A Degeneration Formula of GW-Invariants

By Jun Li. 2001

397 citations recorded by the source corpus.

This is the second part of the paper "A degeneration of stable morphisms and relative stable morphisms", (math.AG/0009097). In this paper, we constructed the relative Gromov-Witten invariants of a pair of a smooth variety and a smooth divisor. We then proved a degeneration formula of Gromov-Witten invariants, in cycle form.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8edb1fa11e09cdf178391a64a080a44490803a07

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In Mathematical Physics

Citations in the corpus

Double ramification cycles on the moduli spaces of curves (2016, background): “We refer the reader to [25, 26] for the foundations.”
Good degeneration of Quot-schemes and coherent systems (2011, methodology): “ O δ∈Λspl β Lδ∼= O Cβ and Y δ∈Λspl β sδ= π∗t; (2) the morphism Φδ factors through s−1 δ (0) ⊂ Cβ and eﬀects an isomorphism C†,δ 0 ∼= s−1 δ (0). The proof of this decomposition is essentially given in [Li02]. Note that this Propositionstates that Cβ 0 ⊂ C βis a complete intersection substack, and the”
Tropical Hurwitz numbers (2008, background): “The theory of relative stable maps [9, 10] extends this scenario to more general Hurwitz numbers, with arbitrary ramification profiles over the branch points.”
K3 surfaces with Picard number one and infinitely many rational points (2005, background): “Crucial to this work are the following identities [Li 2002] taking place in the Chow ring with rational coefficients.”
A mathematical theory of the topological vertex (2004, methodology): “Because C E is T –equivariant, the limiting class of ŒC TR B TR 2H2rC4l E TR B TR 5It was shown in [20] the existence of a global vector bundle E can be replaced by that „ is dominated by a quasi-projective scheme.”
Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces (2014, methodology): “…one can cite methods based on Tropical geometry [Mik05], and on the degeneration of the target space X, such as Li’s degeneration formula [Li02, Li04] in the algebraic setting, symplectic sum formulas in the symplectic setting [IP04, LR01, TZ14], or more generally symplectic field…”
Affine manifolds, log structures, and mirror symmetry (2002, result): “Previous general work in this direction is due to Tian; Li and Ruan [21]; Ionel and Parker [15]; Gathmann [6] and Li [22],[23], who covers cases relative a smooth divisor and of normal crossing varieties with smooth singular locus.”
Virasoro constraints for target curves (2003, background): “The virtual class is defined by an (equivariant) perfect obstruction theory [5, 9].”
The logarithmic cotangent complex (2005, background): “First, we want to generalize the log smooth deformation theory due to K. Kato ([11], 3.14, see also [10]) in order to understand more complicated deformation theory problems in the logarithmic category (see section 5). Such deformation theory problems arise for example in the study of log Gromov–Wit”
Toric degenerations of toric varieties and tropical curves (2004, background): “also [FuOh], [KoSo], and the work on Gromov-Witten invariants for degenerations [LiARu], [IoPa], [ LiJ2 ], [Si]).”
Decomposition of degenerate Gromov–Witten invariants (2017, methodology): “ §5 employs these formulas in the discussion of a number of hopefully instructive examples. §5.1 contains the already announced discussion of our decomposition formula in the traditional situation of [Li]. In §5.2 we retrieve the classical number 12 of nodal plane sections of a cubic surface passing”
Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, background): “See [13, 15, 18, 30] for a technical discussion of relative theories.”
Curve counting via stable pairs in the derived category (2007, background): “28) OX[k] s → F on X[k], the k-step degeneration [36] along S.”
Logarithmic Gromov-Witten invariants (2011, methodology): “is a family of ordinary stable maps in X[n] satisfying certain conditions (predeformability and ﬁniteness of a certain notion of automorphism group), and has speciﬁed tangencies with D[n]. In §1.3 of [Li2], Li observes that X[n] and An carry canonical log structures. The log structure on An is the d”
Torus localization and wall crossing for cosection localized virtual cycles (2015, methodology): “ch as the Gromov-Witten invariant and the Donaldson-Thomas invariant. Quite a few methods for handling the virtual fundamental classes were discovered such as torus localization ([15]), degeneration ([25]), virtual pullback ([28]) and cosection localization ([18]). Often combining these methods turn”
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves (2003, methodology): “We use J. Li’s algebro-geometric definition of this moduli space, and his description of its obstruction theory [Li1, Li2 ].”
Logarithmic Stable Maps (2008, background): “The idea that log structures should be useful in (relative) GromovWitten theory has been around many years ([21, 32]).”