Questions tagged [gromov-witten-theory]
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163 questions
4
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1
answer
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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
186
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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 $\mathbb{P}^4$, ...
12
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1
answer
2k
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How to understand Givental's I- and J-functions?
I am learning about mirror symmetry and having trouble understanding Givental's I- and J-functions. For example the J-function for the quintic threefold $X$ is defined by the formula
$$
J:=e^{(t_0+...
- 121
2
votes
1
answer
202
views
What is the value of this hyperelliptic Hodge-type integral?
Consider the moduli space
$$
\overline{M}_{0,4}(B\mathbb{Z}/2)
$$
This has virtual (and real) dimension one. In a certain sense this moduli space paramaterizes "genus 1 hyperelliptic curves"; that is, ...
- 6,290
4
votes
0
answers
91
views
Topology of a convergent sequence of stable maps on a symplectic manifold
Let $(M,\omega)$ be a compact symplectic manifold. Let $J$ be a compatible almost complex structure. Let $g$ be the Riemannian metric corresponding to $\omega,J$.
Let $f_\nu\colon C_\nu\to M$ be a ...
- 21.8k
5
votes
1
answer
346
views
Looking for a reference (on GW invariants of quintic)
1) Apparently, physicist can calculate the GW invariants of quintic CY 3-fold up to genus 51.
I am looking for a reference that has a table of these number for some low degrees (say up to degree 5) ...
2
votes
0
answers
197
views
Gromov-Witten invariants for arithmetic surfaces counting sections passing through points
Suppose we are given an arithmetic surface, $X\to \text{Spec}\mathbb{Z}[1/N]$ smooth and quasi-projective, and a finite set of closed points all in different vertical fibers.
Can we count the number ...
- 845
5
votes
2
answers
647
views
Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ?
I wanted to know if there is something analogous to Kontsevich's recursion formula for
enumeration of genus zero curves in $\mathbb{C}\mathbb{P}^2$, for higher genus curves.
There is a
similar ...
- 3,245
2
votes
1
answer
235
views
Is there a formula for the number of rational cuspidal curves in surfaces other than P^2?
Let $M$ be a two dimensional compact complex manifold and $A \in H_2(M, \mathbb{Z})$
a fixed homology class. Define a rational curve in $M$ to be $\textit{1-cuspidal}$ if the singularities of the ...
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7
votes
2
answers
892
views
Intuition behind the age grading in quantum cohomology of orbifolds
Let $\mathscr{X}$ be a smooth DM-stack with projective coarse moduli space. I am interested in the orbifold cohomology ring $H^\mathrm{orb}(\mathscr{X})$, as defined by Chen-Ruan (for orbifolds) and ...
- 40.3k
1
vote
0
answers
110
views
Regular almost complex structures on symplectic toric manifolds
Under which assumptions the almost complex structure J defined on a symplectic toric manifold is Fredholm regular for every J-holomorphic sphere?
- 11
6
votes
1
answer
572
views
Obstruction sheaf is a vector bundle when the moduli space is non-singular?
I am working on some basic of Gromov-Witten theory and stuck in understanding obstruction bundle. Recall that a perfect obstruction theory on a scheme or stack $M$ due to Behrend and Fantechi is a ...
- 63
0
votes
1
answer
215
views
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
4
votes
2
answers
957
views
Explicit computation of Gromov-WItten invariants
After studying some foundation of Gromow-Witten invariants, I now would like to see an explicit computation. I heard that one should first take a look at the total space of $\mathcal{O}(-1)^{\oplus2}$ ...
- 241
0
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0
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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
1
vote
0
answers
266
views
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
3
votes
2
answers
744
views
Conics in the quadric line complex
Hello, I apologize in advance if this question is misguided somehow, since my algebraic geometry is pretty shaky.
I am wondering if there is a way to understand all the conics in a generic quadric ...
- 1,129
5
votes
1
answer
1k
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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 ...
