The Griffiths conjecture asserts that every ample vector bundle E over a compact complex manifold S admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on O P(E * ) (1) to induce a Griffiths positive L 2 -metric on the vector bundle E. This result suggests to study the relative Kähler-Ricci flow on O P(E * ) (1) for the fibration P(E * ) → S. We define a flow and give arguments for the convergence.
Let X be a compact Kähler space with klt singularities and vanishing first Chern class.
We prove the Bochner principle for holomorphic tensors on the smooth locus of X: any such tensor is parallel with respect to the singular Ricci-flat metrics.
As a consequence, after a finite quasi-étale cover X splits off a complex torus of the maximum possible dimension.
We then proceed to decompose the tangent sheaf of X according to its holonomy representation.
In particular, we classify those X which have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Calabi–Yau or irreducible holomorphic symplectic. As an application of these results, we show that if X has dimension four, then it satisfies Campana’s Abelianity Conjecture.
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Given a holomorphic family f : X → S of compact complex manifolds and a relatively ample line bundle L → X , the higher direct images R n−p f * Ω p X /S (L) carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images. This generalizes a result of Schumacher [Sch12], where he computed the curvature of R n−p f * Ω p X /S (K ⊗m X /S ) for a family of canonically polarized manifolds. For p = n, it coincides with a formula of Berndtsson obtained in [Be11]. Thus, when L is globally ample, we reprove his result [Be09] on the Nakano positivity of f * (K X /F ⊗ L).
The Griffiths conjecture asserts that every ample vector bundle E over a compact complex manifold S admits a hermitian metric with positive curvature in the sense of Griffiths. In this article, we first give a sufficient condition for a positive hermitian metric on O P(E * ) (1) to induce a Griffiths positive L 2 -metric on the vector bundle E. This result suggests to study the relative Kähler-Ricci flow on O P(E * ) (1) for the fibration P(E * ) → S. We define this flow and prove its convergence.
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