# Questions tagged [gromov-witten-theory]

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140
questions

**4**

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194 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,
\...

**8**

votes

**0**answers

234 views

### The closure of a locus in $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n)$

Consider the closure $K \subset \overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n)$ in the stable maps space of the locus $K_0$ of maps $(f : C \to \mathbb{P}^1, p_1, \ldots, p_n)$ where $C \cong \mathbb{P}...

**4**

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100 views

### The Fock space in Costello's paper “Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products”

Let $X$ be a smooth projective variety. In this Annals paper, Costello expressed the descendent genus $g$ Gromov-Witten (GW) invariants of $X$ in terms of genus zero GW invariants of the symmetric ...

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142 views

### Explicitly computing Donaldson-Thomas invariants (of CY 3-folds)

I am interested in the explicit computation of generating functions of rank 1 and higher rank Donaldson-Thomas (DT) invariants. In particular, I am interested in DT invariants of K3 fibered Calabi-Yau ...

**3**

votes

**1**answer

263 views

### 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 ...

**6**

votes

**1**answer

331 views

### Why is the inertia stack of a smooth Deligne-Mumford stacks called inertia?

Let $\mathcal{X}$ be a smooth Deligne-Mumford stack. Then there is an associated stack $I\mathcal{X}$, called the inertia stack of $\mathcal{X}$.
Why is the inertia stack called "inertia"?
We can ...

**2**

votes

**1**answer

278 views

### 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-...

**8**

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321 views

### The virtual fundamental class as derived intersection

Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...

**3**

votes

**1**answer

119 views

### Localization on varieties with toric singularities

Is there a written version of Atiyah-Bott localization formula for varieties/manifolds with toric singularities? More precisely, suppose $F$ is a fixed locus of a torus action $\mathbb{T}={\mathbb{C}^*...

**4**

votes

**1**answer

170 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 ...

**2**

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118 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**

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118 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 $\displaystyle H^*(X;\mathbb{C})=\bigoplus_{p,q}H^{p,q}(X)$. ...

**7**

votes

**1**answer

216 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 (&...

**3**

votes

**1**answer

187 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 ...

**5**

votes

**1**answer

172 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 ...

**2**

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67 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
...

**4**

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161 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/...

**4**

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145 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://...

**2**

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113 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**

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102 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

237 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/...

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301 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 ...

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107 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**

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148 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 ...

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217 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 ...

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302 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 $S$,...

**4**

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202 views

### 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):\...

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357 views

### 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 ...

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187 views

### De Jonquières formula vs. Relative GW invariants

Background. Let $C$ be a smooth projective curve of genus $g$. Denote by $C^d$ its d-th symmetric product and by $G^r_d(C)$ its associated variety
of linear series of type $\mathfrak{g}^r_d$, i.e.
$$ ...

**1**

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144 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 ...

**5**

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140 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 ...

**4**

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74 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 ...

**4**

votes

**1**answer

202 views

### Gromov-Witten invariant $\langle p, p, \ell\rangle_{0, 1}$ counting degree $1$, genus $0$ curves in $\mathbb{CP}^2$?

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, ...

**2**

votes

**1**answer

245 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 ...

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86 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?

**3**

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174 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 $\mathbb{P}^4$, ...

**5**

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**1**answer

245 views

### 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=(\...

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183 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 ...

**4**

votes

**0**answers

114 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) ...

**5**

votes

**1**answer

426 views

### Computing quantum cohomology for total spaces of vector bundles over $\mathbb{P}^m$

Let $X$ be the total space of $\mathscr{O}(-1)^{\oplus{n}}\rightarrow\mathbb{P}^m$. I think there should be some general way to compute its quantum cohomology $QH^\ast(X)$.
However, since I'm not ...

**10**

votes

**3**answers

717 views

### In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$?

Let $X$ be a Calabi-Yau threefold and let us fix a homology class $\beta\in H_2(X,\mathbb Z)$, just for simplicity. The generating series of Gromov-Witten invariants of $X$ in class $\beta$, $$\mathsf ...

**12**

votes

**1**answer

499 views

### Quantum cohomology of line bundles over $\mathbb P^N$

Let $n,N$ be two positive integers. Consider the total space of the line
bundle $\mathcal O(-n)$ on $\mathbb C\mathbb P^N$. This is an algebraic variety with an action of $G=GL(N+1,\mathbb C)\times \...

**3**

votes

**0**answers

497 views

### What is the formula for the homology class represented by the diagonal?

Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis
for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion).
Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...

**0**

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220 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{...

**9**

votes

**1**answer

565 views

### Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space?

Consider the following question: Let $X$ be a compact complex manifold
and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let
$\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic ...

**0**

votes

**1**answer

192 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 ...

**1**

vote

**0**answers

245 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 ...

**6**

votes

**1**answer

515 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**

votes

**1**answer

270 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
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**4**

votes

**0**answers

109 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 $P_L=L\setminus B$...