- 51
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
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
8
votes
1
answer
818
views
To what extent does Poincare duality hold on moduli stacks?
Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the manifold,...
- 6,290
9
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0
answers
860
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 ones....
- 1,331
4
votes
0
answers
128
views
Is there a formula for the A-model partition function in terms of hyperbolic structure?
The A-model partition function is a topological invariant of any hyperbolic surface. So, what is this invariant?
I'm not sure if there is a way to normalize these since the usual perscription $Z(S^2) ...
- 607
2
votes
1
answer
382
views
Zero and Negative Gromov-Witten invariants in genus 0
I'm working on a project and I've used the Picard-Fuchs equation at a maximally unipotent monodromy point for a certain 1-dimensional family of Calabi-Yau 3-folds to calculate the A-model Yukawa ...
- 2,108
7
votes
1
answer
982
views
Trivial obstructions and virtual fundamental classes
Suppose $X$ is a DM stack, and let $E^\bullet$ be a perfect obstruction theory of $X$ such that the $E^{-1}$ term admits a trivial quotient/sub-bundle. Is it true that the virtual fundamental class $[...
- 6,290
3
votes
1
answer
245
views
Intersection theory on M_{g,n}
Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?
4
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2
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934
views
a question about Gromov-Witten invariant
Do the Gromov-Witten invariants count the morphisms from a curve to a variety over $\mathbb{C}$?
- 1,499
1
vote
1
answer
370
views
enumerative Gromov-Witten invariants
Let $X$ be a sympletic manifold and $A\in H_2(X;\mathbb{Q})$. Let $g$ and $k$ be nonnegative integers.
Assume that $$\mathcal{M}_{g,k}(X;A)$$
is dense in
$$\overline{\mathcal{M}}_{g,k}(X;A).$$
Are ...
- 73
2
votes
0
answers
162
views
Degree 2 curves on a degree d hypersurface in P^(2d+2)/3
One of the foundations of Gromov-Witten theory is the use (due to Kontsevich I think) of localization to calculate the number of degree $n$ curves on a general quintic 3-fold. When calculating the ...
- 265
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 ...
1
vote
0
answers
122
views
Is it possible to find an explicit definition of the "universal" (co)tangent bundle?
Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$
maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point
$y \in \mathbb{P^1} $ $\textit{...
- 3,245
4
votes
0
answers
118
views
Twisting stable maps to C* equivariant space by a line bundle
Let $X$ be a $\mathbb{C}^*$-equivariant algebraic variety. Then there is a notion of a map to $X$ twisted by a line bundle. Namely, let $B$ be a variety and $L/B$ a line bundle. Let $P_L=L\setminus B$...
- 7,972
3
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0
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270
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holomorphic embeddings of the sphere into the quintic in degree 2
Is there an explicit way of
classifying (with regard to their compatibiliy with $\Omega_+$ or $\Omega_-,$ see below)
the various families of
equivariant holomorphic embeddings from $\mathbb{CP}^1$ to ...
3
votes
2
answers
661
views
Quantum cohomology for open varieties
Hi, I know very little about the quantum cohomology (QC for short). I only got interested in the subject as the genus zero part may be relevant to a problem I'm working on. So I hope my question makes ...
- 7,527
6
votes
1
answer
577
views
Gromov-Witten and integrability 2.
This is a followup of my previous question Gromov-Witten and integrability. As I have learned from the answer (but guessed before), GW potentials of the point and $P^1$ (with different modifications) ...
- 1,343
3
votes
0
answers
154
views
G.W. invariants $<[pt],[pt]>_{0,[A]} \neq 0$ such that there exists ample L with $c_1(L)([A])=1$
Do there exist interesting examples of projective algebraic varieties such that the two-point genus 0 Gromov Witten invariants in homology class $[A]$, $GW<pt,pt>_{0,[A]}$, is non-zero, and ...
- 3,758
1
vote
1
answer
110
views
When does a stable map to a special fiber (locally) deform to a family of stable maps?
