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DonaldsonThomas Invariants of 2dimensional sheaves inside threefolds and modular forms
, 1309
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GromovWitten/Pairs descendent correspondence for toric 3folds
, 2013
"... We construct a fully equivariant correspondence between GromovWitten and stable pairs descendent theories for toric 3folds X. Our method uses geometric constraints on descendents, An surfaces, and the topological vertex. The rationality of the stable pairs descendent theory plays a crucial role in ..."
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We construct a fully equivariant correspondence between GromovWitten and stable pairs descendent theories for toric 3folds X. Our method uses geometric constraints on descendents, An surfaces, and the topological vertex. The rationality of the stable pairs descendent theory plays a crucial role in the definition of the correspondence. We prove our correspondence has a nonequivariant limit. As a result of the construction, we prove an explicit nonequivariant stationary descendent correspondence for X (conjectured by MNOP). Using descendent methods, we establish the relative GW/Pairs correspondence for X/D in several basic new log CalabiYau geometries. Among the consequences is a rationality constraint for nonequivariant descendent GromovWitten series for P³.
The GopakumarVafa formula for symplectic manifolds
"... The GopakumarVafa conjecture predicts that the GromovWitten invariants of a CalabiYau 3fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic GromovWitten theory, we prove that the GopakumarVafa formula holds for any symplectic Cala ..."
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The GopakumarVafa conjecture predicts that the GromovWitten invariants of a CalabiYau 3fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic GromovWitten theory, we prove that the GopakumarVafa formula holds for any symplectic CalabiYau 6manifold, and hence for CalabiYau 3folds. The results extend to all symplectic 6manifolds and to the genus zero GW invariants of semipositive manifolds.
ON THE MOTIVIC STABLE PAIRS INVARIANTS OF K3 SURFACES
"... Abstract. For a K3 surface S and a class β ∈ Pic(S), we study motivic invariants of stable pairs moduli spaces associated to 3fold thickenings of S. We conjecture suitable deformation and divisibility invariances for the Betti realization. Our conjectures, together with earlier calculations of Kawa ..."
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Abstract. For a K3 surface S and a class β ∈ Pic(S), we study motivic invariants of stable pairs moduli spaces associated to 3fold thickenings of S. We conjecture suitable deformation and divisibility invariances for the Betti realization. Our conjectures, together with earlier calculations of KawaiYoshioka, imply a full determination of the theory in terms of the Hodge numbers of the Hilbert schemes of points of S. The work may be viewed as the third in asequenceofformulasstartingwith YauZaslowand KatzKlemmVafa (each recoveringthe former). Numerical data suggest the motivic invariants are linked to the Mathieu M24 moonshine phenomena. The KKV formula and the Pairs/NoetherLefschetz correspondencetogetherdeterminetheBPScountsofK3fiberedCalabiYau 3folds in fiber classes in terms of modular forms. We propose a framework for a refined P/NL correspondence for the motivic invariants of K3fibered CY 3folds. For the STU model, a complete
CURVE COUNTING ON ABELIAN SURFACES AND THREEFOLDS
"... Abstract.We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the welldeveloped study of the reduced GromovWitten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The ge ..."
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Abstract.We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the welldeveloped study of the reduced GromovWitten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasimodular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by GöttscheShende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the generating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, Göttsche, and LangeSernesi. Further evidence
A̸=0
"... Abstract. The GopakumarVafa conjecture predicts that the GromovWitten invariants of a CalabiYau 3fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic GromovWitten theory, we prove that the GopakumarVafa formula holds for any sympl ..."
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Abstract. The GopakumarVafa conjecture predicts that the GromovWitten invariants of a CalabiYau 3fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic GromovWitten theory, we prove that the GopakumarVafa formula holds for any symplectic CalabiYau 6manifold, and hence for CalabiYau 3folds. The results extend to all symplectic 6manifolds and to the genus zero GW invariants of semipositive manifolds. The GopakumarVafaconjecture [GV] predicts that the GromovWitten invariants GWA,g of a CalabiYau 3fold can be expressed in terms of some other invariants nA,h, called BPS numbers, by a transform between their generating functions: GWA,gt 2g−2 q A = ∑
NOTES ON THE PROOF OF THE KKV CONJECTURE
"... Abstract. The KatzKlemmVafa conjecture expresses the GromovWitten theory of K3 surfaces (and K3fibred 3folds in fibre classes) in terms of modular forms. Its recent proof gives the first nontoric geometry in dimension greater than 1 where GromovWitten theory is exactly solved in all genera. W ..."
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Abstract. The KatzKlemmVafa conjecture expresses the GromovWitten theory of K3 surfaces (and K3fibred 3folds in fibre classes) in terms of modular forms. Its recent proof gives the first nontoric geometry in dimension greater than 1 where GromovWitten theory is exactly solved in all genera. We survey the various steps in the proof. The MNOP correspondence and a new Pairs/NoetherLefschetz correspondence for K3fibred 3folds transform the GromovWitten problem into a calculation of the full stable pairs theory of a local K3fibred 3fold. The stable pairs calculation is then carried out via degeneration, localisation, vanishing results, and new multiple cover formulae. Contents