Multiplicities of irreducible theta divisors

Let (A,Δ) be a complex principally polarized abelian variety of dimension g⩾4. Based on vanishing theorems, differentiation techniques and intersection theory, we show that whenever the theta divisor Δ is irreducible, its multiplicity at any point is at most g−2. This improves work of Kollár, Smith–Varley, and Ein–Lazarsfeld. We also introduce some new ideas to study the same type of questions for pluri-theta divisors.