Murre’s conjectures and explicit Chow–Künneth projectors for varieties with a nef tangent bundle

Abstract

Abstract. In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow-Künneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle explicit Chow-Künneth projectors are obtained which satisfy Murre's conjectures and the motivic Hard Lefschetz theorem is verified.

Algebraic cycles on a very special EPW sextic

“…Remark 2.5. The following varieties have finite-dimensional motive: abelian varieties, varieties dominated by products of curves [32], K3 surfaces with Picard number 19 or 20 [38], surfaces not of general type with p g = 0 [23, Theorem 2.11], certain surfaces of general type with p g = 0 [23], [40], [55], Hilbert schemes of surfaces known to have finite-dimensional motive [13], generalized Kummer varieties [57, Remark 2.9(ii)], [21], threefolds with nef tangent bundle [27], [47,Example 3.16], fourfolds with nef tangent bundle [28], log-homogeneous varieties in the sense of [12] (this follows from [28,Theorem 4.4]), certain threefolds of general type [49,Section 8], varieties of dimension ≤ 3 rationally dominated by products of curves [47,Example 3.15], varieties X with A Clearly, if Y has finite-dimensional motive then also X = Y /G has finite-dimensional motive. The nilpotence theorem extends to this set-up: Proposition 2.8.…”

Niveau and coniveau filtrations on cohomology groups and Chow groups

“…The corollary above thus settles Murre's conjecture (D) for abelian 3-folds. Let's also mention that Iyer[14] proves Murre's conjectures for 3-folds with a nef tangent bundle. These include abelian 3-folds.…”

“…Finally, in the fourth section, we settle Murre's conjectures (except perhaps for the 'independency' conjecture) in some new cases. Examples of varieties for which Murre's conjectures are known to hold true include uniruled 3-folds [2], 3-folds with a nef tangent bundle [14] and elliptic modular 3-folds [11]. Our new examples include 3-folds rationally dominated by the product of three curves, rationally connected 3-folds, Calabi-Yau 3-folds, abelian 3-folds, rationally connected 4-folds, 4-folds admitting a curve as their base for their maximal rationally connected fibration, rationally connected 5-folds with vanishing Hodge number h 3,1 , and some complete intersections of low degree, for example, cubic 6-folds, quartic 7-folds and the smooth intersection of two quadrics in P 10 .…”

Some cubics with finite-dimensional motive

“…The following varieties have finite-dimensional motive: abelian varieties, varieties dominated by products of curves [15], K3 surfaces with Picard number 19 or 20 [22], surfaces not of general type with p g = 0 [7, Theorem 2.11], certain surfaces of general type with p g = 0 [7], [23], [34], Hilbert schemes of surfaces known to have finite-dimensional motive [4], generalized Kummer varieties [36, Remark 2.9(ii)], threefolds with nef tangent bundle [9] (an alternative proof is given in [30, Example 3.16]), fourfolds with nef tangent bundle [10], log-homogeneous varieties in the sense of [3] Remark 4 It is an embarrassing fact that up till now, all examples of finite-dimensional motives happen to lie in the tensor subcategory generated by Chow motives of curves, i.e. they are "motives of abelian type" in the sense of [30].…”