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Results tagged with abelian-varieties
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user 4144
Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.
16
votes
Does the self-product of a $g$-dimensional abelian variety contain an abelian variety of dim...
In fact, it is no for completely elementary reasons. If $A$ is simple and
$B\subset A^n$ is an abelian variety with $\dim B < g$, then $Hom(B,A^n)=Hom(B,A)^n$ is necessarily zero. So $B=0$.
- 35.2k
1
vote
Algebraic cycles of dimension 2 on the square of a generic abelian surface
Probably you know this, but I might point out that Nori [Proc. Indian Acad, 1989]
proved that the Griffiths group of a generic abelian 3-fold is infinitely generated.
It may be worth looking at, even …
- 35.2k
4
votes
Polarizations on intermediate Jacobians
If enough Hodge numbers vanish so that the Hodge structure $H^{2k+1}(X)$ has level one,
then $J^kX$ should be an abelian variety. This applies to Fano (e.g. cubic) 3-folds for example.
Later that day …
- 35.2k
7
votes
Accepted
complex deformations of abelian varieties
Hugo,
Although this was already discussed in the comments, perhaps I can write few more details
here. The material can be found in many books such as Mumford's Abelian Varieties or
the book on the sa …
- 35.2k
3
votes
Generic Mumford Tate group and algebraic points
For Question 1, there is a countable union of proper analytic subsets in $\mathcal{H}_g$ (so meagre in the sense of Baire category) such that anything in the complement has Mumford-Tate $GSp_{2g}$. Un …
- 35.2k
17
votes
Accepted
The Tate conjecture for abelian varieties
Let me put it this way, Tate's conjecture for abelian varieties is known to imply the Hodge conjecture for abelian varieties, and the last is very much open for this class. For the implication, see th …
- 35.2k
5
votes
Accepted
Families of abelian varieties on the line (or more generally simply connected varieties)
Q2 should be yes for polarized VHS by a rigidity theorem of Schmid [theorem 7.24, Variations of Hodge structure, Inventiones 1973], which says roughly that a PVHS is determined by the Hodge structure …
- 35.2k
8
votes
Is there for every variety X an abelian variety A such that their 1st l-adic cohomologies ar...
Let me slightly expand my comment from yesterday. Unfortunately, because of
various time constraints, this will still be quite sketchy. Note that I'm only addressing the
titular question. I have noth …
- 35.2k
6
votes
Accepted
Homogeneous vector bundles on Abelian varieties
I don't have access to Miyanishi's article either (at least during lockdown), but as Ulrich suggested, one can look at Mukai's paper "Duality between D(X) and $D(\hat X)$...", where he introduces the …
- 35.2k
18
votes
Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
The class of Abelian varieties is the simplest class of varieties where the Hodge conjecture is not known to be true. So naturally a certain amount of effort is directed toward them.
However, it's no …
- 35.2k