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Apr
18
awarded  Popular Question
Apr
17
awarded  Popular Question
Apr
14
answered Deformation long exact sequence of GW theory in the analytical setting
Apr
10
awarded  Yearling
Mar
24
comment Deformation long exact sequence of GW theory in the analytical setting
Thanks Jason, I think this sheaf theoretic description of yours was the key to make sense of Q2. Now I see where I was struggling: to make sense of the map $du\colon \mathbb{C}^\infty(T\Sigma(-p)) \to \mathbb{C}^\infty (u^*TX)$. Originally, $du$ is a map from $T\Sigma$ to $u^*TX$, how do the marked points affect it then?
Mar
24
comment Deformation long exact sequence of GW theory in the analytical setting
Thanks Jason. By $T{\Sigma}(-\sum p_i)$ you mean sheaf of smooth (and not the usual holomorphic) sections of $T\Sigma$ that vanish at those points, correct?
Mar
24
revised Deformation long exact sequence of GW theory in the analytical setting
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Mar
24
revised Deformation long exact sequence of GW theory in the analytical setting
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Mar
24
comment Deformation long exact sequence of GW theory in the analytical setting
I am just trying to write down its analytical description explicitly. Since I don't well understand the short exact sequence from which such long exact sequence is constructed (in the analytical setting) I can't simply follow the algebraic definition of connecting map. $D_u\bar\partial$ is of the form $\bar\partial+A$ where $\bar\partial$ defines a holomorphic structure on $u^*TX$ and $A$ is some degree 0 map (which depends on Nijenhueis tensor). So $Def(u)$ and $Obs(u)$ are deformations of $H^0_{\bar\partial}(TX)$ and $H^1_{\bar\partial}(TX)$. That makes me a bit confused.
Mar
24
revised Deformation long exact sequence of GW theory in the analytical setting
edited body
Mar
24
asked Deformation long exact sequence of GW theory in the analytical setting
Mar
3
accepted Hochschild homology of Fukaya category in mirror symmetry
Feb
25
comment Euler class in the non-compact case
I particularly meant the non-compact case and most importantly, not necessarily orientable but just relatively orientable (where the zero set of a compactly supported transverse section would still be orientable (oriented)).
Feb
24
comment Euler class in the non-compact case
Two questions: What is the reference for this? What if the bundle is relatively orientable? i.e. the real line bundle $det(E^*)\otimes det(TM)$ is trivial. In this case, for a transverse section $s$, the zero set is of $s$ is a compact oriented submanifold of $M$; thus, defines an element in $H_{dim M-rank E}(M)$. However, we dont have Poincare duality to get some cohomology class.
Feb
24
awarded  Popular Question
Feb
23
comment Isotopy extension theorems
By $F(t,f(0,x)) = F(t,x)$, you most probably mean $F(t,f(0,x)) = f(t,x)$. Do you know any reference for an orbifold version of this theorem?
Feb
16
accepted Points with finite stabilizer in Hamiltonian torus actions
Feb
16
comment Points with finite stabilizer in Hamiltonian torus actions
Sounds reasonable, the local example inspired from your solution is the $S^1$-action on $\mathbb{C}^2$, $e^{i\theta}(x,y)=(e^{im\theta}x, e^{in\theta} y)$, where $(m,n)=1$. It is free away intersections with $x=0$ and $y=0$.
Feb
15
asked Points with finite stabilizer in Hamiltonian torus actions
Jan
15
awarded  Popular Question