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.
Multiplicities of irreducible theta divisors
Victor Lozovanu
2024-01-01
Abstract
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.
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