Curvature of higher direct image sheaves. (English) Zbl 1392.32008

Geiger, Thomas; Schumacher, Georg

Oguiso, Keiji (ed.) et al., Higher dimensional algebraic geometry. In honour of Professor Yujiro Kawamata’s sixtieth birthday. Proceedings of the conference, Tokyo, Japan, January 7–11, 2013. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-86497-046-4/hbk). Advanced Studies in Pure Mathematics 74, 171-184 (2017).

Summary: 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 \(R^qp_*\mathcal O(F)\) 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.
For the entire collection see [Zbl 1388.14012].

MSC:

32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
14D20	Algebraic moduli problems, moduli of vector bundles
32G13	Complex-analytic moduli problems
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)

Keywords:

Weil-Petersson metric; families of Hermite-Einstein bundles; stable bundles; curvature of direct image sheaves; moduli spaces