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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.
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
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
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
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
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 …
CommunityBot
- 1
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
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