# Questions tagged [gromov-witten-theory]

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61
questions with no upvoted or accepted answers

**48**

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825 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 ...

**16**

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2k views

### 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 ...

**13**

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306 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 ...

**11**

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428 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 ...

**11**

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560 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 ...

**9**

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363 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 ...

**9**

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689 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....

**8**

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248 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}...

**8**

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478 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 ...

**6**

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131 views

### 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{...

**6**

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338 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 $S$,...

**5**

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

**5**

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155 views

### 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 $\...

**5**

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160 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 ...

**5**

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144 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 ...

**5**

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352 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 ...

**5**

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262 views

### Gromov-Witten theory of equivariant local projective plane

Can I find written explicitly in the literature a formula for the genus zero equivariant Gromov-Witten theory of local $\mathbb{P}^2$?
I understand that the method of Givental will give the answer, ...

**4**

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196 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,
\...

**4**

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122 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 ...

**4**

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

**4**

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164 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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159 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://...

**4**

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211 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):\...

**4**

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379 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 ...

**4**

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75 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**

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115 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) ...

**4**

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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 $P_L=L\setminus B$...

**4**

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139 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 ...

**4**

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190 views

### The hypergeometric pullback conjecture

Here arXiv:math/0510287, Golishev proposed the following conjecture:
The hypergeometric pullback conjecture. Let $X$ be a Fano variety. Then,for any constituent $C$ of the quantum D-module $Q$ there ...

**3**

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79 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 ...

**3**

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105 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 ...

**3**

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223 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 ...

**3**

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

**3**

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534 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 $M\times M$. ...

**3**

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261 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**

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150 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**

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265 views

### 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**

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204 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 ...

**3**

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191 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 ...

**3**

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243 views

### Transitive action on moduli space of holomorphic curves.

If $G$ is a complex semi simple Lie group and $P$ is a parabolic subgroup of it, the quotient $G/P$ is endowed with a Kahler structure and the action of $G$ on $G/P$ is holomorphic with respect to ...

**3**

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355 views

### genus one Gromov-Witten invariants of Calabi-Yau 3-folds

In
http://arxiv.org/PS_cache/hep-th/pdf/9302/9302103v1.pdf
physicists calculate (predict) genus one GW invariants of quintic (Table 1) and some other cases (Table 2).
Can any body explain to me (...

**2**

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127 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 ...

**2**

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121 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 ...

**2**

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75 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
...

**2**

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121 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 ...

**2**

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**0**answers

149 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 ...

**2**

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146 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 ...

**2**

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181 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,...

**2**

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165 views

### Are there ways to make low degree checks for enumerative formulas except for curves in CP^2?

This is a concrete question in Enumerative geometry. Let $S$ be a compact
complex surface and $L\rightarrow S$ a holomorphic line bundle. Let
$$ \delta_d = \text{dim}~ \mathbb{P}(H^0(S,L^d)) $$
...

**2**

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702 views

### a question on Costello's theorem

Costello's theorem,"open TCFT=Calabi Yau A-infinity category",he also mentions when applied to Fukaya category we can recover Gromov-Witten theory, but I see that it needs some assumption,also even if ...