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

Event Category:
Valley Geometry Seminar
Eyal Markman

A $2n$-dimensional abelian variety $A$ is of Weil-type, if it admits an embedding $\eta:K\rightarrow \mathrm{End}_\mathbb{Q}(A)$, where $K=\mathbb{Q}[\sqrt{-d}]$ for some positive integer $d$, such that each of the eigenspaces $W$ and $\bar{W}$ of $\eta(\sqrt{-d})$, with eigenvalues $\sqrt{-d}$ and $-\sqrt{-d}$, intersects $H^{1,0}(A)$ in an $n$-dimensional subspace. In that case $HW:=\wedge^{2n}W\oplus \wedge^{2n}\bar{W}$ is a $Gal(K/\mathbb{Q})$-invariant $2$-dimensional subspace of $H^{2n}(A,K)$ spanned by rational $(n,n)$-classes called Hodge-Weil classes. The Hodge conjecture predicts that $HW$ 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 $X$ be an abelian $n$-fold, $n\geq 2$, and $\hat{X}$ its dual abelian $n$-fold. Endow $V\!\!=\!\!H^1(X,\mathbb{Z})\!\oplus \!H^1(\hat{X},\mathbb{Z})$ with the natural symmetric bilinear pairing. The even cohomology $H^{ev}(X,\mathbb{Z})$ is the half spin representation of $\mathrm{Spin}(V)$.

A coherent sheaf $F$ on $X$ is a $K$-secant sheaf, if $ch(F)$ belongs to a $2$-dimensional subspace $P$ of $H^{ev}(X,\mathbb{Q})$ spanned by Hodge classes, such that the line $\mathbb{P}(P)$ intersects the spinorial variety in $\mathbb{P}[H^{ev}(X,K)]$ along two distinct complex conjugate pure spinors. The $K$-secant $P$ determines an embedding $\eta:K\rightarrow \mathrm{End}_\mathbb{Q}(X\!\times \!\hat{X})$ and a non-degenerate $2$-form $h$ on $X\!\times \!\hat{X}$. The triple $(X\!\times \!\hat{X},\eta,h)$ is a polarized abelian variety of Weil type, for a non-empty open subset of such $K$-secants.

We consider a sheaf $E$ over $X\!\times\!\hat{X}$, which is the image via Orlov's equivalence $\Phi:D^b(X\!\times \!X)\rightarrow D^b(X\!\times \!\hat{X})$ of the outer tensor product $F_1\boxtimes F_2$ of two $K$-secant sheaves. We prove that the characteristic class of $E$ remains of Hodge-type under all deformations of $(X\!\times \!\hat{X},\eta,h)$. When $X$ is the Jacobian of a genus $3$ curve, we reduce the proof of algebraicity of the Hodge Weil classes of deformations of $(X\!\times \!\hat{X},\eta,h)$ to a conjecture that an unobstructedness theorem of Buchweitz-Flenner for deformations of semiregular coherent sheaves generalizes to semiregular twisted reflexive sheaves.

Friday, February 23, 2024 - 4:00pm
LGRT 1681

Refreshments at 3:30PM.