All Questions
Tagged with complex-geometry abelian-varieties
59
questions
4
votes
1
answer
227
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Homogeneous polynomials cutting out complex abelian varieties
This is an update to a previous question of mine. The more clarified questions, results and definitions make me feel like this warrants a separate post instead of a large edit of the original one.
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- 1,755
2
votes
0
answers
161
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Product subvariety of a simple abelian variety
Terminology: A subvariety $V$ of an abelian variety $A/\mathbb{C}$ is called a product if there are integral closed subvarieties $U,W\subset A$ such that $\dim U,\dim W>0$ and the sum morphism $U\...
- 545
3
votes
0
answers
263
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Explicit family of polynomials describing embedded torus in complex projective space
This question is cross-posted (with modifications) from MSE. The original question is probably unfit for MathOverflow (although a professor I asked said that this is very nontrivial), but I'm hoping ...
- 1,755
3
votes
1
answer
194
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How to determine the type of a divisor on a product of elliptic curves?
I already asked this on Math.SE, but didn't receive an answer yet.
Say $E_1, \dotsc, E_n$ are elliptic curves (everything over $\mathbb C$), and $D \subset E_1 \times \dotsc \times E_n$ is an ...
- 1,141
1
vote
0
answers
136
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Smooth symmetric divisors in abelian varieties without points of order $2$
Let $X=V/\Lambda$ be a complex abelian variety of dimension $g$, endowed with a polarization $M$ of type $(d_1, \ldots, d_g)$. A divisor $D \in |M|$ is called symmetric if $(-1)_X^*D=D$, namely if it ...
- 65.7k
3
votes
0
answers
288
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Ampleness of the normal bundle to the Albanese image
Let $X$ be a projective surface of general type over $\mathbb{C}$, and assume that $\Omega_X$ is globally generated. Then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is a local embedding ...
- 65.7k
1
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0
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115
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Abelian subvarieties corresponding to vector subspaces
Let $S$ be a connected smooth projective surface.
Let $C$ a smooth curve on $S$
In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following:
Let
\begin{equation*}
r: ...
- 519
4
votes
1
answer
272
views
Difference between stabilizer and automorphism group of subvariety of an abelian variety
Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega_X$ ample.
Often, people speak about the stabilizer $\mathrm{...
- 303
4
votes
0
answers
197
views
𝔾ₘ extensions vs line bundles over abelian varieties
Given a complex polarized abelian variety $V$, we can define a map $$\operatorname{Ext}^1\left(V, \mathbb{G}_m\right) \to \operatorname{Pic}\left(V\right)$$
by viewing an extension as a $\mathbb{G}_m$-...
- 752
3
votes
0
answers
185
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Endomorphisms ring of complex abelian variety under isogenies
I’m trying to understand if over $\mathbb{C}$ two abelian varieties have the same complex multiplication if and only if they are isogenous. Is it true?
If it is true this means that if I consider $A$ ...
8
votes
0
answers
199
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Monodromy groups that are profinitely dense in Sp(2g,Z)
$\DeclareMathOperator\Sp{Sp}$Assume $g\geq 2$. It is known that there exist finitely generated subgroups of $\Sp(2g,\mathbb{Z})$ of infinite index that surject onto all finite quotients of $\Sp(2g,\...
1
vote
0
answers
152
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From a factor of automorphy on an abelian variety to a divisor
Given a complex abelian variety $A = V/\Gamma$ (for $\Gamma$ being a lattice in the complex vector space $V$), one knows how to describe a holomorphic line bundle in terms of factors of automorphy: By ...
- 12k
1
vote
0
answers
221
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Harmonic forms on a complex torus
Let $T=\mathbb{C}^3/\Lambda$ be a complex torus of our interest and $L$ be a holomorphic line bundle on $T$, I am interested in $H^{0,2}_{\bar\partial_L}(T,L)$, i.e., the $(0,2)$ harmonic forms taking ...
- 923
3
votes
1
answer
225
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Is the Ueno fibration smooth?
Let $A$ be an abelian variety over $\mathbb{C}$ and let $X\subset A$ be a closed subvariety. Let $X\to Y$ be the Ueno fibration. (That is, $Y$ is of general type and a closed subvariety of $A/B$ where ...
- 303
2
votes
0
answers
508
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Polarizations in algebraic and symplectic geometry
In context of Abelian varieties there are a couple of equivalent ways to
introduce the polarization of a algebraic variety. One way is to
choose a line bundle $\mathcal{L}$ which satisfies certain ...
- 5,780