. Kaledin, Misha Verbitsky|ArXiv.org|May 1, 2000

Geometry and complex manifolds4 references3 citations

TL;DR

This paper establishes a period map for non-compact holomorphically symplectic manifolds under mild cohomological assumptions, showing that the coarse moduli space of formal symplectic deformations is isomorphic to the formal completion of $ H^2(M) $ at the class of the symplectic form. The key result extends the Bogomolov-Tian-Todorov theorem to non-compact symplectic manifolds using a symplectic adaptation of Ran's algebraic deformation theory, leveraging the vanishing of obstructions via the flatness of a certain sheaf complex.

ABSTRACT

We study the deformations of a holomorphic symplectic manifold $M$, not necessarily compact, over a formal ring. We show (under some additional, but mild, assumptions on $M$) that the coarse deformation space exists and is smooth, finite-dimensional and naturally embedded into $H^2(M)$. For a holomorphic symplectic manifold $M$ which satisfies $H^1(M,{\cal O}_M) = H^2(M,{\cal O}_M)=0$, the coarse moduli of formal deformations is isomorphic to $\Spec\C[[t_1, ..., t_n]]$, where $t_1$, ... $t_n$ are coordinates in $H^2(M)$. This revised version contains one minor improvement: exposition in Subsection 5.1 has been made more detailed and rigourous.

Motivation & Objective

• To develop a deformation theory for non-compact holomorphically symplectic manifolds, where standard methods fail due to non-degenerate Dolbeault spectral sequences.
• To extend the period map construction—previously valid for compact Calabi-Yau and holomorphically symplectic manifolds—to non-compact settings under mild cohomological assumptions.
• To show that the coarse moduli space of formal symplectic deformations is isomorphic to the formal completion of $ H^2(M) $ at $[Ω]$, generalizing the compact case.
• To overcome obstructions in deformation theory by focusing on deformations of the pair $(M, \Omega)$ rather than $M$ alone, using the symplectic structure to simplify the cohomological framework.

Proposed method

• Adopt a symplectic adaptation of Z. Ran's algebraic proof of the Tian-Todorov theorem, replacing the Calabi-Yau condition with holomorphic symplectic structure.
• Use the Kodaira-Spencer class to classify deformations, but restrict to symplectic deformations to eliminate obstructions.
• Employ elementary extensions of rings to inductively lift deformations, using the exact sequence of differentials to control lifting conditions.
• Define the period map $ \operatorname{Per}: \operatorname{Spl}(M,\Omega) \to H^2(M) $, showing it is an isomorphism onto the formal neighborhood of $[\Omega]$ under the assumption $ H^i(\mathcal{O}_M) = 0 $ for $ i \geq 1 $.
• Prove flatness of the sheaf $ R^2\pi_*F^1\Omega^{\bullet}(\widetilde{X}/S) $, which ensures the absence of obstructions in the deformation process.
• Use the operadic deformation theory framework to interpret the symplectic tangent complex as $ F^1\Omega^{\bullet}(X) $, justifying the cohomological simplifications.

Experimental results

Research questions

• RQ1 Can a period map be constructed for non-compact holomorphically symplectic manifolds, analogous to the compact case?
• RQ2 Under what conditions does the coarse moduli space of formal symplectic deformations exist and become smooth and finite-dimensional?
• RQ3 Why do obstructions vanish in the deformation theory of holomorphic symplectic manifolds even when the Dolbeault spectral sequence does not degenerate?
• RQ4 How does the period map relate to the cohomology $ H^2(M) $ in the non-compact setting?
• RQ5 Can the algebraic deformation theory of Ran be adapted to the symplectic case by focusing on the pair $ (M, \Omega) $ rather than $ M $ alone?

Key findings

• For a holomorphic symplectic manifold $ M $ with $ H^i(\mathcal{O}_M) = 0 $ for all $ i \geq 1 $, the coarse moduli space of formal symplectic deformations $ \operatorname{Spl}(M,\Omega) $ is isomorphic to the formal completion of $ H^2(M) $ at $[\Omega]$.
• The period map $ \operatorname{Per}: \operatorname{Spl}(M,\Omega) \to H^2(M) $ is an isomorphism in this case, generalizing the compact case result.
• The deformation theory remains unobstructed because the sheaf $ R^2\pi_*F^1\Omega^{\bullet}(\widetilde{X}/S) $ is flat over the base, ensuring liftings at each step.
• The key technical innovation is the use of elementary extensions and the lifting of the Kodaira-Spencer class through the differential exact sequence, avoiding circular dependencies.
• The result holds for a broader class of non-compact manifolds beyond the $ H^i(\mathcal{O}_M) = 0 $ case, with the period map remaining an immersion into $ H^2(M) $, though the moduli space is no longer isomorphic to $ H^2(M) $.
• The symplectic structure allows the replacement of the problematic $ H^1(\Omega^{n-1}(M)) $ term in Calabi-Yau theory with $ H^1(\Omega^1(M)) $, whose obstructions vanish under the given assumptions.