I'm sure the answer to my question is well-known -- I'm mostly looking for a reference.
Suppose I have a nonsingular variety $X$ which fibers over $\mathbb{A}^1$. Moreover, suppose I have a stable ...
- 111
4
votes
1
answer
852
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 ...
- 1,499
2
votes
0
answers
187
views
bijection of moduli space of equivariant holomorphic embeddings
Consider the moduli space $\mathcal{M}$ of equivariant holomorphic embeddings of closed oriented Riemann surfaces into a generic quintic three-fold $X$ in $\mathbb{P}^4,$ of given degree $d \in H_2(X,...
- 533
3
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0
answers
245
views
Are there any results on stable maps to Artin stacks with infinite stabilizers?
The Abramovich-Vistoli/Chen-Ruan theory of twisted stable maps into Deligne-Mumford stacks is extremely useful, as is the generalization to tame Artin stacks in positive characteristic. I am ...
- 142
5
votes
1
answer
911
views
What is the Gromov-Witten potential associated to String Topology?
Kevin Costello's article on the Gromov-Witten potential associated to a TCFT constructs for each TCFT, i.e. a functor from chains on Riemann surfaces with boundary ...
- 8,167
10
votes
0
answers
651
views
gromov witten donaldson thomas correspondence
Let $X$ be a nonsingular projective 3-fold. I am trying to understand the proof of the GW/DT correspondence as presented in Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds. I would ...
- 513
5
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0
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372
views
Deformation theory with a view toward GW theory and DT theory
I am studying GW theory (and DT theory) in algebraic geometry. I now understand the heuristic "Aut, Def, Obs" argument written in Mirror Symmetry book (by Hori et al.), but it is too hard for me to ...
- 349
7
votes
1
answer
460
views
Some questions on moduli of stable maps
Let $\overline{M}_{0,k}(\mathbb{P}^n,d)$
denote the moduli space of genus zero degree $d$ stable maps with $k$ marked points. This is an orbifold of expected dimension. Let $\overline{U}_{0,k}(\...
3
votes
1
answer
488
views
Curve Splitting in the Degeneration Formula for Relative GW Invariants (MNOP II)
I am attempting to understand and use the degeneration formula for GW invariants as stated in 'Gromov-Witten theory and Donaldson-Thomas theory II' (Maulik, Nekrasov, Okounkov, Pandharipande). The ...
- 437
16
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0
answers
3k
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MNOP conjecture
Let $X$ be a smooth, projective, Calabi-Yau 3-fold (CY makes the exposition more elegant, I don't think it is necessary).
To define Gromov-Witten invariants, we consider moduli spaces of stable ...
- 2,218
3
votes
0
answers
199
views
Questions about the details in the construction of virtual fundamental class
Let $\pi :D \subset \mathcal{X} \to S$ be a flat family of stable curves of genus $g$ with marked points $D$. Let $\mathcal{X} \to X$ be a flat family of stable morphisms in the sense of Kontsevich ...
- 233
8
votes
0
answers
549
views
Description of virtual fundamental class
For some concrete examples, is there an easy way to describe the virtual fundamental class (say, by capping off the moduli pace with an obstruction bundle ). Consider the moduli space of stable maps ...
- 147
5
votes
0
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165
views
question about relative stable maps
Let $C$ be a connected smooth curve, $0\in C$ a closed point and $W\rightarrow C$ a family of projective schemes. Assume that the fibers $W_t$ of $W$ are smooth for all $t\neq 0$ and that $W_0=Y_1\cup ...
- 101
1
vote
1
answer
500
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Does any one understand the details of M Kazarian's work in enumerative geometry of $\mathbb{C}\mathbb{P}^2$ ?
I wanted to know if anyone understood the details of the paper
"Multisingularities, cobordisms, and enumerative geometry" available at the site
http://www.mi.ras.ru/~kazarian/.
In particular does ...
- 3,245