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32
MIRROR SYMMETRY FOR LOG CALABIYAU SURFACES I
, 2011
"... We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anticanonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is con ..."
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Cited by 35 (7 self)
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We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anticanonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga’s 1981 cusp conjecture.
GromovWitten/Pairs correspondence for the quintic 3fold
, 2012
"... We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of ..."
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Cited by 18 (6 self)
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We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role. The GW/P correspondence for CalabiYau complete intersections provides a structure result for the GromovWitten invariants in a fixed curve class. After change of variables, the GromovWitten series is a rational function in the variable −q = e iu invariant under q ↔ q −1.
FUNCTORIAL TROPICALIZATION OF LOGARITHMIC SCHEMES: THE CASE OF Constant Coefficients
, 2014
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Comparison theorems for GromovWitten invariants of smooth pairs and of degenerations
, 2013
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Logarithmic stable maps
"... Abstract. We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semistable variety of form xy = 0. We study the compactification of the moduli spaces of such maps and provide a perfect obstruction theory, applicable to the moduli spaces of (un)ramif ..."
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Cited by 5 (0 self)
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Abstract. We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semistable variety of form xy = 0. We study the compactification of the moduli spaces of such maps and provide a perfect obstruction theory, applicable to the moduli spaces of (un)ramified stable maps and stable relative maps. As an application, we obtain a modular desingularization of the main component of Kontsevich’s moduli space of elliptic stable maps to a projective space. 1.
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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Cited by 5 (2 self)
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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.
INVARIANCE OF QUANTUM RINGS UNDER ORDINARY FLOPS
"... ABSTRACT. For ordinary flops over a smooth base, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. If the flop is of splitting type, the big quantum cohomology ring is also shown to b ..."
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Cited by 4 (3 self)
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ABSTRACT. For ordinary flops over a smooth base, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. If the flop is of splitting type, the big quantum cohomology ring is also shown to be invariant after an analytic continuation in the Kähler moduli space. Viewed from the context of the Kequivalence (crepant transformation) conjecture, there are two new features of our results. First, there is no semipositivity assumption on the varieties. Second, the local structure of the exceptional loci can not be deformed to any explicit (e.g. toric) geometry and the analytic continuation is nontrivial. This excludes the possibility of an ad hoc comparison by explicit computation of both sides. To achieve that, we have to clear a few technical hurdles. One technical breakthrough is a quantum Leray–Hirsch theorem for the local models (a certain toric bundle) which extends the quantum D modules of Dubrovin connection on the base by a Picard–Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray– Hirsch theorem will be discussed in subsequent papers.