Advanced Studies in Pure Mathematics

Curvature of higher direct image sheaves

Thomas Geiger and Georg Schumacher

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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

Source
Higher Dimensional Algebraic Geometry: In honour of Professor Yujiro Kawamata's sixtieth birthday, K. Oguiso, C. Birkar, S. Ishii and S. Takayama, eds. (Tokyo: Mathematical Society of Japan, 2017), 171-184

Dates
Received: 13 November 2013
Revised: 13 September 2014
First available in Project Euclid: 23 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1540319487

Digital Object Identifier
doi:10.2969/aspm/07410171

Mathematical Reviews number (MathSciNet)
MR3791213

Zentralblatt MATH identifier
1392.32008

Subjects
Primary: 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30] 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}

Keywords
Weil-Petersson metric Families of Hermite-Einstein bundles Stable bundles Curvature of direct image sheaves Moduli spaces

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


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