# Questions tagged [gromov-witten-theory]

The gromov-witten-theory tag has no usage guidance.

156
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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 ...

2
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### 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 ...

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### 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 ...

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1
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### 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 ...

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### 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 ...

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### 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 ...

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282
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $H^*(X,\mathbb{C)}$. ...

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### Computing Gromov-Witten invariant of $4$ lines in $\mathbb{C}P^3$

I'm trying to understand what the number of genus 0 curves through four lines in $\mathbb{C}P^3$ is i.e $Gr_{0,4}^{\mathbb{C}P^3, L}(PD(L),PD(L),PD(L),PD(L))$ where $L$ is the class of a line $\mathbb{...

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### Algebraic deformation invariance of Gromov-Witten invariants

Let $\mathcal{X}\to S$ be a smooth family of projective varieties over a smooth curve $S$. Let $\beta$ be a curve class supported in some fiber and consider the relative moduli space of stable maps $\...

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### 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 ...

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221
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### 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 ...

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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 ...

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### 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 ...

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### 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,
\...

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346
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### 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}...

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### 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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### 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 ...

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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 ...

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### 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 ...

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1
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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-...

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458
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### 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 ...

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### 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}^*...

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234
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### 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 ...

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### 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 ...

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

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### 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
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1
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293
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### 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 ...

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### 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 ...

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### 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
...

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### 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/...

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### 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://...

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### 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 ...

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### 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 ...

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### 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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### 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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### 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)...

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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 ...

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### 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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### 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 $S$,...

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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):\...

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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 ...

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### 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.
$$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...