Polarizations, torsors and theta groups
Jef Laga
TL;DR
The paper develops a theta-group framework to decide when a polarization $\lambda$ on an abelian variety $A/k$ is realized by a line bundle on an $A$-torsor, reducing the problem to a lifting issue for $\big(A[\lambda],e_{\lambda}\big)$. It proves a general criterion: the existence of an $(X,L)$ with $\phi_L=\lambda$ is equivalent to the existence of a linear theta group for $(A[\lambda],e_{\lambda})$, and this can be checked via cohomological obstructions. The authors establish positive results for odd degree polarizations and several small even-degree types, and they construct high-dimensional counterexamples (dimension $\ge 7$) where such a torsor does not exist, using Totaro’s Chow-theoretic framework and recent negligible-cohomology results of Merkurjev–Scavia. The work connects moduli of polarized abelian varieties, Galois embedding problems, and obstruction theories, with implications for the geometry of the moduli stacks and the arithmetic of polarizations. Overall, it clarifies when polarizations can be lifted to actual line bundles on torsors and highlights subtle obstructions that arise in higher dimensions or with complex $2$-primary structure.
Abstract
Let $λ\colon A\rightarrow A^{\vee}$ be a polarization on an abelian variety over a field $k$. If $k$ is not algebraically closed, there might not exist an ample line bundle on $A$ defined over $k$ that represents $λ$. To remedy this, Poonen and Stoll have asked the following question: does there exist a line bundle on an $A$-torsor that represents $λ$? We give a criterion for the existence of such a torsor and line bundle which only depends on the kernel of $λ$. Using this criterion, we show that the answer to the question is yes when the polarization has odd or small even degree. On the other hand, we show that for every $g\geq 7$, there exists a polarized $g$-dimensional abelian variety for which the answer to the question is no.