All Questions
Tagged with toric-varieties dg.differential-geometry
12
questions
2
votes
0
answers
193
views
Understanding projective space as fibrations of tori over spaces with boundaries
The toric manifold $\mathbb{CP}^1$ can be understood as a circle fibration over an interval $I$, with the circles having zero radius at the boundaries of the interval. How does one generalize this ...
- 1,155
3
votes
1
answer
296
views
To what extent are toric manifolds and principal torus bundles "the same thing"?
I am a little confused by the different definitions for toric manifolds/varieties. Depending on the definition of toric manifolds and principal torus bundles that one chooses, when is a toric manifold ...
- 31
0
votes
0
answers
231
views
What is the symplectic manifold whose Delzant polytope is a trapezoid?
What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...
- 1
6
votes
0
answers
121
views
Geodesic rays in a toric variety
Let $\{\alpha_0, \ldots, \alpha_r\} \subset \mathbb{Z}^n$ be a finite subset of lattice points and let $\Phi: (\mathbb{C}^*)^n \to \mathbb{C}\mathbb{P}^r$ be the corresponding map from the algebraic ...
- 883
1
vote
1
answer
265
views
Moment map coordinates in tours action
I am trying to understand the proof of lemma 3.1, in this paper
In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality.In second they say, $g(dz_i,...
- 75
3
votes
1
answer
515
views
Kahler-Einstein metrics on Toric manifolds are Torus-invariant?
let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus $\mathbb{T}^{m}...
- 1,707
1
vote
1
answer
150
views
Legal potentials on delzant polytopes
Let $P \subset \mathbb R^n$ be a Delzant polytope defined by inequalities $\ell_i(x) \geq 0, i=1, \ldots, d$.
Of course, from the symplectic point of view, the inequalities $a_i \ell_i \geq 0$ still ...
- 585
4
votes
1
answer
493
views
Toric Fano Kahler manifolds and Delzant polytopes
Let $P$ be a Delzant polytope in $\mathbb R^n$, given by a set of inequalities $\ell_i(x) > 0$ where $\ell_i(x) = \sum_k \mu_k^i x_k - \lambda_i$.
In his paper http://arxiv.org/abs/0803.0985 ...
- 585
5
votes
2
answers
628
views
Is there an extremal metric on toric Fano manifolds which have nonzero Futaki invariant?
According the work by Wang & Zhu, on toric Fano manifolds there exist Kaehler-Ricci solitons. If Futaki=0, there also exist CSCK metrics. But if the Futaki invariant does not vanish, what about ...
- 53
3
votes
0
answers
503
views
"Step-by-Step" toric resolution process?
WLOG the fan $\Sigma$ of our toric variety $X_{\Sigma}$ is simplicial. (So $X_{\Sigma}$ has at worst orbifold singularities and all cones $\sigma \in \Sigma$ are simplicial).
The classical toric ...
- 1,929
4
votes
1
answer
298
views
Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?
My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...
7
votes
2
answers
2k
views
Deformations of Hirzebruch surfaces and toric action
Hi,
the Hirzebruch surface $F_n$ admits a deformation for $0\leq m\leq n$ defined by the equation
$$
\mathcal{M}=\{ ([x_0:x_1],[y_0:y_1:y_2],t) \in \mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{C}...
- 289