Curvatures of direct image sheaves of vector bundles and applications. (English) Zbl 1293.32031

Liu, Kefeng; Yang, Xiaokui

J. Differ. Geom. 98, No. 1, 117-145 (2014).

Summary: Let \(p: \mathcal X\to S\) be a proper Kähler fibration and \(\;E \to \mathcal X\) a Hermitian holomorphic vector bundle. As motivated by the work of B. Berndtsson [Ann. Math. (2) 169, No. 2, 531–560 (2009; Zbl 1195.32012)], by using basic Hodge theory, we derive several general curvature formulas for the direct image \(p_*(K_{X/S} \otimes \mathcal E)\) for a general Hermitian holomorphic vector bundle \(\mathcal{E}\) in a simple way. A straightforward application is that, if the family \(\mathcal{X}_{\mathcal S}\) is infinitesimally trivial and the Hermitian vector bundle \(\mathcal{E}\) is Nakano-negative along the base \(\mathcal{S}\), then the direct image \(p_* (K_{X/S} \otimes \mathcal E)\) is Nakano-negative. We also use these curvature formulas to study the moduli space of projectively flat vector bundles with positive first Chern classes and obtain that, if the Chern curvature of direct image \(p_*(K_X \otimes E)\) – of a positive projectively flat family \((E, h(t))_{t \in \mathbb{D}} \to X\) – vanishes, then the curvature forms of this family are connected by holomorphic automorphisms of the pair \((X,E)\).

MSC:

32L05	Holomorphic bundles and generalizations
32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
32L15	Bundle convexity

Keywords:

Hermitian holomorphic vector bundle; curvature; direct image

Citations:

Zbl 1195.32012