Symplectic gluing and family Gromov-Witten invariants

Lee, Jun-Ho; Parker, Thomas H.

doi:10.1090/fic/047/10

Cited by 3 publications

(2 citation statements)

References 16 publications

(23 reference statements)

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“…The J α -holomorphic map equation can also be used to define a set of "Family GW invariants" that are directly related to enumerative invariants. That context is explained in [17], [18] and [19].…”

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confidence: 99%

Family Gromov-Witten invariants for Kähler surfaces

Lee¹

2004

Duke Math. J.

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We prove a structure theorem for the Gromov-Witten invariants of compact Kähler surfaces with geometric genus p g > 0. Under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components, the theorem shows that the GW invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic Euler characteristic of the surface. We compute special cases of these universal functions.Much of the work on the Gromov-Witten invariants of Kähler surfaces has focused on rational and ruled surfaces, which have geometric genus p g = 0. This paper focuses on surfaces with p g > 0, a class that includes most elliptic surfaces and most surfaces of general type. In this context we prove a general "structure theorem" that shows (with one technical assumption) how the GW invariants are completely determined by the local geometry of a generic canonical divisor.The structure theorem is a consequence of a simple fact: the "Image Localization Lemma" of Section 3. Given a Kähler surface X and a canonical divisor D ∈ |K X |, this lemma shows that the complex structure J on X can be perturbed to a non-integrable almost complex structure J D with the property the image of all J D -holomorphic maps lies in the support of D. This immediately gives some striking vanishing theorems for the GW invariants of Kähler surfaces (see Section 3). More importantly, it implies that the Gromov-Witten invariant of X for genus g and n marked points is a sum GW g,n (X, A) = GW loc g,n (D k , A k ) over the connected components D k of D of "local invariants" that count the contribution of maps whose image lies in or (after perturbing to a generic moduli space) near D k . These local invariants have not been previously defined. The proof of their existence relies on using non-integrable structures and geometric analysis techniques. Our structure theorem characterizes the local invariants by expressing them in terms of usual GW invariants of certain standard surfaces. For this we make the mild assumption that one can deform the Kähler structure on X and choose a canonical divisor D so that all components of D are smooth. With this assumption, the restriction of K X to each component D k of multiplicity m k is the normal bundle N k to D k , and N m k +1 k = K D k , * partially supported by the N.S.F.

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Family Gromov-Witten invariants for Kähler surfaces

Lee¹

2004

Duke Math. J.

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“…have been defined, calculated and used in many places in the literature including [4,11,13,14,15,16,17,18,19,20,24]. Below, we explain the special cases we require.…”

Section: Family Gromov-witten Invariants Family Gromov-witten Invarimentioning

confidence: 99%

Pseudoholomorphic tori in the Kodaira–Thurston manifold

Evans

Kędra²

2015

Compositio Math.

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The Kodaira–Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov–Witten invariants which count pseudoholomorphic tori in the Kodaira–Thurston manifold. For a fixed symplectic form the Gromov–Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov–Witten computation for a non-Kähler manifold.

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A structure theorem for the Gromov-Witten invariants of Käler surfaces

Lee¹,

Parker²

2007

J. Differential Geom.

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Symplectic gluing and family Gromov-Witten invariants

Cited by 3 publications

References 16 publications

Family Gromov-Witten invariants for Kähler surfaces

Family Gromov-Witten invariants for Kähler surfaces

Pseudoholomorphic tori in the Kodaira–Thurston manifold

A structure theorem for the Gromov-Witten invariants of Käler surfaces

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