All Questions
33 questions
1
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70
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Degree axiom for P1 or P2
I am getting stuck on equation (7.33) on p. 192 of Cox-Katz's Mirror Symmetry and Algebraic Geometry. This concerns the degree of a cohomology class used as input for a Gromov-Witten invariant.
Let $X$...
- 511
1
vote
1
answer
237
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Generalization of Gromov-Witten theory counting surfaces, 3-folds, etc
I don't work on the Gromov-Witten theory, but I find that I need to study the following problem, which seems to be similar to the Gromov-Witten theory:
Let $X$ be a variety and $\alpha_{1}, \cdots, \...
- 305
2
votes
0
answers
158
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Divisibility of Gromov-Witten invariants
We know if a smooth complex projective variety $X$ is Fano, when the insertions of Gromov-Witten invariants are integral cohomology classes $\alpha_i \in H^*(X; {\mathbb Z})$, in genus zero the (...
- 803
11
votes
2
answers
1k
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Simple examples of Gromov-Witten invariants not being enumerative
I understand why Gromov-Witten invariants in general are not enumerative, so it's not necessary to explain this. However to test something I'm working on, I'm looking for examples of concrete ...
- 111
3
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0
answers
159
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Gromov-Witten invariants of cocharacter closures in toric varieties
$\require{AMScd}$
Let $X$ be a toric projective variety with dense algebraic torus $\iota:(\mathbb{C}^\times)^n \to X$, and let $u:\mathbb{C}^\times \to X$ be a cocharacter, by which I mean a map ...
- 1,231
3
votes
1
answer
458
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Are Gromov-Witten invariants birational invariants?
Let $X$ and $Y$ be two smooth complex projective varieties. Then they are also symplectic manifolds. We know Gromov-Witten (GW) invariants are symplectic invariants. That means if there exists a a ...
- 1,159
2
votes
1
answer
618
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Reference request for Gromov-Witten Invariants of non compact manifolds
The title of my question essentially explains what I am looking for, but let elaborate a bit, to put it in a more specific context.
There are quite a few papers, where the authors compute Gromov-...
- 3,245
2
votes
0
answers
206
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Are rational varieties symplectically rationally connected?
Was it proven already that smooth rational complex projecitve varieties are symplectically rationally connected? I.e. some GW invariant with two point insertions is non zero. What about smooth toric ...
- 14.3k
7
votes
0
answers
475
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Blow-up/Blow-down correspondence via Hodge Mirror Symmetry?
Let $X$ be a projective variety. Let $S \subset X$ be the nonsingular complete intersection of $k$ nonsingular divisors of $X$ of codimension $2k>2$. Denote $\tilde{X}$ the blow up of $X$ along $S$,...
- 1,981
4
votes
0
answers
354
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Higher genus Cohen-Jones-Segal's conjecture?
Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism
$$(ev,\Phi):\...
- 1,981
5
votes
0
answers
433
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What is the fundamental group of Kontsevich's space of stable maps?
... at least in the case where the target is a rationally connected variety.
This question is a follow-up to question
Constructing embedded families of curves with general moduli
and Jason Starr's ...
- 1,981
3
votes
0
answers
185
views
Abstract VFC vs. what people actually use for Quintic 3-fold
Moduli space of genus $0$ degree $d$ maps in a quintic Calabi-Yau threefold $X$, written as $\overline{\mathcal{M}}_{0,0}(X,[d])$, can be embedded in the corresponding moduli space of $\mathbb{P}^4$, ...
6
votes
1
answer
288
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Deformation long exact sequence of GW theory in the analytical setting
Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume $C=(\...
0
votes
0
answers
231
views
Is the complex structure on a del-Pezzo surface a regular complex structure?
Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost
complex structure $J$ is said to be $\textit{...
- 3,245
0
votes
1
answer
215
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When is the normal neighbourhood of the boundary of the moduli space of cuvres parametrized by exactly one branch?
Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z}) $
a fixed homology class that is $\textit{decomposable}$. Let
$$ \overline{\mathcal{M}}_{0,n}(X, \beta) $$
denote the stable ...
- 3,245
1
vote
0
answers
266
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How does one define Moduli spaces in Symplectic Geometry and naively interpret higher genus GW Invariants?
This is a very basic question about the definition of Moduli space of maps.
My reason for asking this question is because I haven't actually seen this
definition explicitly given anywhere, and hence ...
- 3,245
7
votes
1
answer
706
views
Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?
Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to
$8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...
- 3,245
4
votes
1
answer
305
views
What are the indecomposable classes on a del-Pezzo surface?
Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...
- 3,245
3
votes
0
answers
290
views
Do J-holomorphic curves "very nearly" fail to be an immersion near the bubbling points?
Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, $u_t$...
- 3,245
1
vote
1
answer
412
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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 ...
20
votes
3
answers
3k
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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 ...
- 241
3
votes
1
answer
1k
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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 ...
- 3,245
13
votes
3
answers
2k
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 ...
- 1,398
13
votes
1
answer
1k
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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 ...
- 2,927
16
votes
2
answers
2k
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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 "...
- 21k
2
votes
2
answers
874
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.
- 1,499
7
votes
2
answers
3k
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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 H_2(X;\mathbb{Z})}...
- 21k
15
votes
1
answer
3k
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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 ...
- 21k
20
votes
1
answer
4k
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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 ...
- 21k
3
votes
1
answer
249
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?
4
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2
answers
757
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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 ...
- 21k
14
votes
4
answers
2k
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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 ...
- 21k
26
votes
2
answers
2k
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Manifolds distinguished by Gromov-Witten invariants?
What is a simplest example of a manifold $M^{2n}$ that admits two symplectic structures with isotopic almost complex structures, and such that Gromov-Witten invariants of these symplectic structures ...
- 28.9k