Multiplicities of irreducible theta divisors

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