# Questions tagged [gromov-witten-theory]

The tag has no usage guidance.

156 questions
Filter by
Sorted by
Tagged with
10 votes
2 answers
853 views

### 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 ...
• 101
2 votes
1 answer
78 views

### Question on Gromov-Witten invariants when $A=0$

I started trying to learn about Gromov-Witten invariants by reading the book "$J$-holomorphic curves and Symplectic Topology" and I have a doubt in an example the authors provide. It's ...
• 671
2 votes
0 answers
122 views

### Some relative GW calculations

I have a question about the $\psi$ class in the following paper of Graber and Vakil: https://arxiv.org/abs/math/0309227 For $k,d\geq 2$, and a partition $d=d_1+\cdots+d_k$ of $d$ into positive ...
2 votes
1 answer
127 views

### Counting maximally tangent conics relative to a cubic

Is it possible to count the number of conics in $\mathbb{P}^2$ that are fully tangent at one point to a given (generic) cubic curve using basic intersection theory calculations? The corresponding ...
1 vote
0 answers
72 views

### symplectic gromov witten invariants of weighted projective space

Does anyone know if the symplectic (genus zero) GW invariants have been computed for weighted projective spaces? It seems there are already algebraic computations https://arxiv.org/abs/math/0608481 Is ...
• 113
2 votes
0 answers
142 views

### Is there a degeneration formula for Gromov-Witten K-theoretic invariants?

By Gromov-Witten K-theoretic invariants (call them KGW) I mean the invariants defined by Givental and Lee. I expect the formula expresses the KGW of the generic fiber of a given degeneration in terms ...
• 21
2 votes
1 answer
282 views

### admissible covers vs. stable maps to P^1

The paper Stable maps and tautological classes of Faber-Pandharipande shows that the Gromov-Witten classes on spaces of relative stable maps of the projective line push forward to tautological classes ...
• 735
11 votes
0 answers
553 views

### Does quantum cohomology have an $E_\infty$-ring structure?

Consdier the classical singular cohomology ring $H^*(X, \mathbb{Z})$ of a manifold $X$. The $H^n(X, \mathbb{Z})$'s are the cohomology groups of a chain complex and it turns out that the multiplication ...
• 111
3 votes
0 answers
140 views

### How to enumerate branched covers of $\mathbb{P}^1$ branched over $0,1$ and $\infty$?

Setup: Let $u:\Sigma \to X$ be a holomorphic map of closed Riemann surfaces with branch points $P \subset X$. For each branch point $p \in P$, we have a partition $\Gamma_p$ of $\text{deg}(u)$ given ...
• 1,121
6 votes
0 answers
156 views

### Descendent Gromov-Witten invariants and Frobenius manifolds

I've heard it said that genus $0$ descendent Gromov-Witten invariants of a smooth projective variety $X$ can be encoded in the structure of a Frobenius manifold on the cohomology $H^*(X,\mathbb{C)}$. ...
6 votes
0 answers
165 views

• 1,443
50 votes
0 answers
1k views

### What is the current status of derived differential geometry?

I hope you will excuse this naive and general question. I've read from many places (e.g. Dominic Joyce's website, John Pardon's thesis, etc.) that the/a "right" foundations for many ...
• 1,785
5 votes
0 answers
221 views

### Connected relative Gromov Witten invariants

I am currently interested to compute relative Gromov Witten invariants(GW) over $\mathbb{P}^1$. In the paper https://arxiv.org/pdf/math/0204305.pdf eq 3.1 gives the count of relative disconnected GW ...
• 675
3 votes
0 answers
150 views

### 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 ...
• 1,121
1 vote
0 answers
114 views

### Uniqueness in Mare combinatorics and bounds on Gromov-Witten invariants

Let $R$ be the root system of a Weyl group $W$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. the set of positive roots $\alpha$ such that $\ell(s_\alpha)=2\operatorname{ht}\alpha-1$. Based ...
• 962
4 votes
0 answers
200 views

### A conjectural inequality of the constant terms of functions

Could someone help me with the following question? This is equivalent to my previous question A conjecture about the barycenter of a polytope Let $D$ be a differential operator defined as follows, \...
• 1,091
8 votes
1 answer
346 views

4 votes
1 answer
234 views

### Total Chern Class of Hodge Bundle via CohFT

I am interested in the calculation of total Chern class of Hodge bundle. I am aware that there is a way by the Grothendieck-Riemann-Roch formula, however, I read that this is also a cohomological ...
• 253
2 votes
0 answers
125 views

### Coarse underlying curve of a smooth stable curve

In the theory of moduli spaces of smooth stable curves with $n$-marked points, I have come across the notion of the coarse underlying curve. Let $C$ be a smooth stable genus $g$ curve with $n$-marked ...
4 votes
0 answers
187 views

### Quantum cup product and Dolbeault cohomology

Let $X$ be a smooth projective variety over $\mathbb{C}$. We consider the small quantum cup product $\star$ on the deRham cohomology ring $\displaystyle H^*(X;\mathbb{C})=\bigoplus_{p,q}H^{p,q}(X)$. ...
• 441
7 votes
1 answer
280 views

