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Singularities of base loci on abelian varieties

Giuseppe Pareschi

TL;DR

The paper proves that the log canonical threshold of the base ideal of a complete linear system on a complex abelian variety is at least 1, with equality precisely when the base locus contains divisorial components. It leverages the Fourier–Mukai framework, generic vanishing, and Chen–Jiang decompositions to analyze multiplier ideals of base loci, showing that for $c<1$ the multiplier ideal is trivial and that nontriviality of the base ideal multiplier corresponds to divisorial base components. The results connect base-locus geometry with a refined GV/IT0/M-regular decomposition, yielding a precise divisorial criterion and suggesting radicality and reducedness questions for base schemes. The methods fuse Nadel vanishing, theta-group actions, and Chen–Jiang theory in the setting of abelian varieties to achieve a sharp threshold statement.

Abstract

We prove that the log canonical threshold of the base ideal of a complete linear system on an abelian variety is $\ge 1$, and equality holds if and only if the base locus has divisorial components.

Singularities of base loci on abelian varieties

TL;DR

The paper proves that the log canonical threshold of the base ideal of a complete linear system on a complex abelian variety is at least 1, with equality precisely when the base locus contains divisorial components. It leverages the Fourier–Mukai framework, generic vanishing, and Chen–Jiang decompositions to analyze multiplier ideals of base loci, showing that for the multiplier ideal is trivial and that nontriviality of the base ideal multiplier corresponds to divisorial base components. The results connect base-locus geometry with a refined GV/IT0/M-regular decomposition, yielding a precise divisorial criterion and suggesting radicality and reducedness questions for base schemes. The methods fuse Nadel vanishing, theta-group actions, and Chen–Jiang theory in the setting of abelian varieties to achieve a sharp threshold statement.

Abstract

We prove that the log canonical threshold of the base ideal of a complete linear system on an abelian variety is , and equality holds if and only if the base locus has divisorial components.

Paper Structure

This paper contains 8 sections, 2 theorems, 18 equations.

Table of Contents

  1. Background material on the Fourier-Mukai transform and generic vanishing
  2. Notation
  3. (Symmetric) Fourier-Mukai transform associated to the Poincarè line bundle
  4. Vanishing conditions
  5. Chen-Jiang decompositions and theorem
  6. The proof
  7. A preliminary lemma.
  8. Proof of the Theorem

Key Result

Theorem 1

$lct(\mathfrak{b}_L)\ge 1$ and equality holds if and only if the zero locus of $\mathfrak{b}_L$ has divisorial components.

Theorems & Definitions (3)

  • Theorem
  • Lemma 2.1.1
  • proof