Multiplicities of irreducible theta divisors

Abstract

Let $(A, \Delta)$ be a complex principally polarized abelian variety of dimension $g \geqslant 4$. Based on vanishing theorems, differentiation techniques and intersection theory, we show that whenever the theta divisor $\Delta$ is irreducible, its multiplicity at any point is at most $g-2$. This improves work of Kollár $\href{https://doi.org/10.1515/9781400864195}{[23]}$, Smith–Varley $\href{https://doi.org/ 10.1215/S0012-7094-96-08214-9}{[37]}$, and Ein–Lazarsfeld $\href{ https://www.jstor.org/stable/2152909}{[13]}$. We also introduce some new ideas to study the same type of questions for pluri-theta divisors.