Positivity of twisted relative pluricanonical bundles and their direct images

Abstract

Our main goal in this article is to establish a metric version of the positivity properties of twisted relative pluricanonical bundles and their direct images. Some of the important technical points of our proof are an L 2 / m L^{2/m} -extension theorem of Ohsawa-Takegoshi type which is derived from the original result by a simple fixed point method and the notion of “singular Hermitian metric” on vector bundles, together with an appropriate definition of positivity of the associated curvature. Part of this article is based on the joint work of the first-named author with Bo Berndtsson, and it can be seen as an expanded and updated version of it.

Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

“…As the complex structure deforms, for pseudoconvex domains the variation of the Bergman kernels was initially studied by Maitani & Yamaguchi [20] and generalized to higher dimensional cases by Berndtsson [1]. For general results on arbitrary dimensional Stein manifolds and complex projective algebraic manifolds, see Berndtsson [2], Tsuji [26], Berndtsson & Pǎun [6] and Pǎun & Takayama [25], etc. These important results indicating semi-positivity properties of the relative canonical bundles recently turn out to have close relations with the Ohsawa-Takegoshi L 2 extension theorem [8,21], space of Kähler metrics [3], etc.…”

Complex Brunn–minkowski Theory and Positivity of Vector Bundles

“…Since negativity (and positivity) of the curvature is defined in terms of plurisubharmonic functions, and since plurisubharmonic functions extend over varieties of codimension at least 2, it turns out that this is a useful definition. Given all this, we have the following theorem of Paun-Takayama, [54], and Hacon-Popa-Schell, [39], which seems to be the most general theorem on (metric) positivity of direct images. Theorem 6.1.…”

On semipositivity theorems

“…Therefore, we can easily see that the Hodge metric of F b in Theorem 1.1 is a semipositively curved singular hermitian metric in the sense of Pȃun-Takayama (see [PȃT,Definition 2.3.1] and [HPS,Lemma 18.2]). Moreover, in Corollary 1.6, the induced metric on A is a seminegatively curved singular hermitian metric in the sense of Pȃun-Takayama (see [PȃT,Definition 2.3.1] and [HPS,Lemma 18.2]). For the details of singular hermitian metrics on vector bundles and some related topics, see [PȃT] (see also [HPS] and [B1]).…”