Rational maps and $\mathrm{K3}$ surfaces

TL;DR

This paper proves that generic complex projective K3 surfaces cannot be dominated by non-isomorphic rational maps from surfaces with trivial canonical class, highlighting a rigidity property of K3 surfaces.

Contribution

It establishes a new restriction on dominant rational maps to K3 surfaces from surfaces with trivial canonical class, showing such maps are essentially isomorphisms.

Findings

No dominant rational map from a surface with trivial canonical class to a generic K3 surface unless it is an isomorphism.

Supports the rigidity of K3 surfaces in the context of rational maps.

Advances understanding of the structure of rational maps involving K3 surfaces.

Abstract

We prove that generic complex projective $\mathrm{K3}$ surface $S$ does not admit a dominant rational map $A\, -\!\to S$, which is not an isomorphism, from a surface $A$ with trivial canonical class.