Fourier-stable subrings in the Chow rings of abelian varieties (Q2518156)

The author aims to generalize some results for Jjacobian varieties by \textit{A. Beauville} [Compos. Math. 140, No. 3, 683--688 (2004; Zbl 1062.14011)] and by himself [J. Alg. Geom. 16, No. 3, 459--476 (2007; Zbl 1123.14002)] to the case of an arbitrary abelian variety. Let \(A\) be an abelian variety of dimension \(g\) and let \({\mathcal{QT}}(A)\) denote the subring of the Chow ring \(CH^*(A)_Q\) with respect to the Pontryagin product generated by \(CH^g(A)_\mathbb{Q}\) and \(CH^{g-1}(A)_\mathbb{Q}\). The author shows that \({\mathcal{QT}}(A)\) is stable under the usual product and under the Fourier transform with respect to any polarization of \(A\). For the proof he employs an \(sl_2\)-triple associated with the polarization as in [\textit{K. Künnemann}, Invent. Math. 113, 85--102 (1993; Zbl 0806.14001)], and shows that \({\mathcal {QT}}(A)\) is stable under the \(sl_2\)-action. Furthermore, inspired by the appearance of differential operators, he proves that the intersection-product with any divisor class is a differential operator of order \(\leq 2\) on \(CH^*(A)\) with respect to the Pontryagin product.