Strict and nonstrict positivity of direct image bundles

By B. Berndtsson. 2010

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This paper is a sequel to (Berndtsson in Ann Math 169:531–560, 2009). In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kähler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Nakano). Here we address the question if the curvature is strictly positive when the Kodaira–Spencer class does not vanish. We prove that this is so provided the twisting line bundle is strictly positive along fibers, but not in general.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/402ac8ade2b1c3270cf24fc08f9a272589c5fdbd

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry (2013, methodology): “The results in [6] and [8] deal with more general line bundles L overX and also more general fibrations thanX × Ω, see section 3.”
The openness conjecture and complex Brunn-Minkowski inequalities (2014, background): “To understand the meaning of this statement it is not absolut ely necessary to resort to the technical definition of curvature and its extension to bundl es of infinite rank - although there are explicit formulas for the curvature that are sometimes o f interest, see [5].”
Positivity and vanishing theorems for ample vector bundles (2010, background): “∗ ⊗ (detE)r are Nakano-positive if r > 1([12]). Recently, Berndtsson proved in [2] that SkE⊗detE is Nakano-positive as soon as E is ample. For more related results, we refer the reader to [2], [3] [4], [27], [28] and [33] and references therein. 2 Let hFS be the Fubini-Study metric on TPn and Skh”
K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics (2012, background): “The previous proposition can be seen as a partial generalization to singular fibrations of a result in [9] (see Theorem cor1.”