Pullback of a vector bundle on Abelian variety via $(-1)$

Question

Let $A$ be an abelian variety over some field $k$ and $(-1) : A \to A$ is the inverse map of $A$ as an algebraic group. If $V$ is a vector bundle over $A$ what is $(-1)^* V$? In other words, is there a way to describe $(-1)^* V$ in more standart functors? If it is not possible in general, what about special cases of $\operatorname{dim} A=1$ or $\operatorname{rk} V=1$?

For example, for elliptic curves, using Atiyah classification, it is easy to see that if $c_1(V)=0$ for some vector bundle $V$ then $(-1)^*(V) \cong V^*$. Is it true for higher dimensions?

Also, it is very interesting to understand $(-1)_*$, in particular, what is $(-1)_*\mathcal{O}$?

Answer 1

I'm not quite sure for general vector bundles, but for line bundles over $\mathbb{C}$, one knows by the Appel-Humbert Theorem that if $L$ is a line bundle on $A=\mathbb{C}^n/\Lambda$, then it is characterized by a positive definite Hermitian form $H:\mathbb{C}^2\to\mathbb{C}$ and a semi-character $\chi:\Lambda\to S^1$. One then has that $(-1)^*L(H,\chi)\simeq L(H,\chi^{-1})$ (see, for example, Birkenhake-Lange).

Answer 2

For any morphism $f:X \to Y$ of varieties (or schemes), one has $f^*\mathcal O_Y = \mathcal O_X$.

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A line bundle on an ellipitic curve of degree $d$ is of the form $\mathcal O(D)$ where $D = (d-1)O + P$, where $O$ is the origin and $P$ is a point on the curve. Applying $[-1]^*$ takes this to $\mathcal O(D')$, where $D' = (d-1)O + (-P).$