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Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)$
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Claire Voisin

Print publication date: 2014

Print ISBN-13: 9780691160504

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691160504.001.0001

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On the Chow ring of K3 surfaces and hyper-Kahler manifolds

On the Chow ring of K3 surfaces and hyper-Kahler manifolds

Chapter:
(p.88) Chapter Five On the Chow ring of K3 surfaces and hyper-Kahler manifolds
Source:
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)
Author(s):

Claire Voisin

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691160504.003.0005

This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was discovered that they have a canonical 0-cycle o of degree 1 with the property that the product of two divisors of S is a multiple of o in CH₀(S). This result would later be extended to Calabi–Yau hypersurfaces in projective space. The chapter also considers a decomposition in CH(X × X × X)ℚ of the small diagonal Δ‎ ⊂ X × X × X that was established for K3 surfaces, and is partially extended to Calabi–Yau hypersurfaces. Finally, the chapter uses this decomposition and the spreading principle to show that for families π‎ : XB of smooth projective K3 surfaces, there is a decomposition isomorphism that is multiplicative over a nonempty Zariski dense open set of B.

Keywords:   K3 surfaces, hyper-Kähler manifolds, abelian varieties, Calabi–Yau hypersurfaces, projective space, decomposition, spreading principle, decomposition isomorphism

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