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Lefschetz Defect in Families

Matteo Verni

TL;DR

The paper establishes that the Lefschetz defect $\delta_X$ of smooth Fano varieties is invariant under smooth deformations, enabling deformation-stable classifications within each family. It provides a cohomological characterization of $\delta_X(D)$ via kernels in $H^2$ and $N^1$, enabling a purely cohomological perspective and extending Voisin's ideas. The work also analyzes the Lefschetz defect for abelian varieties, tying $\delta_A$ to isogeny factors and multiplicities, and describes its behavior in families via $\Lambda$-polarizations, addressing subtleties from Noether-Lefschetz phenomena. Together, these results offer deformation-stable invariants for Fano and abelian settings and illuminate how the defect interacts with linear equivalence, cohomology, and isogeny decompositions.

Abstract

We provide a cohomological characterization of the Lefschetz defect of smooth complex projective varieties. As a consequence, we deduce that the Lefschetz defect of a smooth Fano variety is invariant under smooth deformation. We also characterize the Lefschetz defect of an abelian variety in terms of its isogeny factors, and study it in families.

Lefschetz Defect in Families

TL;DR

The paper establishes that the Lefschetz defect of smooth Fano varieties is invariant under smooth deformations, enabling deformation-stable classifications within each family. It provides a cohomological characterization of via kernels in and , enabling a purely cohomological perspective and extending Voisin's ideas. The work also analyzes the Lefschetz defect for abelian varieties, tying to isogeny factors and multiplicities, and describes its behavior in families via -polarizations, addressing subtleties from Noether-Lefschetz phenomena. Together, these results offer deformation-stable invariants for Fano and abelian settings and illuminate how the defect interacts with linear equivalence, cohomology, and isogeny decompositions.

Abstract

We provide a cohomological characterization of the Lefschetz defect of smooth complex projective varieties. As a consequence, we deduce that the Lefschetz defect of a smooth Fano variety is invariant under smooth deformation. We also characterize the Lefschetz defect of an abelian variety in terms of its isogeny factors, and study it in families.
Paper Structure (11 sections, 12 theorems, 63 equations)

This paper contains 11 sections, 12 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Outline of the article
  3. Lefschetz Defect
  4. Basic definitions and remarks
  5. Another cohomological characterization of $\delta_X(D)$
  6. Lefschetz defect in families of Fano Manifolds
  7. Extending line bundles
  8. Effectivity of Cartier divisors in a family
  9. Invariance of d_X for Fano families
  10. Lefschetz defect on abelian varieties
  11. Families of abelian varieties

Key Result

Theorem 1.1

Let $X_1$ and $X_2$ be two smooth Fano varieties which are deformation equivalent via a smooth deformation. Then $\delta_{X_1}=\delta_{X_2}.$

Theorems & Definitions (25)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Theorem 2.6
  • proof
  • ...and 15 more