Donaldson invariants of symplectic manifolds

TL;DR

This paper proves that certain symplectic 4-manifolds have nonzero Donaldson invariants and characterizes their basic classes, especially in the context of Lefschetz fibrations, advancing understanding of their topology.

Contribution

It establishes nonvanishing Donaldson invariants for symplectic 4-manifolds with specific Betti numbers and characterizes basic classes in Lefschetz fibrations, providing new insights into their structure.

Findings

Symplectic 4-manifolds with $b_1=0$ and $b^+>1$ have nonvanishing Donaldson invariants.

The canonical class is always a basic class in these manifolds.

Basic classes of Lefschetz fibrations over the sphere are characterized when they evaluate maximally on a generic fiber.

Abstract

We prove that symplectic 4-manifolds with $b_1 = 0$ and $b^+ > 1$ have nonvanishing Donaldson invariants, and that the canonical class is always a basic class. We also characterize in many situations the basic classes of a Lefschetz fibration over the sphere which evaluate maximally on a generic fiber.