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Question from Taubes' SW$\Rightarrow$ Gr

I am trying to understand Taubes' paper on SW$\Rightarrow$ Gr. I don't understand how either of the equations 2.16 or 2.17 appears, I would be happy to understand how the curvature term $F_a$ appears ...
Partha's user avatar
  • 883
4 votes
1 answer
187 views

Taubes' SW$\Rightarrow$ Gr

I am reading Taubes' paper on SW$\Rightarrow$ Gr and lost in some analysis, can anyone help me to see how to get equation 2.19 from equation 2.18? Is this some version of Kato for the Laplacian?
Partha's user avatar
  • 883
1 vote
0 answers
69 views

Enumerative or Gromov-Witten invariants from derived category of coherent sheaves

Let $X$ be a smooth projective toric Fano surface over $\mathbb{C}$. Suppose I have a nice presentation of $D^b_{Coh}(X)$ given by a full, strong exceptional collection $\mathcal{E} = \{E_i\}_{i\in I}$...
locally trivial's user avatar
1 vote
0 answers
108 views

Degeneration differential form nodal curve

I have a (possibly very basic) question about differential forms on nodal curves. After reading Witten's survey "Two-dimensional gravity and intersection theory on moduli space", I am ...
Dizbro's user avatar
  • 11
10 votes
2 answers
1k views

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 ...
user290289's user avatar
2 votes
1 answer
99 views

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 ...
Someone's user avatar
  • 781
2 votes
0 answers
131 views

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 ...
Mohammad Farajzadeh-Tehrani's user avatar
2 votes
1 answer
146 views

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 ...
Mohammad Farajzadeh-Tehrani's user avatar
1 vote
0 answers
91 views

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 ...
Yuan Yao's user avatar
  • 113
2 votes
0 answers
172 views

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 ...
jimmy's user avatar
  • 21
2 votes
1 answer
332 views

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 ...
Hans Sachs's user avatar
11 votes
0 answers
600 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 ...
QuantumRing's user avatar
3 votes
0 answers
146 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 ...
Julian Chaidez's user avatar
6 votes
0 answers
173 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 $H^*(X,\mathbb{C)}$. ...
John Rached's user avatar
6 votes
0 answers
170 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{...
cr1t1cal's user avatar
  • 755
6 votes
0 answers
265 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 $\...
Philip Engel's user avatar
  • 1,493
53 votes
0 answers
2k 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 ...
davik's user avatar
  • 2,035
5 votes
0 answers
258 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 ...
GGT's user avatar
  • 685
3 votes
0 answers
156 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 ...
Julian Chaidez's user avatar
1 vote
0 answers
122 views

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 ...
Christoph Mark's user avatar
4 votes
0 answers
204 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, \...
user42804's user avatar
  • 1,091
8 votes
1 answer
353 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}...
Dori Bejleri's user avatar
  • 2,900
4 votes
0 answers
168 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 ...
Yuhang Chen's user avatar
  • 1,099
4 votes
2 answers
505 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 ...
user avatar
3 votes
1 answer
425 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 ...
Yuhang Chen's user avatar
  • 1,099
7 votes
1 answer
791 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 ...
Yuhang Chen's user avatar
  • 1,099
2 votes
1 answer
569 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-...
Ritwik's user avatar
  • 3,235
11 votes
0 answers
544 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 ...
Dmitry Vaintrob's user avatar
3 votes
1 answer
225 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}^*...
Mohammad Farajzadeh-Tehrani's user avatar
4 votes
1 answer
268 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 ...
Peter Liu's user avatar
  • 253
2 votes
0 answers
131 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 ...
user avatar
5 votes
0 answers
202 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 $\displaystyle H^*(X;\mathbb{C})=\bigoplus_{p,q}H^{p,q}(X)$. ...
ChiHong Chow's user avatar
7 votes
1 answer
290 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 (&...
Arun Debray's user avatar
  • 6,766
4 votes
1 answer
333 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 ...
Filip's user avatar
  • 1,627
5 votes
1 answer
205 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 ...
Hans Sachs's user avatar
2 votes
0 answers
95 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 ...
GGT's user avatar
  • 685
4 votes
0 answers
208 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/...
Ritwik's user avatar
  • 3,235
4 votes
0 answers
231 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://...
Chitrabhanu's user avatar
2 votes
0 answers
156 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 ...
Linax Dio's user avatar
3 votes
0 answers
124 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 ...
Andrey Feldman's user avatar
6 votes
1 answer
346 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/...
user113988's user avatar
13 votes
0 answers
348 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 ...
user avatar
1 vote
0 answers
156 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)...
mamiemamie's user avatar
2 votes
0 answers
198 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 ...
aglearner's user avatar
  • 14k
3 votes
0 answers
283 views

Nefness property for symplectic equivalency of Moishezon manifolds

Definition: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 same ...
user avatar
7 votes
0 answers
464 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 $S$,...
Nati's user avatar
  • 1,971
4 votes
0 answers
320 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):\...
Nati's user avatar
  • 1,971
5 votes
0 answers
416 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 ...
Nati's user avatar
  • 1,971
1 vote
0 answers
230 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. $$ ...
Nati's user avatar
  • 1,971
1 vote
0 answers
159 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 ...
Nati's user avatar
  • 1,971