Advanced Studies in Pure Mathematics

Curvature of higher direct image sheaves

Thomas Geiger and Georg Schumacher

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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

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

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

Permanent link to this document euclid.aspm/1540319487

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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}

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


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.

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