Seiberg-Witten invariants for 4-manifolds with b_+=0

Andrei Teleman (1) , Christian Okonek (1)

1 Institut für Mathematik [Zürich]

We extend Seiberg-Witten theory to 4-manifolds with b_+=0. The moduli space of irreducible monopoles depends in this case on a parameter varying in an infinite dimensional space which is divided into a countable family of chambers by a codimension-1 wall. The Seiberg-Witten invariant associated with a class of Spin^c-structures is a functorial map which assigns an integer to every chamber and satisfies a universal wall-crossing formula for transversal wall-crossing; it should be regarded as a distinguished element in the Z-torsor of functions satisfying the wall-crossing formula. The Seiberg-Witten invariant for a 3-manifold with b_1=0 is completely determined by the 4-dimensional invariant of the product with a circle. When the base manifold is a complex surface, a Kobayashi-Hitchin-type correspondence allows us to compute the invariants in the so-called complex geometric chambers by counting complex curves.

Mots clés: Seiberg-Witten invariants; Inoue surfaces; Hopf surfaces

Domaines: Géométrie différentielle [math.DG]; Topologie géométrique [math.GT]; Variables complexes [math.CV]

Andrei Teleman, Christian Okonek. Seiberg-Witten invariants for 4-manifolds with b_+=0. Complex Analysis and Algebraic Geometry. A Volume in Memory of Michael Schneider (Th. Peternell, F-O. Schreyer, editors), de Gruyter, pp.347-357, 2000. ⟨hal-00881762⟩