Instanton counting and wall-crossing in Donaldson invariants
Donaldson invariants of 4-manifolds were originally
introduced via the moduli spaces of anti-self-dual connections, but
can be approached also by moduli spaces of stable bundles when the
underlying manifolds are complex projective surfaces. Based on this
approach, we compute the wall-crossing terms of Donaldson invariants,
describing what happens when the ample line bundle is changed. This is
an old subject, but a recent advance, the `instanton counting',
enables us to compute them in the torus equivariant setting. Then the
terms are naturally expressed in terms of the so-called Seiberg-Witten
curves. Finally, we express the blow-up formula of Donaldson
invariants, as a wall-crossing formula via moduli spaces of perverse
coherent sheaves on the blow-up.
Talks are based on series of joint works with Kota Yoshioka, Lothar
G\"ottsche, and Takuro Mochizuki.