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0
votes
0answers
128 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 ...
7
votes
1answer
250 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
1answer
149 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 ...
0
votes
0answers
138 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 ...
5
votes
1answer
213 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
1answer
204 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 $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ ...
2
votes
0answers
80 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 ...
0
votes
0answers
99 views

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} $ ...
3
votes
0answers
181 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, ...
6
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0answers
158 views

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 ...
2
votes
1answer
365 views

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?
2
votes
0answers
85 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
votes
1answer
181 views

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 ...
2
votes
1answer
138 views

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, ...
16
votes
3answers
974 views

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 ...
2
votes
1answer
196 views

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 ...
2
votes
0answers
120 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 ...
12
votes
1answer
611 views

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 ...
3
votes
1answer
196 views

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 $k=0,1,2$ ?
2
votes
0answers
174 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 ...
1
vote
1answer
79 views

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 ...
7
votes
0answers
267 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 ...
2
votes
2answers
251 views

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 ...
3
votes
0answers
251 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 ...
2
votes
0answers
136 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 ...
4
votes
1answer
244 views

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) ...
3
votes
0answers
95 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 ...
0
votes
1answer
181 views

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 ...
1
vote
0answers
113 views

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 ...
3
votes
0answers
164 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 ...
9
votes
1answer
548 views

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 $$ ...
1
vote
1answer
205 views

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 ...
2
votes
1answer
214 views

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 ...
6
votes
1answer
223 views

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 ...
22
votes
2answers
892 views

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 ...
5
votes
0answers
256 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 ...
3
votes
2answers
432 views

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 $\mathcal{O}(-1)^{\oplus2}$ ...
17
votes
3answers
1k views

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 ...
3
votes
2answers
435 views

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 ...
9
votes
0answers
359 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 ...
0
votes
0answers
100 views

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] = ...
3
votes
1answer
596 views

Is P^2 important in Kontsevich's recursion formula?

There is a famous recursion formula by Kontsevich to find the number of genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points. My question is the following: Let $S$ be a complex ...
2
votes
0answers
148 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 $$ \delta_d = \text{dim}~ \mathbb{P}(H^0(S,L^d)) $$ ...
4
votes
1answer
659 views

Trivial obstructions and virtual fundamental classes

Suppose $X$ is a DM stack, and let $E^\bullet$ be a perfect obstruction theory of $X$ such that the $E^{-1}$ term admits a trivial quotient/sub-bundle. Is it true that the virtual fundamental class ...
3
votes
0answers
262 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 ...
3
votes
0answers
233 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 ...
4
votes
3answers
962 views

Minimal genus, adjunction inequality

Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$. As I know ...
7
votes
1answer
483 views

To what extent does Poincare duality hold on moduli stacks?

Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the ...
7
votes
2answers
540 views

Intuition behind the age grading in quantum cohomology of orbifolds

Let $\mathscr{X}$ be a smooth DM-stack with projective coarse moduli space. I am interested in the orbifold cohomology ring $H^\mathrm{orb}(\mathscr{X})$, as defined by Chen-Ruan (for orbifolds) and ...
12
votes
1answer
2k views

Donaldson-Thomas Invariants in Physics

First of all, I am sorry for there are a bunch of questions (though all related)and may not be well framed. What are the DT invariants in physics. When one is computing DT invariants for a Calabi-Yau ...