6
votes
0answers
163 views

Question on Ionel and Parker's paper: Relative Gromov Witten Invariants

In defining Gromov-Witten invariants using symplectic geometry, most of the trouble is to achieve transversality for moduli spaces of pseudo-holomorphic curves which are multiple covers of simple ...
1
vote
1answer
170 views

Enumerativity of Gromov-Witten invariants of orbifolds

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf. Is there some sense, or some ...
16
votes
3answers
1k views

Why are Gromov-Witten invariants of K3 surfaces trivial?

Why is GW invariants of K3 surfaces are trivial? My naive guess is that GW invariants are deformation invariant and you can always deform your K3 surface to non-projective one, which has no subcomplex ...
3
votes
1answer
497 views

Is P^2 important in Kontsevich's recursion formula?

There is a famous recursion formula by Kontsevich to find the number of genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points. My question is the following: Let $S$ be a complex ...
9
votes
3answers
1k views

Computation of Gromov-Witten invariants for symplectic manifolds

According to references, Gromov-Witten invariants were first defined for symplectic manifolds and later for projective varieties algebraically, and they coincide on the overlap. Because I thought ...
7
votes
1answer
599 views

Gromov-Witten invariants counting curves passing through two points

Let us say that a closed symplectic manifold $X$ is $GW_g$-connected if there is a nonvanishing Gromov-Witten invariant of the form $GW_{g,n}^{X,A}(\beta,point, point,\alpha_3,\ldots,\alpha_n)$ --in ...
4
votes
1answer
752 views

Relative Gromov-Witten Invariants

A central issue in defining relative GW-invariants on a symplectic manifold is the possibility that a sequence of relative pseudoholomorphic curves can degenerate in such a manner, that components lie ...
11
votes
2answers
1k views

Deformation quantization and quantum cohomology (or Fukaya category) — are they related?

Good afternoon. Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of ...
2
votes
2answers
646 views

Why are people interested in defining GW invariant in algebraic geometry category

Originally it is in symplectic geometry. Is it just curosity or any other special reason? Thank you for clarifying.
3
votes
1answer
562 views

Do the virtual fundamental classes satisfy functorial properties?

In Gromov–Witten theory, if the symplectic virtual fundamental classes constructed by B.Siebert satisfy functorial properties, i.e., if $f\colon X\to Y$ is an appropriate map between symplectic ...
3
votes
1answer
749 views

Different definitions of Novikov ring?

Following, e.g., Wikipedia's definitions, the (small) quantum cohomology ring of $X$ is defined over a "Novikov ring" consisting of formal power series of the form $$ \sum_{\beta \in ...
3
votes
2answers
1k views

Are Fukaya categories Calabi-Yau categories?

Let X be a compact symplectic manifold. There is an idea, I think probably originally due to Kontsevich, that we should be able to get Gromov-Witten invariants of X out of the Fukaya category of X. ...
10
votes
1answer
2k views

Where does the Givental reconstruction formula come from?

In (for example) Semisimple Frobenius structures at higher genus (section 1.2) and Gromov-Witten invariants and quantization of quadratic Hamiltonians (section 6.8), Givental gives a conjectural ...
9
votes
1answer
2k views

Hochschild (co)homology of Fukaya categories and (quantum) (co)homology

There is a conjecture of Kontsevich which states that Hochschild (co)homology of the Fukaya category of a compact symplectic manifold $X$ is the (co)homology of the manifold. (See page 18 of ...
3
votes
1answer
209 views

GW invariants for varieties with negative first Chern class

Does there exist any theorem claiming that if a variety with negative first Chern class has no rational curves then every GW invariant is zero?
3
votes
2answers
533 views

Convergence of quantum cohomology

For which manifolds or varieties is quantum cohomology known to converge? Are there any manifolds for which quantum cohomology is known to not converge? I seem to have the impression that quantum ...
9
votes
4answers
1k views

Negative Gromov-Witten invariants

I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to ...
17
votes
2answers
1k views

Manifolds distinguished by Gromov-Witten invariants?

What is the simplest example of a manifold M^2n that admits two different symplectic structrues with isotopic almost complex structures, and such that Gromov Witten invariants of these symplectic ...