[Paper Review] Extension of cohomology classes and holomorphic sections defined on subvarieties

Xiangyu Zhou, Langfeng Zhu|arXiv (Cornell University)|Sep 19, 2019

Geometry and complex manifolds40 references4 citations

TL;DR

This paper establishes two new extension theorems for cohomology classes and holomorphic sections on non-reduced analytic subvarieties defined by multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. By introducing a novel simultaneous approximation of weight functions and a refined limit process in solving $\bar{\partial}$-equations, the authors prove injectivity of natural restriction maps, resolving a question by Cao-Demailly-Matsumura and generalizing Demailly's $L^2$ extension theorem with optimal estimates.

ABSTRACT

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura, and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general $L^2$ extension theorem obtained by Demailly.

Motivation & Objective

• To resolve a question posed by Cao-Demailly-Matsumura regarding extension of cohomology classes on analytic subvarieties defined by multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities.
• To unify and generalize existing injectivity theorems in complex geometry by establishing a new extension result for cohomology classes on non-reduced subvarieties.
• To optimize Demailly's general $L^2$ extension theorem by incorporating singular weights and achieving optimal $L^2$ estimates under minimal curvature assumptions.
• To develop a new technical framework involving simultaneous approximation of two weight functions and a reordering of limit processes in $\bar{\partial}$-equation solving.

Proposed method

• Introduce a double approximation technique for two weight functions $\psi$ and $\phi$ in the context of singular metrics on holomorphic line bundles.
• Apply a novel limit process that performs the convergence of weight functions prior to solving $\bar{\partial}$-equations, ensuring better control of $L^2$ norms.
• Use a cutoff function $\sigma_{\varepsilon,t}$ to localize solutions and construct a sequence of $\bar{\partial}$-closed forms $v_{k,\varepsilon,t}$ with controlled $L^2$-norms.
• Construct a holomorphic limit $F_{k,\varepsilon,t}$ via $\bar{\partial}$-solutions $w_{k,\varepsilon,t}$ and use Montel's theorem to extract uniform limits on compact sets.
• Employ a singular metric $h$ with curvature current satisfying $\sqrt{-1}\Theta_{L,h} + (1+\alpha)\sqrt{-1}\partial\bar{\partial}\psi \geq 0$ to ensure integrability and injectivity.
• Replace the original metric $h$ with $h_1 = h e^{-\Phi(\Psi)}$ using a convex increasing function $\Phi$ to ensure the existence of a global holomorphic extension $F$ with optimal $L^2$-bounds.

Experimental results

Research questions

• RQ1 Can cohomology classes defined on a non-reduced analytic subvariety, arising as the support of a quotient of multiplier ideal sheaves, be extended to the ambient manifold under minimal curvature assumptions?
• RQ2 Does the injectivity of the restriction map $H^q(X, \mathcal{O}_X(K_X \otimes L) \otimes \mathcal{I}(he^{-\psi})) \to H^q(X, \mathcal{O}_X(K_X \otimes L) \otimes \mathcal{I}(h))$ hold under the curvature condition $\sqrt{-1}\Theta_{L,h} + (1+\alpha)\sqrt{-1}\partial\bar{\partial}\psi \geq 0$?
• RQ3 Can Demailly's general $L^2$ extension theorem be optimized to achieve sharp $L^2$ estimates when the weight function $\psi$ has arbitrary singularities?
• RQ4 Is it possible to construct a global holomorphic extension $F$ of a section $f$ defined on a subvariety $Y$ such that $F - f \in \mathcal{I}(he^{-\psi})$ and $\|F\|^2_{L^2}$ is bounded by a constant multiple of $\|f\|^2_{L^2}$?

Key findings

• The natural homomorphism $H^q(X, \mathcal{O}_X(K_X \otimes L) \otimes \mathcal{I}(he^{-\psi})) \to H^q(X, \mathcal{O}_X(K_X \otimes L) \otimes \mathcal{I}(h))$ is injective for all $q \geq 0$ under the curvature conditions $\sqrt{-1}\Theta_{L,h} + \sqrt{-1}\partial\bar{\partial}\psi \geq 0$ and $\sqrt{-1}\Theta_{L,h} + (1+\alpha)\sqrt{-1}\partial\bar{\partial}\psi \geq 0$.
• The quotient sheaf $\mathcal{I}(h)/\mathcal{I}(he^{-\psi})$ is supported on an analytic subvariety $Y$, and the induced map $H^q(X, \mathcal{O}_X(K_X \otimes L) \otimes \mathcal{I}(h)) \to H^q(X, \mathcal{O}_X(K_X \otimes L) \otimes \mathcal{I}(h)/\mathcal{I}(he^{-\psi}))$ is surjective for all $q \geq 0$.
• A global holomorphic extension $F \in H^0(X, \mathcal{O}_X(K_X \otimes L) \otimes \mathcal{I}'_\psi(h))$ exists such that $F - f_i \in \mathcal{I}(he^{-\psi})_x$ for all $x \in U_i \cap X$, ensuring the extension respects the ideal sheaf structure.
• The extension satisfies the optimal $L^2$ estimate $\int_X \frac{|F|^2_{\omega,h}}{e^{\psi} R(\psi)} dV_{X,\omega} \leq \left(\frac{1}{\alpha R(\alpha_0)} + C_R\right) \int_Y |f|^2_{\omega,h} dV_{X,\omega}[\psi]$.
• The proof establishes the existence of a uniform constant $C_0$ such that $\varlimsup_{t \to -\infty} C(t) \leq C_0$, which is essential for the convergence of the sequence of approximating solutions.
• By replacing $h$ with $h_1 = h e^{-\Phi(\Psi)}$ for a suitable convex increasing $\Phi$, the authors ensure the existence of a global holomorphic extension $F$ with optimal $L^2$-bounds, confirming the optimality of the estimate.