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

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)
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