All Questions
Tagged with toric-varieties sg.symplectic-geometry
36
questions
1
vote
0
answers
136
views
Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form
Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...
- 191
0
votes
0
answers
59
views
existence of moment maps for non-nef toric varieties
The noncompact toric variety $X_1 = \operatorname{Tot} \mathcal{O}(-1) + \mathcal{O}(-1) \to \mathbb{CP}^1$, the total space of the sum of two line bundles over the complex line, is defined as the ...
- 533
3
votes
0
answers
55
views
Complex structures compatible with a symplectic toric manifold
Let $(M^{2m},\omega)$ be a compact symplectic manifold with equipped with an effective Hamiltonian torus $\mathbb T^m$ action.
Suppose $J_0$ and $J_1$ are two $\mathbb T^m$-invariant compatible ...
- 1,401
2
votes
1
answer
225
views
Why are symplectic toric varieties projective?
Let $X$ be a symplectic toric manifold meaning a compact symplectic manifold $(X, \omega)$ with $\dim{X} = 2n$ equipped with a Hamiltonian action of a maximal-dimension torus $\mathbb{T} = (\mathbb{S}^...
- 3,393
1
vote
0
answers
100
views
Seeing $\mathbb{CP}^2 \mathbin\# \overline{\mathbb{CP}^2}$ as a symplectic reduction of different manifolds
I have been reading the paper "Remarks on Lagrangian intersections on toric manifolds" by Abreu and Macarini, which gives several non-displaceability results by avoiding the use of ...
- 791
1
vote
0
answers
124
views
Doubt in the proof of Mcduff''s method of probes
I have been reading the paper "Displacing Lagrangian toric fibers by probes" by Dusa Mcduff, here is the arxiv link https://arxiv.org/pdf/0904.1686.pdf.
I have a doubt in the proof of lemma $...
- 791
4
votes
1
answer
121
views
Almost toric mutations
I'm trying to understand the details of the almost toric mutation process as explained in Section 8.4 in https://arxiv.org/pdf/2110.08643.pdf. More specifically, given an almost toric fibration $f: (M,...
- 755
2
votes
1
answer
268
views
Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$
I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$.
Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical ...
- 1,037
4
votes
1
answer
193
views
If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic?
Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?
- 143
6
votes
2
answers
375
views
From Delzant polytope to lattice polytope
By definition, an $n$-dimensional Delzant polytope $P$ is not necessarily a lattice polytope. But
is there a natural way (or operations) to turn $P$ into a lattice polytope using the fact that the ...
- 143
14
votes
0
answers
455
views
How should we think about the algebraic moment map?
My question is about the "algebraic moment map", as discussed by Frank Sottile in the final section of this paper, or by Bill Fulton in his Introduction to Toric Varieties, where he referes ...
- 6,210
3
votes
1
answer
322
views
An example in symplectic geometry
$\DeclareMathOperator\SU{SU}$Let $M$ be a coadjoint orbit of dimension 6 of $\SU(3)$, and let $T$ be the maximal torus in $\SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map ...
- 133
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 ...
- 1,241
2
votes
0
answers
97
views
Explicit formula for the moment map of toric manifold
Let $P$ be a Delzant polytope in $M\otimes{\mathbb R}\cong \mathbb R^n$, and it is well-known that we can associate to it a toric manifold $X=X_P$ with the moment map $\pi: X\to P$.
I would like to ...
- 2,739
3
votes
1
answer
352
views
Is the minimal Chern number of a toric manifold at least 2?
I would like to show that the minimal Chern number $N_M$ of a toric manifold $M$ is at least $2$, where
$$
N_M := \underset{l>0}{\min} \lbrace \exists \ A \in H_2(M;\mathbb{Z}) \ : \ \langle c, A \...
- 1,227