Reduced classes and curve counting on surfaces II: calculations

Abstract

We calculate the stable pair theory of a projective surface $S$. For fixed curve class $\beta \in H^2(S)$ the results are entirely topological, depending on ${\beta }^2$, $\beta .c_1(S)$, $c_1(S{)}^2$, $c_2(S)$, $b_1(S)$ \emph{and} invariants of the ring structure on $H^{\ast }(S)$ such as the Pfaffian of $\beta$ considered as an element of ${\Lambda }^2{H}^1(S{)}^{\ast }$. 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 $K_S$. 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.

19 pages. Minor corrections

References in corpus (1)

• Reduced classes and curve counting on surfaces I: theory

Cited by in corpus (16)

• Reduced classes and curve counting on surfaces I: theory
• The Katz-Klemm-Vafa conjecture for K3 surfaces
• Semiregularity and obstructions of complete intersections
• Degeneracy loci, virtual cycles and nested Hilbert schemes I
• Stable pair invariants of local Calabi-Yau 4-folds
• Degeneracy loci, virtual cycles and nested Hilbert schemes II
• Stable pairs with descendents on local surfaces I: the vertical component
• Moduli spaces of stable pairs
• Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes
• Virtual ${\chi }_{-y}$-genera of Quot schemes on surfaces
• Sheaves on surfaces and virtual invariants
• Stable pair invariants of surfaces and Seiberg-Witten invariants
• Enumerative invariants and wall-crossing formulae in abelian categories
• Notes on the proof of the KKV conjecture
• Rank-one sheaves and stable pairs on surfaces
• The Kleiman-Piene Conjecture and node polynomials for plane curves in ${\mathbb{P}}^3$