Modular forms and Donaldson invariants for 4-manifolds with $b_+=1$

TL;DR

This paper investigates the Donaldson invariants of 4-manifolds with $b_+=1$, deriving a generating function for wall-crossing terms using modular forms, and explicitly computes invariants for the projective plane.

Contribution

It provides a modular form-based generating function for wall-crossing terms and computes all Donaldson invariants of the projective plane, assuming a conjecture now proven.

Findings

Derived a generating function for wall-crossing terms using modular forms.

Determined all Donaldson invariants of the projective plane.

Established recursive relations via blowup formulas.

Abstract

We study the Donaldson invariants of simply connected $4$-manifolds with $b_+=1$, and in particular the change of the invariants under wall-crossing. We assume the conjecture of Kotschick and Morgan about the shape of the wall-crossing terms (which Oszva\'th and Morgan are now able to prove), and are determine a generating function for the wall-crossing terms in terms of modular forms. As an application we determine all the Donaldson invariants of the projective plane in terms of modular forms. The main tool are the blowup formulas, which are used to obtain recursive relations.