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arXiv · 2407.11412

Singular Nakano positivity of direct image sheaves of adjoint bundles

Takahiro Inayama·Shin-ichi Matsumura·Yuta Watanabe
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Abstract

In this paper, we consider a proper K\"ahler fibration $f \colon X \to Y$ and a singular Hermitian line bundle $(L, h)$ on $X$ with semi-positive curvature. We prove that the direct image sheaf $f_{*}(\mathcal{O}_{X}(K_{X/Y}+L) \otimes \mathcal{I}(h))$, equipped with the Narasimhan-Simha metric, is singular Nakano semi-positive in the sense that the $\overline{\partial}$-equation can be solved with optimal $L^{2}$-estimate. Our proof does not rely on the theory of Griffiths positivity for the direct image sheaf.

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Takahiro Inayama, Shin-ichi Matsumura, Yuta Watanabe. 2024-07-16. Singular Nakano positivity of direct image sheaves of adjoint bundles. https://arxiv.org/abs/2407.11412

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