Reduced classes and curve counting on surfaces I: theory

Abstract

We develop a theory of \emph{reduced} Gromov-Witten and stable pair invariants of surfaces and their canonical bundles. We show that classical Severi degrees are special cases of these invariants. This proves a special case of the MNOP conjecture, and allows us to generalise the Göttsche conjecture to the non-ample case. In a sequel we prove this generalisation. We prove a remarkable property of the moduli space of stable pairs on a surface. It is the zero locus of a section of a bundle on a smooth compact ambient space, making calculation with the reduced virtual cycle possible.

54 pages. Typo fixed

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• The Katz-Klemm-Vafa conjecture for K3 surfaces
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• Gromov-Witten invariants of varieties with holomorphic 2-forms
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Cited by in corpus (36)

• Vafa-Witten invariants for projective surfaces I: stable case
• Holomorphic anomaly equations and the Igusa cusp form conjecture
• Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds
• Equivariant K-theory and refined Vafa-Witten invariants
• The Katz-Klemm-Vafa conjecture for K3 surfaces
• Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
• Semiregularity and obstructions of complete intersections
• Reduced classes and curve counting on surfaces II: calculations
• Gromov-Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations
• Stable pair invariants of local Calabi-Yau 4-folds
• Semiregularity as a consequence of Goodwillie's theorem
• Torus localization and wall crossing for cosection localized virtual cycles
• Degeneracy loci, virtual cycles and nested Hilbert schemes II
• Stable pairs with descendents on local surfaces I: the vertical component
• Curve counting and S-duality
• Semiorthogonal decompositions of stable pair moduli spaces via d-critical flips
• Moduli spaces of stable pairs
• Gopakumar-Vafa invariants via vanishing cycles
• A support theorem for Hilbert schemes of planar curves, II
• Notes on the proof of the KKV conjecture
• Gromov-Witten theory of $K3\times {\mathbb{P}}^1$ and quasi-Jacobi forms
• Stable pair invariants of surfaces and Seiberg-Witten invariants
• Sheaves on surfaces and virtual invariants
• Enumerative invariants and wall-crossing formulae in abelian categories
• Log BPS numbers of log Calabi-Yau surfaces
• Quasimaps to moduli spaces of sheaves on a $K3$ surface
• Rank-one sheaves and stable pairs on surfaces
• The chi-y genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces
• Stable maps and singular curves on K3 surfaces
• Gromov-Witten classes of K3 surfaces
• Higher rank flag sheaves on Surfaces and Vafa-Witten invariants
• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
• Semiregularity via derived deformation theory
• Equivariant Segre and Verlinde invariants for Quot schemes
• Stable pairs on nodal K3 fibrations
• Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten theories