### Gromov-Witten invariants and the mod 2 spectral flow

I'm reading Lee-Parker, “A structure theorem for the Gromov-Witten invariants of Kähler surfaces”, which studies Gromov-Witten invariants within symplectic geometry. Lee-Parker write (&...
• 6,646
3 votes
1 answer
293 views

### Equivariant quantum cohomology of conical symplectic resolutions

There is a couple of papers on this [Braverman, Maulik, Oblomkov, Okounkov, Pandharipande] where authors calculate quantum cohomology for various conical symplectic resolutions. The language in these ...
• 1,537
5 votes
1 answer
192 views

### Curve-counting with fixed source

Suppose I fix a smooth projective curve $C$ of positive genus, and I have a smooth projective variety $X$. Do standard tools from GW theory (or any curve-counting theory for that matter) allow me to ...
• 735
2 votes
0 answers
90 views

### Diagonal operator and infinite wedge space formalism

Let $\bigwedge^{\infty /2}V$ denote semiinfinte wedge space. The followin article section 2 gives a good description about the space and the operator on it. https://arxiv.org/pdf/math/0207233.pdf ...
• 675
4 votes
0 answers
191 views

### How does one obtain the formula for the number of genus one curves in P^2 using Getzler's relation?

I am trying to get the formula for the number of degree $n$ genus one curves in $\mathbb{P}^2$ passing through 3n generic points, by following the procedure in Getzler's paper https://arxiv.org/pdf/...
• 3,225
4 votes
0 answers
211 views

### Virtual fundamental class of Moduli space of stable maps in genus 1

What is the virtual fundamental class of $\overline{M}_{1,n}(\mathbb{P}^2,d)$? In general the virtual fundamental class is difficult to compute I guess. But if you look at Proposition 2.5 of https://...
• 547
2 votes
0 answers
140 views

### Reference request: explicit equivariant localization formula on toric complete intersections

This post is about an equivariant integration formula in a famous paper https://arxiv.org/pdf/alg-geom/9701016.pdf by Alexander Givental, where the author presented the formula without proof or ...
3 votes
0 answers
120 views

### Open Gromov-Witten invariants via lozalization with $\mathbb{C}^{*}$ (not $S^1$) action

Amplitudes of open A-model on a Calabi-Yau 3-fold $X$ with branes are given by the open Gromov-Witten invariants of $X$. It is known how to compute them if there is a toric action on a manifold, which ...
6 votes
1 answer
322 views

### Toda Hierarchy and Quantum Cohomology of $\mathbb{P}^1$ Frobenius manifolds

People usually say that the quantum cohomology of $\mathbb{P}^1$ Frobenius manifold $QH^*(\mathbb{P}^1)$, corresponds to dispersionless extended Toda hierarchy (e.g. page 6 of https://arxiv.org/pdf/...
• 385
13 votes
0 answers
338 views

### Log symplectic vortex equations in Hamiltonian log GW theory

Hamiltonian Gromov-Witten theory(see Mundet-Tian paper) corresponds to a new type of Symplectic vortex equations: Such type of models gives a connection to Hitchin-Kobayashi correspondence and Floer ...
1 vote
0 answers
139 views

### Basic property of Gromov-witten invariant

I am reading 'An invitation to Quantum cohomology by J.Kock, I.vainsencher'. I added a picture of the page on which I have question. The projection is flat and therefore has positive relative (fiber)...
2 votes
0 answers
167 views

### 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 ...
• 13.8k
3 votes
0 answers
237 views

### Nefness property for symplectic equivalency of Moishezon manifolds

Definition:The two symplectic manifolds $(X,\omega_X)$ and $(Y,\omega_Y)$ are defined to be symplectically equivalent if there exists a diffeomorphism $\phi:X\to Y$ such that $\phi^∗ω_Y$ is in the ...
6 votes
0 answers
432 views

### 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 $S$,...
• 1,961
4 votes
0 answers
296 views

• 1,961
1 vote
0 answers
153 views

### Cycles in the Chow ring of the moduli of curves coming from Hurwitz theory

Are there interesting cycles (other then the famous ones such as: Harris and Mumford's gonality divisor, the Gieseker-Petri divisor [which can be realized as the branch locus of a forgetful map from ...
• 1,961
5 votes
0 answers
151 views

### Virasoro constraints for parametrized GW invariants

Gromov-Witten invariants count isolated stable maps from Riemann surfaces to a fixed symplectic manifold $(M,\omega)$ subject to some incidence conditions. If we instead replace the target manifold ...
• 1,961
4 votes
0 answers
81 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 ...
• 20.3k
4 votes
1 answer
220 views

Let $p \in H^4(\mathbb{CP}^2)$ and $\ell \in H^2(\mathbb{CP}^2)$ be the cohomology classes Poincaré dual to a point and a line respectively. Question. What is the Gromov-Witten invariant $\langle p, ... • 193 2 votes 1 answer 303 views ### Gromov compactness theorem for genus$g >0\$ Riemann surfaces

In McDuff & Salamon's book "J-holomorphic Curves and Symplectic Topology", only the Gromov compactness for spheres has been proved, and I wonder whether or not this compactness result is still ...
• 2,661