Dual Nakano positivity and singular Nakano positivity of direct image sheaves

TL;DR

The paper studies positivity properties of direct image sheaves $f_*(K_{X/Y}\otimes L)$ and its multiplier-ideal twists under a projective morphism $f:X\to Y$, for both smooth and singular Hermitian metrics on $L$. It extends Berndtsson’s L^2-methods to show dual Nakano semi-positivity of the smooth canonical metric on the direct image in the zero-variation (no deformation) setting, and develops a theory for singular metrics yielding locally Nakano semi-positivity in a precise $L^2$-type sense, together with coherence results for extended $L^2$-subsheaves. The work also analyzes projectivized bundles and special cases (e.g., $X$ a product or a projectivized bundle) where dual Nakano positivity can be obtained under vanishing Kodaira-Spencer data. Moreover, it clarifies the relation between the minimal extension property and Nakano positivity, supplying counterexamples to emphasize limits, and provides a framework to define and study a canonical singular metric on direct image sheaves with its $L^2$-positivity properties. These results have implications for the study of deformations, multiplier ideals, and the construction of positively curved direct images in complex geometry.

Abstract

Let $f:X\to Y$ be a surjective projective map and $L$ be a holomorphic line bundle on $X$ equipped with a (singular) semi-positive Hermitian metric $h$. In this article, by studying the canonical metric on the direct image sheaf of the twisted relative canonical bundles $K_{X/Y}\otimes L\otimes\mathscr{I}(h)$, we obtain that this metric has dual Nakano semi-positivity when $h$ is smooth and there is no deformation by $f$ and that this metric has locally Nakano semi-positivity in the singular sense when $h$ is singular.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Theorem 2.5
• Lemma 2.6