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Multiplicative Chow-Künneth decompositions and varieties of cohomological \(K3\) type. (English) Zbl 1475.14008

A Chow-Künneth decomposition of a smooth projective variety \(X\) is a direct-sum decomposition of its rational Chow motive. This in particular says that the diagonal in \(X\times X\) can be written as a sum of orthogonal projectors to the graded parts of the Chow ring. By Murre’s conjecture, every smooth projective variety is supposed to have a Chow-Künneth decomposition. There is a natural notion of multiplicativity of a Chow-Künneth decomposition and admitting a multiplicative Chow-Künneth decomposition (MCK) is a cycle theoretic property. The question arises which varieties admit an MCK decomposition and the article under review contributes to it.
M. Shen and C. Vial conjectured in [The Fourier transform for certain hyperkähler fourfolds. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1386.14025)] that hyperkähler varieties admit an MCK. On the other hand, it is known that Fano varieties and canonically polarized varieties do not in general admit an MCK. Examples can be found in the present article (Example 2.11 due to Beauville and Example 3.3 due to Fu-Vial and Ceresa). The question thus becomes to find classes of varieties with ample or anti-ample canonical sheaf that do admit an MCK decomposition. In the present article, the authors collect evidence that on the Fano side, varieties which are cohomologically of \(K3\) type have an MCK decomposition. This is proven for cubic fourfolds (in fact, for cubic hypersurfaces in general, see Corollary 1.3 and Theorem 5.4) and for Küchle fourfolds of type c7 (Theorem 6.2). Also for canonically polarized varieties, the authors establish the existence of an MCK decomposition for two families of Todorov surfaces (Theorem 1.4). Contrary to previously known examples, these surfaces are not birational to products of curves.

MSC:

14C15 (Equivariant) Chow groups and rings; motives
14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J45 Fano varieties
14J42 Holomorphic symplectic varieties, hyper-Kähler varieties

Citations:

Zbl 1386.14025
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References:

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