Some Motivic Properties of Gushel-Mukai Sixfolds

Abstract

Gushel-Mukai sixfolds are an important class of so-called Fano-K3 varieties. In this paper we show that they admit a multiplicative Chow-Künneth decomposition modulo algebraic equivalence and that they have the Franchetta property. As side results, we show that double EPW sextics and cubes have the Franchetta property, modulo algebraic equivalence, and some vanishing results for the Chow ring of Gushel-Mukai sixfolds.

Zero-cycles on double EPW sextics

“…We are thus reduced to proving the Franchettina property for Y 2 . This is readily done; the argument is very similar to that for Gushel-Mukai sixfolds in [5].…”

“…Notation 2.6. Let F (G) denote the Hilbert scheme parametrizing conics contained in Gr (2,5). Given an ordinary Gushel-Mukai fourfold Y , let F = F (Y ) be the Hilbert scheme of conics contained in Y .…”

Topics on Fano varieties of K3 type

“…This has been verified for some of the FK3 constructed in [FM21b] see e.g. [Lat20b,Lat21,BL22]. To this day, his work is still in progress, and we look forward to seeing more experimental verification of this conjecture.…”

The Tate Conjecture for even dimensional Gushel-Mukai varieties in characteristic $p\geq 5$

“…Again we write down all weights that occur; there are sixteen of them, two of which occur with multiplicity 3. We know we can only get a contribution to H j (Gr, Ω 3 Gr ) with j ≤ 2 if there exists an element w ∈ W of length at most 2 for which w • λ ∈ A. This happens for eight of the weights but in none of these cases w • λ is a dominant weight.…”

“…(Non-zero Plücker coordinates in positions 12 and 13, and again Q is singular in this point. )(3) with A = diag(−2, −3, 0, −2, −4). A singular point of Gr ∩Q is the point (0 : t 4 : 0 : 0 : 0 : 0 : 0 : −t 7 : 0 : 0).…”