Curvature of higher direct image sheaves

TL;DR

This paper computes the curvature of higher direct image sheaves of Hermite-Einstein bundles over a compact K"ahler manifold, providing explicit formulas and discussing applications to moduli spaces of stable bundles.

Contribution

It derives explicit curvature formulas for higher direct image sheaves of Hermite-Einstein bundles, including special cases like fixed determinant and endomorphism bundles.

Findings

Curvature formulas are simplified for fixed determinant families.

Explicit metric computations for moduli spaces of stable bundles.

Analysis of curvature behavior on the complement of an analytic subset.

Abstract

Given a family $(F,h) \to X \times S$ of Hermite-Einstein bundles on a compact K\"ahler manifold $(X,g)$ we consider the higher direct image sheaves $R^q p_* \mathcal{O}(F)$ on $S$, where $p: X \times S \to S$ is the projection. On the complement of an analytic subset these sheaves are locally free and carry a natural metric, induced by the $L_2$ inner product of harmonic forms on the fibers. We compute the curvature of this metric which has a simpler form for families with fixed determinant and families of endomorphism bundles. Furthermore, we discuss the metric for moduli spaces of stable vector bundles.