The abelian-varieties tag has no wiki summary.
3
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0answers
39 views
Triviality of the tangent space of an abelian variety
The tangent bundle of an abelian variety $A/K$ is trivial. Mumford's proof of this fact goes something like this: given the (non-zero) value of a section in a fiber, one can determine the value in all ...
5
votes
2answers
140 views
Translation of a Paper of Tate
I'm just wondering if anyone knows if the paper "Classes d'isogenie des varietes abeliennes sur un corps fini" by John Tate has been translated into English. My French is not that good and I found it ...
3
votes
1answer
79 views
Does the Euler characteristic increase if I add an effective divisor
Let $D$ be an effective divisor on a smooth projective connected complex algebraic variety $X$. Suppose that $D\leq E$. Is it true that $$\chi(X,\mathcal{O}_X(D)) \leq \chi(X,\mathcal{O}_X(E))?$$
1
vote
1answer
89 views
Linear system of divisors on complete variety
I am currently reading Mumford's abelian varieties and Milne's notes on them and I have a problem understanding the proof that they are projective. Both of them use that a complete linear system of ...
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0answers
31 views
Theta group representation
Let $(X,L)$ be a polarized abelian variety over $k=\overline{k}$, and let $K(L)$ be the kernel of the isogeny $X\to X^\vee$ that sends $x$ to $t_x^*L\otimes L^{-1}$. The theta group $\mathscr{G}(L)$ ...
2
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31 views
Is there a construction known for associating a K3 surface to a curve or cover of curves
Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian.
Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
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0answers
19 views
Are there generalizations of Prym varieties to higher dimensions
Prym varieties are abelian varieties that are associated to a double cover of algebraic curves.
Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way?
...
4
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0answers
114 views
If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$
Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
2
votes
1answer
70 views
Abelian subvarieties of a principally polarized abelian variety are principally polarized
Let $A$ be a principally polarized abelian variety. Let $X\subset A$ be an abelian subvariety. Is $X$ also principally polarized? Here's what I think should be a proof. Is it correct?
We may and do ...
3
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0answers
56 views
Is $M_g$ NEVER proper? And why does $T_g$ contain products?
Work over an algebraically closed field ($\mathbb C$, if you prefer) and fix $g\geq 2$. By $M_g$ I mean, of course, the moduli space of smooth projective curves of genus $g$. I know it is in general ...
6
votes
1answer
213 views
What is the Albanese map good for?
I am reading a textbook on complex manifolds and come across Albanese map. For a compact Kaehler manifold $X$,
$$
T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z})
$$
is a complex torus, called the Albanese ...
3
votes
1answer
64 views
(Non)-Isomorphic Jacobians
I've started reading Hindry-Silverman's Diophantine Geometry and I got a little ahead of myself. Non-isomorphic elliptic curves have non-isomorphic Jacobians, that is because elliptic curves are ...
4
votes
2answers
129 views
Subvariety of Product of Elliptic Curves
This is almost certainly known (and maybe written down somewhere?). Is there an example of two elliptic curves $C, E/k$ that are not isomorphic, yet there is an embedding $C\hookrightarrow E\times E$ ...
3
votes
1answer
134 views
Jacobian of a curve
Let $C$ be a curve and $J$ be its Jacobian.
What is the relation between $H^1(C,\mathcal{O}_C)$ and $H^1(J,\mathcal{O}_J)$ ?
Can someone point me to an easy reference for this subject?
3
votes
1answer
133 views
Canonical divisor of an abelian variety
Let $A$ be an abelian variety over a field $k$ and let $K_A$ be its canonical divisor. Then I'm almost certain that $K_A$ is trivial, but I can't seem to prove it, nor find a counter example, nor ...
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votes
0answers
69 views
Some complex torus is not an abelian variety
Let $e_1,...,e_4\in\mathbb{C}^2$ whose coordinates are all algebraically independent. Let $\Lambda$ be the lattice spanned by these vectors. Why is $\mathbb{C}^2/\Lambda$ not an abelian variety?
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0answers
69 views
Why is the trace map on an abelian variety continuous
Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$.
Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i ...
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0answers
87 views
Notation for a canonical quotient of an abelian variety in positive characteristic
This may be a somewhat silly question, but there it goes.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian variety of dimension $g\geq1$. ...
