Finite group actions on symplectic Calabi-Yau $4$-manifolds with $b_1>0$

TL;DR

This paper studies finite cyclic group actions on symplectic Calabi-Yau 4-manifolds with positive first Betti number, analyzing fixed-point sets and topological structures, and introduces methods for classifying symplectic surface embeddings.

Contribution

It provides a complete fixed-point set classification for cyclic actions and explores the topology of these manifolds, including conditions for them to be torus bundles over tori.

Findings

Fixed-point set structures are fully determined for cyclic actions.

Certain involutions fix 2-dimensional surfaces, leading to torus bundle structures.

Maximal number of disjoint symplectic (-2)-spheres in rational 4-manifolds is established.

Abstract

This is the first of a series of papers devoted to the topology of symplectic Calabi-Yau $4$-manifolds endowed with certain symplectic finite group actions. We completely determine the fixed-point set structure of a finite cyclic action on a symplectic Calabi-Yau $4$-manifold with $b_1>0$. As an outcome of this fixed-point set analysis, the $4$-manifold is shown to be a $T^2$-bundle over $T^2$ in some circumstances, e.g., in the case where the group action is an involution which fixes a $2$-dimensional surface in the $4$-manifold. Our project on symplectic Calabi-Yau $4$-manifolds is based on an analysis of the existence and classification of disjoint embeddings of certain configurations of symplectic surfaces in a rational $4$-manifold. This paper lays the ground work for such an analysis at the homological level. Some other result which is of independent interest, concerning the maximal number of disjointly embedded symplectic $(-2)$-spheres in a rational $4$-manifold, is also obtained.