[Paper Review] Curvature of vector bundles and subharmonicity of Bergman kernels

Bo Berndtsson|ArXiv.org|May 23, 2005

Geometry and complex manifolds8 references3 citations

TL;DR

This paper establishes that the Bergman kernel associated with a family of weighted $L^2$-spaces of holomorphic functions is strictly plurisubharmonic, by proving that the corresponding infinite-rank vector bundle of holomorphic sections carries a Hermitian metric with strictly positive curvature in the sense of Nakano. The result extends to holomorphic vector bundles: if a finite-rank holomorphic vector bundle $V$ is ample in the sense of Hartshorne, then $S^m(V) igotimes \det V$ admits a Hermitian metric with strictly positive Nakano curvature for all $m \geq 0$. This generalizes earlier results on Bergman kernel subharmonicity and provides a new curvature criterion for ampleness.

ABSTRACT

In a previous paper, \cite{Berndtsson}, we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted $L^2$-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of $L^2$-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sections to line bundles over a locally trivial fibration we also prove that if a holomorphic vector bundle, $V$, over a complex manifold is ample in the sense of Hartshorne, then $V\gr\det V$ has an Hermitian metric with curvature strictly positive in the sense of Nakano.

Motivation & Objective

• To establish the strict Nakano positivity of the vector bundle of holomorphic $L^2$-sections over a family of weighted spaces.
• To derive the subharmonicity of Bergman kernels as a corollary of the curvature result.
• To extend the curvature positivity result to holomorphic fibrations and line bundles over projective bundles.
• To prove that if a holomorphic vector bundle $V$ is ample in the Hartshorne sense, then $S^m(V) \otimes \det V$ admits a Hermitian metric with strictly positive Nakano curvature.

Proposed method

• Uses the curvature formula of Griffiths for subbundles of holomorphic vector bundles, relating the curvature of the subbundle to the ambient bundle and the second fundamental form.
• Applies Hörmander's $L^2$-estimate for the $\bar{\partial}$-equation to control the second fundamental form of the $L^2$-section bundle.
• Constructs the bundle $E$ of holomorphic sections of $L \otimes K_{X_t}$ over fibers of a locally trivial fibration to define a canonical $L^2$-metric.
• Employs the Kodaira-Nakano-Hörmander estimate for line bundles on compact manifolds to control curvature in the fibration setting.
• Analyzes the Chern connection and curvature of the line bundle $O_{\mathbb{P}(V)}(l)$ over projective bundles to relate the curvature of $E(l)$ to the geometry of $V$.
• Uses the fact that $E(r) = \det V$ and $E(r+1) = V \otimes \det V$ to identify the relevant bundles and apply the curvature result.

Experimental results

Research questions

• RQ1 Is the Bergman kernel function $K_t(z,z)$ strictly plurisubharmonic in the parameter $t$?
• RQ2 Does the vector bundle of holomorphic $L^2$-sections over a family of weighted spaces carry a Hermitian metric with strictly positive curvature in the sense of Nakano?
• RQ3 Can the positivity of the curvature of $E = \Gamma(X_t, L|X_t \otimes K_{X_t})$ be established for holomorphic fibrations with positive line bundles?
• RQ4 If a holomorphic vector bundle $V$ is ample in the Hartshorne sense, does $S^m(V) \otimes \det V$ admit a Hermitian metric with strictly positive Nakano curvature?

Key findings

• The vector bundle $E$ of holomorphic $L^2$-sections over a family of weighted spaces has a Hermitian metric with strictly positive curvature in the sense of Nakano.
• As a consequence, the Bergman kernel function $K_t(z,z)$ is strictly plurisubharmonic in the parameter $t$.
• For a locally trivial holomorphic fibration with a positive line bundle $L$ satisfying the local triviality condition, the bundle $E$ of holomorphic sections of $L \otimes K_{X_t}$ has strictly positive Nakano curvature.
• If a holomorphic vector bundle $V$ is ample in the Hartshorne sense, then $S^m(V) \otimes \det V$ admits a Hermitian metric with strictly positive Nakano curvature for all $m \geq 0$.
• The bundle $E(r)$ of holomorphic sections of $O_{\mathbb{P}(V)}(r) \otimes K_{\mathbb{P}(V^*_t)}$ is isomorphic to $\det V$, and $E(r+1)$ is isomorphic to $V \otimes \det V$.