Reduced classes and curve counting on surfaces II: calculations

الأصل · EN

We calculate the stable pair theory of a projective surface S. For fixed curve class β∈ H²(S) the results are entirely topological, depending on β², β.c₁(S), c₁(S)², c₂(S), b₁(S) and invariants of the ring structure on H*(S) such as the Pfaffian of β considered as an element of Λ² H¹(S)*. Amongst other things, this proves an extension of the Göttsche conjecture to non-ample linear systems. We also give conditions under which this calculates the full 3-fold reduced residue theory of Kₛ. This is related to the reduced residue Gromov-Witten theory of S via the MNOP conjecture. When the surface has no holomorphic 2-forms this can be expressed as saying that certain Gromov-Witten invariants of S are topological. Our method uses the results of KT1 to express the reduced virtual cycle in terms of Euler classes of bundles over a natural smooth ambient space.