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Rational maps and $\mathrm{K3}$ surfaces

Ilya Karzhemanov, Grisha Konovalov

TL;DR

The paper proves a rigidity phenomenon for dominant rational maps from surfaces with trivial canonical class to complex projective K3 surfaces: if such a map $\phi$ is non-isomorphic and has degree at least two, then the target K3 surface $S$ must have Picard number $\rho(S) > 1$. The argument employs a concise birational setup with a resolution of indeterminacy and Stein factorization, together with differential-form considerations and ramification data, to constrain the exceptional loci and the possible images of curves. It connects with lattice-theoretic perspectives on K3–Abelian maps and corroborates special cases via known constructions (Kummer, Shioda) and symplectic automorphisms, while also aligning with Voisin's IVHS viewpoint for certain quartics. The results yield a clearer picture of the rigidity of generic K3 geometry under dominant rational maps and suggest further questions for nonprojective compact complex 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.

Rational maps and $\mathrm{K3}$ surfaces

TL;DR

The paper proves a rigidity phenomenon for dominant rational maps from surfaces with trivial canonical class to complex projective K3 surfaces: if such a map is non-isomorphic and has degree at least two, then the target K3 surface must have Picard number . The argument employs a concise birational setup with a resolution of indeterminacy and Stein factorization, together with differential-form considerations and ramification data, to constrain the exceptional loci and the possible images of curves. It connects with lattice-theoretic perspectives on K3–Abelian maps and corroborates special cases via known constructions (Kummer, Shioda) and symplectic automorphisms, while also aligning with Voisin's IVHS viewpoint for certain quartics. The results yield a clearer picture of the rigidity of generic K3 geometry under dominant rational maps and suggest further questions for nonprojective compact complex surfaces.

Abstract

We prove that generic complex projective surface does not admit a dominant rational map , which is not an isomorphism, from a surface with trivial canonical class.

Paper Structure

This paper contains 2 sections, 7 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Suppose there exists a dominant rational map $\phi \colon A \dasharrow S$ from a surface $A$ with trivial canonical class to a $\mathrm{K3}$ surface $S$. Suppose also that $\operatorname{deg} \phi \ge 2$. Then the Picard number $\operatorname{\rho}(S)$ is strictly greater than $1$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Example 1.2: Kummer surfaces
  • Example 1.3: Shioda surfaces
  • Example 1.4: Symplectic automophisms
  • Example 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 9 more