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Secant sheaves and Weil classes on abelian varieties

Eyal Markman

TL;DR

This paper develops a comprehensive strategy to prove the algebraicity of Hodge Weil classes on abelian varieties of Weil type by combining spin representations, pure spinor techniques, and derived category methods. The core approach constructs a CM endomorphism and a semi-regular secant sheaf whose Chern character yields an algebraic Weil class that remains algebraic under deformations within the split Weil type locus. Specializing to imaginary quadratic CM-fields, the author proves the algebraicity of Weil classes on polarized abelian sixfolds of split Weil type and deduces the Hodge conjecture for abelian fourfolds, thereby establishing the conjecture for all abelian varieties of dimension at most 5. The work integrates Orlov's derived equivalences, the semi-regularity theorem, and Chevalley spinor theory to produce a deformation invariant subalgebra whose Weil components are algebraic, with potential extensions to higher dimensions under stronger semi-regularity hypotheses.

Abstract

Let K be a CM-field, i.e., a totally complex quadratic extension of a totally real field F. Let X be a g-dimensional abelian variety admitting an algebra embedding of F into the rational endomorphisms of X. Let A be the product of X and Pic^0(X). We construct an embedding e of K into the rational endomorphism algebra of A associated to a choice of an F-blilinear polarization on X and a totally imaginary element q in K. We get the [K:Q]-dimensional subspace HW(A,e) of Hodge Weil classes in the d-th cohomology of A, where d:=4g/[K:Q]. We detail a strategy for proving the algebraicity of the Weil classes on all deformation of (A,e,h) as a polarized abelian variety of split Weil type, where h is an e(K) compatible polarization. We then specialize to the case F=Q, so that K is an imaginary quadratic number field. We survey how the above strategy was used to prove the algebraicity of the Weil classes on polarized abelian sixfolds of split Weil type. The algebraicity of the Weil classes on all abelian fourfold of Weil type follows. The Hodge conjecture for abelian varieties of dimension at most 5 is known to follow from the latter result.

Secant sheaves and Weil classes on abelian varieties

TL;DR

This paper develops a comprehensive strategy to prove the algebraicity of Hodge Weil classes on abelian varieties of Weil type by combining spin representations, pure spinor techniques, and derived category methods. The core approach constructs a CM endomorphism and a semi-regular secant sheaf whose Chern character yields an algebraic Weil class that remains algebraic under deformations within the split Weil type locus. Specializing to imaginary quadratic CM-fields, the author proves the algebraicity of Weil classes on polarized abelian sixfolds of split Weil type and deduces the Hodge conjecture for abelian fourfolds, thereby establishing the conjecture for all abelian varieties of dimension at most 5. The work integrates Orlov's derived equivalences, the semi-regularity theorem, and Chevalley spinor theory to produce a deformation invariant subalgebra whose Weil components are algebraic, with potential extensions to higher dimensions under stronger semi-regularity hypotheses.

Abstract

Let K be a CM-field, i.e., a totally complex quadratic extension of a totally real field F. Let X be a g-dimensional abelian variety admitting an algebra embedding of F into the rational endomorphisms of X. Let A be the product of X and Pic^0(X). We construct an embedding e of K into the rational endomorphism algebra of A associated to a choice of an F-blilinear polarization on X and a totally imaginary element q in K. We get the [K:Q]-dimensional subspace HW(A,e) of Hodge Weil classes in the d-th cohomology of A, where d:=4g/[K:Q]. We detail a strategy for proving the algebraicity of the Weil classes on all deformation of (A,e,h) as a polarized abelian variety of split Weil type, where h is an e(K) compatible polarization. We then specialize to the case F=Q, so that K is an imaginary quadratic number field. We survey how the above strategy was used to prove the algebraicity of the Weil classes on polarized abelian sixfolds of split Weil type. The algebraicity of the Weil classes on all abelian fourfold of Weil type follows. The Hodge conjecture for abelian varieties of dimension at most 5 is known to follow from the latter result.

Paper Structure

This paper contains 19 sections, 19 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Abelian varieties of Weil type
  3. Organization of the paper
  4. The Semi-regularity Theorem
  5. Abelian varieties and spin groups
  6. A strategy for proving the algebraicity of Weil classes
  7. Pure spinors
  8. Complex Multiplication
  9. The ${\rm Spin}(V_{\mathbb Q})_B$-invariant subalgebra of $H^*(X\times\hat{X},{\mathbb Q})$
  10. Examples of polarized $X\times\hat{X}$ of split Weil type
  11. Orlov's derived equivalence $\Phi:D^b(X\times X)\rightarrow D^b(X\times\hat{X})$
  12. The homomorphism $B\otimes B\rightarrow HW(X\times\hat{X},\eta)$
  13. A semi-regular sheaf over a sixfold $(X\times\hat{X},\eta)$ of split Weil type
  14. Two secant sheaves
  15. The transform of $F_2\boxtimes F_1$ to $X\times\hat{X}$
  16. ...and 4 more sections

Key Result

Theorem 1.2

markman-sixfolds The Weil classes for abelian fourfolds of Weil type and abelian sixfolds of split Weil type with complex multiplication by a quadratic imaginary number field $K$ are algebraic.

Theorems & Definitions (29)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Example 5.1
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof : Sketch of proof that $\eta(K)\subset {\rm End}_{Hdg}(V_{\mathbb Q})$
  • Lemma 7.1
  • ...and 19 more