A decomposition theorem for singular spaces with trivial canonical class of dimension at most five

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

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Polynomial identities related to special Schubert varieties

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00496-6 ◽

2021 ◽

Author(s):

Francesca Cioffi ◽

Davide Franco ◽

Carmine Sessa

Keyword(s):

Arbitrary Number ◽

Schubert Variety ◽

Decomposition Theorem ◽

Point Of View ◽

Explicit Description ◽

Intersection Cohomology ◽

Schubert Varieties ◽

Polynomial Identities ◽

Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

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On complete intersections with trivial canonical class

Advances in Mathematics ◽

10.1016/j.aim.2014.08.013 ◽

2015 ◽

Vol 268 ◽

pp. 339-349

Author(s):

Lev A. Borisov ◽

Zhan Li

Keyword(s):

Complete Intersections ◽

Canonical Class

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Type II degenerations of K3 surfaces

Nagoya Mathematical Journal ◽

10.1017/s0027763000021462 ◽

1985 ◽

Vol 99 ◽

pp. 11-30 ◽

Cited By ~ 3

Author(s):

Shigeyuki Kondo

Keyword(s):

Number Field ◽

Complex Manifold ◽

Complex Number ◽

Three Dimensional ◽

K3 Surfaces ◽

Type Ii ◽

Holomorphic Map ◽

Canonical Class ◽

Proper Holomorphic Map ◽

Complex Number Field

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

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Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

manuscripta mathematica ◽

10.1007/s00229-021-01294-7 ◽

2021 ◽

Author(s):

V. Cortés ◽

A. Saha ◽

D. Thung

Keyword(s):

Curvature Tensor ◽

Decomposition Theorem ◽

Kähler Manifolds ◽

Kahler Manifolds ◽

Riemann Curvature ◽

Riemann Curvature Tensor ◽

Cohomogeneity One ◽

Civita Connection ◽

Levi Civita Connection ◽

Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

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