# All questions with tag [math: abelian-varieties]

1

## Jacobian of a curve

Let $C$ be a contour and also $J$ be its Jacobian. What is the relationship in between $H^1(C,\mathcal{O}_C)$ and also $H^1(J,\mathcal{O}_J)$? Can a person aim me to a very easy reference for this topic?
2022-07-22 18:32:42
1

## Canonical divisor of an abelian variety

Let $A$ be an abelian selection over an area $k$ and also allow $K_A$ be its approved divisor. After that I'm virtually particular that $K_A$ is unimportant, yet I can not appear to confirm it, neither find a counter instance, neither find any kind of reference on abelian varieties that also states approved divisors. Does any person recognize ...
2022-07-12 14:26:12
0

## Some complex torus is not an abelian variety

Let $e_1,...,e_4\in\mathbb{C}^2$ whose works with are all algebraically independent. Allow $\Lambda$ be the latticework extended by these vectors. Why is $\mathbb{C}^2/\Lambda$ not an abelian selection?
2022-07-11 23:27:44
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## 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 $$f(a) = \sum_{i} (-1)^i \mathrm{Tr}(t_a^\ast, H^i(X,\mathbf{C}))$$ continuous? Here I consider the usual singular cohomology with $\mathbf{C}$-coefficients. (The coefficients d...
2022-07-06 23:53:44
0

## 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 $p&gt;0$ and let $A=A_{/k}$ be an ordinary abelian variety of dimension $g\geq1$. One knows that the $p$-torsion of $A$ is a product: $$A[p]=\hat A[p]\times T_p(A)\otimes(\Bbb Z_p/p\Bbb Z_p).$$ Here $\hat A[p]$ is the maxima...
2022-07-04 21:12:36