Curvature of vector bundles associated to holomorphic fibrations

Abstract

Let L be a (semi)-positive line bundle over a Kähler manifold, X, fibered over a complex manifold Y . Assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle E over Y whose fibers over points y are the spaces of global sections over X y to L ⊗ K X/Y , endowed with the L 2 -metric, is (semi)-positive in the sense of Nakano. We also discuss various applications, among them a partial result on a conjecture of Griffiths on the positivity of ample bundles.

Representability of Chern–Weil forms

“…In the case of curves this was proven in [20,4]. Somewhat strong evidence for this conjecture in the general case is provided by the fact that if E is Hartshorne-ample, then E ⊗ det(E) is Nakano-positive (stronger than Griffiths positive) [1,16]. It is also known [2,11] that the Schur polynomials of Hartshorne-ample bundles are numerically positive (and in fact the only numerically positive characteristic classes are positive linear combinations of the Schur polynomials [11]).…”

“…In particular, X is projective by the Kodaira embedding theorem. 1 It follows from a theorem proven in [12,8] that this equality holds even at the level of Chern-Weil forms for Griffiths-positive bundles.…”

“…As a consequence (see lemma 4.1.1 from [13] for instance) it follows that c 1 (E) and c 2 1 (E) − c 2 (E) have positive representatives 1 . In particular, X is projective by the Kodaira embedding theorem.…”

“…Let Σ be a Riemann surface and X = Σ × P 1 . Let SU (2) act trivially on X and in the standard manner on P 1 = SU(2)/U (1). Let L be a holomorphic line bundle on Σ and E be a rank 2 holomorphic bundle over X which is an extension : (1) For any Ω > 0 and η > 0 satisfying…”

Strong openness of multiplier ideal sheaves and optimal L 2 extension

“…The optimal L 2 extension theorem (Theorem 1.4 [21]) gives unified optimal estimate versions of various well-known L 2 extension theorems in [39,35,10,52,44,4,38,16], etc. Some interesting relations between the optimal L 2 extension and some questions are found, so that the questions are solved in [21] by using optimal L 2 extension (Theorem 1.4), such as Suita's conjecture (see [57] [45]), L-conjecture (see [59]), extended Suita conjecture (see [59]), and an open question posed by Ohsawa (see [42]), etc.…”