Reduced classes and curve counting on surfaces II: calculations

Details Reduced classes and curve counting on surfaces II: calculations Reduced classes and curve counting on surfaces II: calculations

2011-12-13

arxiv.org/abs/1112.3070v1

Mathematics

Algebraic Geometry

18 pages

Scientific paper

We calculate the "surface part" of the reduced residue stable pair theory on the canonical bundle $K_S$ of a projective surface $S$. For fixed curve class $\beta\in H^2(S)$ the results are entirely topological, depending on $\beta^2, \beta.c_1(S), c_1(S)^2, c_2(S), b_1(S)$ and invariants of the ring structure on $H^*(S)$ such as the Pfaffian of $\beta$ considered as an element of $\Lambda^2 H^1(S)^*$. We also give conditions under which this calculates the full 3-fold reduced residue theory of $K_S$. 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 \cite{KT1} to express the reduced virtual cycle in terms of Euler classes of bundles over a natural smooth ambient space.

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