On the Rigidity of Analytic Mappings in Complex Analysis and Geometry

TL;DR

This paper proves various rigidity results for holomorphic and plurisubharmonic mappings in complex geometry, showing under certain conditions they are globally determined or biholomorphic.

Contribution

It establishes new rigidity theorems for holomorphic maps, fiber-wise maps, and group actions, extending classical results in complex analysis and geometry.

Findings

Gradient of U(1)-invariant plurisubharmonic functions has finite fibers.

Fiber-wise holomorphic maps with degree 1 are biholomorphisms under certain conditions.

Holomorphic Lie group actions with large orbits restrict the critical locus of invariant functions.

Abstract

We establish rigidity results for holomorphic mappings and plurisubharmonic functions in complex geometry. First, under mild conditions, we show that the gradient of a $\operatorname{U}(1)$-invariant strictly plurisubharmonic function in $\mathbb{C}^2$ possesses finite fibers and induces a analytic mapping of topological degree $1$ on the symplectic quotient. Second, we prove that continuous fiber-wise holomorphic maps on proper fibrations elevate to global holomorphic maps when anchored by mutually disjoint sections, yielding rigidity for homomorphisms between elliptic fibrations and Abelian schemes. Third, we demonstrate that a fiber-wise holomorphic map of mapping degree $1$ from a fibered compact Kobayashi hyperbolic manifold to a projective variety is a biholomorphism, provided it is injective on a very ample hypersurface. Finally, we prove that a holomorphic Lie group action with sufficiently large orbits confines the critical locus of a proper invariant strictly plurisubharmonic function to the fixed-point set, guaranteeing a unique global minimum and yielding a sharp differential topological obstruction on the orbit dimensions of compact Lie group actions.