## Advanced Studies in Pure Mathematics

### Curvature of higher direct image sheaves

#### Abstract

Given a family $(F,h) \to X \times S$ of Hermite-Einstein bundles on a compact Kähler 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.

#### Article information

Dates
Revised: 13 September 2014
First available in Project Euclid: 23 October 2018

https://projecteuclid.org/ euclid.aspm/1540319487

Digital Object Identifier
doi:10.2969/aspm/07410171

Mathematical Reviews number (MathSciNet)
MR3791213

Zentralblatt MATH identifier
1392.32008

#### Citation

Geiger, Thomas; Schumacher, Georg. Curvature of higher direct image sheaves. Higher Dimensional Algebraic Geometry: In honour of Professor Yujiro Kawamata's sixtieth birthday, 171--184, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07410171. https://projecteuclid.org/euclid.aspm/1540319487