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Results tagged with abelian-varieties
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user 3847
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.
15
votes
Accepted
Canonical lift of the deformation of an ordinary abelian variety
No. The picture over a general base is this: let $A_0\to S_0$ be an ordinary abelian variety over a characteristic $p$ scheme $S_0$, and let $S_n$ ($n\geq 0$) be compatible flat liftings of $S_0$ over …
- 16.2k
5
votes
Accepted
Theta group representation
I think the answer is no due to the following counterexample (there might be a mistake somewhere though):
Take an elliptic curve $X$ with $L = O_X(3P_0)$ (characteristic $\neq 2, 3$), then $K$ will b …
- 16.2k
3
votes
Accepted
Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abe...
Posting my comment as an answer:
Since $T_{A/S}$ is trivial locally on $S$, we have $T_{A/S} = p^* p_* T_{A/S} = p^* {\rm Lie}_S A$. By the projection formula, we get
$$ R^1 p_* T_{A/S} = R^1 p_* p^* …
- 16.2k
9
votes
Accepted
Why is, for a group scheme of finite type, "smooth" (resp. irreducible) equivalent to "geome...
Let $G/K$ be a group scheme of finite type.
$G/K$ is smooth if and only if $\bar G / \bar K$ is smooth. Suppose $\bar G$ is reduced, then it has a smooth $\bar K$-point $x$ (because we are over an a …
- 16.2k
3
votes
Accepted
Complexification of Néron models of Abelian varieties
No. Take an abelian variety $A_0$ over $\mathbf{Q}$ and let $A = A_0\otimes K$ where $R = \mathbf{Q}[\pi]_{(\pi)}$ and $K = R[1/\pi]$. Then $K=K'$, $T'=0$, $N(A')_s^\circ = A\otimes R$. Then you are a …
- 16.2k
4
votes
Accepted
Lifting of Frobenius on torsors over abelian varieties
The canonical lifting $\mathcal{A}$ of $A$ has a canonical lift of the relative Frobenius $F_{\mathcal{A}/W}:\mathcal{A}\to \mathcal{A}'$, where $\mathcal{A}'=F_W^* \mathcal{A}$ (=the canonical lift …
- 16.2k
4
votes
Accepted
Lifting of Frobenius on semi-abelian varieties
My answer to your other question seems to show that if $B$ is ordinary, $A$ has a natural lift over $W$ as a scheme, together with a lift of the Frobenius. I don't see however why the group structure …
- 16.2k