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

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### Canonical model on Kontsevich's moduli space

Motivation: Work of Eisenbud, Harris, and Mumford shows that
$M_g$ is of general type when $g≥24$, a naive question is what the same question about Kontsevich's moduli space?
Let $X$ be a smooth ...

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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 ...

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### Any natural examples of infinite dimensional Cohomological Field Theories?

Cohomological Field Theories as defined by Kontsevich and Manin are a class of linear maps from a vector space $V$ to the cohomology of the Deligne-Mumford moduli space of curves ...

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### 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 ...

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### 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) ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $M\times M$. ...

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### 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 ...

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### 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 ...

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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 ...

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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 ...

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

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

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### 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 ...

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### 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} $ ...

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### 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, ...

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### Reference or explanation: Cup products, deformations of complex structure and Mirror Symmetry

In section 0.3. of their paper "Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields," Barannikov and Kontsevich discuss the fact that Kontsevich's formality morphism (from his paper ...

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### Higher genus Gromov-Witten potential

Is it known if the higher genus (gravitational) Gromov-Witten potential is split in a classical and quantum part like the genus 0 Gromov-Witten potential? If so, Could someone give a reference?

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### 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 ...

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### 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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### 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, ...

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### How mirror of quintic was originally found?

In the 90-91 pager
"A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY",
Candelas, De La Ossal, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically ...

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### Definition of Givental $J$-function of cotangent bundle of flag variety

I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function ...

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### 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 ...

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### Curves on K3 and modular forms

The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing ...

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### 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 $k=0,1,2$ ?

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### 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 ...

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### 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 ...

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### 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 ...

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### Moduli space of stable maps into very ample hypersurfaces!

Let $X$ be a smooth complex projective variety and $L$ be some ample divisor.
For a holomorphic map $u:\Sigma \to X$, we define its degree to be $deg(u^*L)$.
Question: For a given positive integer ...

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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 ...

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### 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 ...

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### 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) ...

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### 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 ...

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### 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 ...

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### Is Gromov-Witten theory of Calabi-Yau threefolds of Type A trivial?

There are some Calabi-Yau threefolds that do not contain any rational curves, e.g. Calabi-Yau threefold of type A in the paper "Calabi-Yau threefolds of quotient type" by Oguiso and Sakurai.
My ...

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### 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 ...

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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
$$
...

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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 ...

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### 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 ...

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### 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 ...

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### Cobordism of orbifolds?

Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...

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### 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 ...

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### 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 $\mathcal{O}(-1)^{\oplus2}$ ...

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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 ...

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### 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 ...

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### 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 ...

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### Obstruction theories on non-smooth spaces with smooth fibres

Given a perfect obstruction theory $E^\bullet$ over a space $X$, we know that if $X$ is smooth, that the virtual fundamental class $[X, E^\bullet]$ is given by
$$[X, E^\bullet] = ...