Cycles on abelian 2n -folds of Weil type from secant sheaves on abelian n -folds.

A 2n2𝑛-dimensional abelian variety A𝐴 is of Weil-type, if it admits an embedding η:K→Endℚ(A)𝜂:𝐾→End𝑄(𝐴), where K=ℚ[−d‾‾‾√]𝐾=𝑄[−𝑑] for some positive integer d𝑑, such that each of the eigenspaces W𝑊 and W¯𝑊¯ of η(−d‾‾‾√)𝜂(−𝑑), with eigenvalues −d‾‾‾√−𝑑 and −−d‾‾‾√−−𝑑, intersects H1,0(A)𝐻1,0(𝐴) in an n𝑛-dimensional subspace. In that case HW:=∧2nW⊕∧2nW¯𝐻𝑊:=∧2𝑛𝑊⊕∧2𝑛𝑊¯ is a Gal(K/ℚ)𝐺𝑎𝑙(𝐾/𝑄)-invariant 22-dimensional subspace of H2n(A,K)𝐻2𝑛(𝐴,𝐾) spanned by rational (n,n)(𝑛,𝑛)-classes called Hodge-Weil classes. The Hodge conjecture predicts that HW𝐻𝑊 is spanned by classes of algebraic cycles.

We present a general strategy for proving the algebraicity of the Hodge-Weil classes. If you prefer short abstracts stop reading here.

Let X𝑋 be an abelian n𝑛-fold, n≥2𝑛≥2, and X̂ 𝑋^ its dual abelian n𝑛-fold. Endow V=H1(X,ℤ)⊕H1(X̂ ,ℤ)𝑉=𝐻1(𝑋,𝑍)⊕𝐻1(𝑋^,𝑍) with the natural symmetric bilinear pairing. The even cohomology Hev(X,ℤ)𝐻𝑒𝑣(𝑋,𝑍) is the half spin representation of Spin(V)Spin(𝑉).

A coherent sheaf F𝐹 on X𝑋 is a K𝐾-secant sheaf, if ch(F)𝑐ℎ(𝐹) belongs to a 22-dimensional subspace P𝑃 of Hev(X,ℚ)𝐻𝑒𝑣(𝑋,𝑄) spanned by Hodge classes, such that the line ℙ(P)𝑃(𝑃) intersects the spinorial variety in ℙ[Hev(X,K)]𝑃[𝐻𝑒𝑣(𝑋,𝐾)] along two distinct complex conjugate pure spinors. The K𝐾-secant P𝑃 determines an embedding η:K→Endℚ(X×X̂ )𝜂:𝐾→End𝑄(𝑋×𝑋^) and a non-degenerate 22-form hℎ on X×X̂ 𝑋×𝑋^. The triple (X×X̂ ,η,h)(𝑋×𝑋^,𝜂,ℎ) is a polarized abelian variety of Weil type, for a non-empty open subset of such K𝐾-secants.

We consider a sheaf E𝐸 over X×X̂ 𝑋×𝑋^, which is the image via Orlov's equivalence Φ:Db(X×X)→Db(X×X̂ )Φ:𝐷𝑏(𝑋×𝑋)→𝐷𝑏(𝑋×𝑋^) of the outer tensor product F1⊠F2𝐹1⊠𝐹2 of two K𝐾-secant sheaves. We prove that the characteristic class of E𝐸 remains of Hodge-type under all deformations of (X×X̂ ,η,h)(𝑋×𝑋^,𝜂,ℎ). When X𝑋 is the Jacobian of a genus 33 curve, we reduce the proof of algebraicity of the Hodge Weil classes of deformations of (X×X̂ ,η,h)(𝑋×𝑋^,𝜂,ℎ) to a conjecture that an unobstructedness theorem of Buchweitz-Flenner for deformations of semiregular coherent sheaves generalizes to semiregular twisted reflexive sheaves.

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