Curvature strict positivity of direct image bundles associated to pseudoconvex families of domains

TL;DR

This paper proves that the direct image bundle associated with certain pseudoconvex families of domains exhibits strict Nakano positivity, providing new geometric insights and applications in complex and convex analysis.

Contribution

It establishes the strict Nakano positivity of direct image bundles for strictly pseudoconvex circular or Reinhardt domains, extending known results and offering new geometric interpretations.

Findings

Direct image bundles are strictly Nakano positive for certain pseudoconvex families.

The result applies even without strictly plurisubharmonic weight functions.

Implications include positivity properties of symmetric powers of ample vector bundles.

Abstract

We consider the curvature strict positivity of the direct image bundle associated to a pseudoconvex family of bounded domains. The main result is that the curvature of the direct image bundle associated to a strictly pseudoconvex family of bounded circular domains or Reinhardut domains are strictly positive in the sense of Nakano, even if the weight functions are not strictly plurisubharmonic. This result gives a new geometric insight about the property of strict pseudoconvexity, and has some applications in complex analysis and convex analysis. We investigate that the main result implies a remarkable result of Berndtsson which states that, for an ample vector bundle $E$ over a compact complex manifold $X$ and any $k\geq 0$, the bundle $S^kE\otimes\det E$ admits a Hermitian metric whose curvature is strictly positive in the sense of Nakano, where $S^kE$ is the $k$-th symmetric product of $E$. The two main ingredients in the argument of the main theorems are Berndtsson's estimate of the lower bound of curvature of direct image bundles and Deng-Ning-Wang-Zhou's characterization of the curvature Nakano positivity of Hermitian vector bundles in terms of $L^2$-estimate of $\bar\partial$.