All Questions
Tagged with toric-varieties complex-geometry
20
questions
1
vote
0
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136
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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
1
vote
1
answer
326
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Three-dimensional analogues of Hirzebruch surfaces
There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
- 173
1
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0
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67
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Embedding toric varieties in other toric varieties as a real algebraic hypersurface
In the question On a Hirzebruch surface, I've seen that the $n$-th Hirzebruch surface is isomorphic to a surface of bidegree $(n,1)$ in $\mathbb{P}^1\times \mathbb{P}^2$. I am trying to answer the ...
- 173
3
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0
answers
55
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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
3
votes
0
answers
183
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Smooth toric compactification of $\mathbb C^n$
By a compactification $(X,Y)$ of $\mathbb C^n$, we mean an irreducible compact complex space $X$ and a closed analytic subspace $Y\subset X$ such that $X\setminus Y$ is biholomorphic to $\mathbb C^n$. ...
- 2,739
3
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0
answers
329
views
When are two complex Tori biholomorphic
Let $g \ge 1$ be a natural number and $\mathbb{C}^g$ complex vector space which
is isomorphic to $\mathbb{R}^{2g}$ is real vector space.
An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a
...
- 5,780
1
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0
answers
105
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Isomorphic Jacobians for different choices of basis of $1$-forms
In Otto Forster's Lectures on Riemann Surfaces on page 170 Jacobi Variety is defined in 21.6:
Suppose $X$ is a compact
Riemann surface of genus $g$ and $ \omega_1,..., \omega_g $ is a basis
of $\Omega ...
- 5,780
4
votes
0
answers
309
views
Toric Fan for the Du Val's singularities D_n and E_n
Let us consider the Du Val's singularities.
i.e. https://en.wikipedia.org/wiki/Du_Val_singularity.
It is well known that they are classified by ADE, because the exceptional divisors arising in the ...
- 489
2
votes
0
answers
112
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Holomorphic line bundles on arbitrary simplicial toric varieties as restrictions
In the question titled "Line bundles and vector bundles on $\mathbb P^1 \times \mathbb P^1$" it was explained how any holomorphic line bundle on $\mathbb P^1 \times \mathbb P^1$ is of the form $\...
- 1,155
1
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0
answers
52
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Connection on line bundle over general simplicial toric variety
In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form
$$
(\mathbb{C}^N \backslash U)/(\...
- 1,155
0
votes
1
answer
423
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Can any simplicial toric variety be embedded in a product of projective spaces?
In this question - On a Hirzebruch surface. , the Hirzebruch surface is shown to be isomorphic to a hypersurface in $\mathbb{P}^1\times \mathbb{P}^2$.
My question is, does such an isomorphism exist ...
- 1,155
8
votes
1
answer
2k
views
Picard group of toric varieties
I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...
- 1,155
1
vote
1
answer
150
views
Find the Picard Fuchs operator of a four parameter fundamental period
In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say
\...
- 2,971
1
vote
0
answers
278
views
An example of threefold
Its description is a little bit complicated but it would be great if anyone can give an example.
I tried to construct it as a toric variety (See the previous question) but did not succeed.
I am ...
- 441
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,...
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