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Non-trivial smooth families of $K3$ surfaces

David Baraglia

Abstract

Let $X$ be a complex $K3$ surface, ${\rm Diff}(X)$ the group of diffeomorphisms of $X$ and ${\rm Diff}_0(X)$ the identity component. We prove that the fundamental group of ${\rm Diff}_0(X)$ contains a free abelian group of countably infinite rank as a direct summand. The summand is detected using families Seiberg--Witten invariants. The moduli space of Einstein metrics on $X$ is used as a key ingredient in the proof.

Non-trivial smooth families of $K3$ surfaces

Abstract

Let be a complex surface, the group of diffeomorphisms of and the identity component. We prove that the fundamental group of contains a free abelian group of countably infinite rank as a direct summand. The summand is detected using families Seiberg--Witten invariants. The moduli space of Einstein metrics on is used as a key ingredient in the proof.

Paper Structure

This paper contains 3 sections, 14 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. The families Seiberg--Witten invariant revisited
  3. The Einstein family

Key Result

Theorem 1.1

Let $X$ be a $K3$ surface. Then $\pi_1({\rm Diff}_0(X))$ contains a free abelian group of countably infinite rank as a direct summand.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 21 more