Singular Nakano positivity of direct image sheaves of adjoint bundles

Abstract

In this paper, we consider a proper Kähler fibration $f:X\to Y$ and a singular Hermitian line bundle $(L,h)$ on $X$ with semi-positive curvature. We prove that the direct image sheaf $f_{\ast }({\mathcal{O}}_X(K_{X/Y}+L)\otimes \mathcal{I}(h))$, 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.

v2: 24 pages; added Section 4 to discuss the direct images of puri-canonical bundles; included an Appendix to summarize approximations of singular Hermitian metrics