On the power structure over the Grothendieck ring of varieties and its applications (original) (raw)

Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points

The Michigan Mathematical Journal, 2006

The power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety of dimension d as an exponent of that for the complex affine space A d . Specializations of this relation give formulae for generating series of such invariants of the Hilbert schemes of points as the Euler characteristic and the Hodge-Deligne polynomial.

A power structure over the Grothendieck ring of varieties

Mathematical Research Letters, 2004

Let R be either the Grothendieck semiring (semigroup with multiplication) of complex quasi-projective varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class L of the complex affine line. We define a power structure over these (semi)rings. This means that, for a power series A(t) = 1 + ∞ i=1 [A i ]t i with the coefficients [A i ] from R and for [M ] ∈ R, there is defined a series (A(t)) [M ] , also with coefficients from R, so that all the usual properties of the exponential function hold. In the particular case A(t) = (1 − t) −1 , the series (A(t)) [M ] is the motivic zeta function introduced by M. Kapranov. As an application we express the generating function of the Hilbert scheme of points, 0-dimensional subschemes, on a surface as an exponential of the surface.

Grothendieck Ring of Varieties with Actions of Finite Groups

Proceedings of the Edinburgh Mathematical Society, 2019

We define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized (‘motivic’) Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher-order Euler characteristics of wreath products.

Grothendieck ring of varieties with finite groups actions

2017

We define a Grothendieck ring of varieties with finite groups actions and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural lambda\lambdalambda-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized ("motivic") Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher order Euler characteristics of wreath products.

Generating series of classes of Hilbert schemes of points on orbifolds

Proceedings of the Steklov Institute of Mathematics

Let K 0 (V C) be the Grothendieck ring of complex quasi-projective varieties. This is the abelian group generated by isomorphism classes [X] of such varieties modulo the relation [X] = [X − Y ] + [Y ] for a Zariski closed subvariety Y ⊂ X; the multiplication is defined by the Cartesian product: [X 1 ] • [X 2 ] = [X 1 × X 2 ]. Let Hilb n X , n ≥ 1, be the Hilbert scheme of zero-dimensional subschemes of length n of a complex quasi-projective variety X. According to [4], Hilb n * Math. Subject Class.: 14C05, 14G10

Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups

Symmetry, 2019

The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order versions of the orbifold Euler characteristic and generalized (“motivic”) versions of them. In a previous paper, the authors defined a notion of the Grothendieck ring K 0 fGr ( Var C ) of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here, we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from K 0 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.

Characteristic classes of Hilbert schemes of points via symmetric products, Geometry & Topology 17

2016

We obtain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. This result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as on a formula for the generating series of the Hirzebruch homology characteristic classes of symmetric products. We apply the same methods for the calculation of generating series formulae for the Hirzebruch classes of the push-forwards of "virtual motives" of Hilbert schemes of a threefold. As corollaries, we obtain counterparts for the MacPherson (and Aluffi) Chern classes of Hilbert schemes of a smooth quasi-projective variety (resp. for threefolds). For a projective Calabi-Yau threefold, the latter yields a Chern class version of the dimension zero MNOP conjecture.

Characteristic classes of Hilbert schemes of points via symmetric products

2012

On Generating Series of Classes of Equivariant Hilbert Schemes of Fat Points

Moscow Mathematical Journal

In previous papers the authors gave formulae for generating series of classes (in the Grothendieck ring K 0 (V C) of complex quasiprojective varieties) of Hilbert schemes of zero-dimensional subschemes on smooth varieties and on orbifolds in terms of certain local data and the, so called, power structure over the ring K 0 (V C). Here we give an analogue of these formulae for equivariant (with respect to an action of a finite group on a smooth variety) Hilbert schemes of zerodimensional subschemes and compute some local generating series for an action of the cyclic group on a smooth surface.

Moduli of Representations, Quiver Grassmannians, and Hilbert Schemes

2015

It is a well established fact, that any projective algebraic variety is a moduli space of representations over some finite dimensional algebra. This algebra can be chosen in several ways. The counterpart in algebraic geometry is tautological: every variety is its own Hilber scheme of sheaves of length one. This holds even scheme theoretic. We use Beilinson's equivalence to get similar results for finite dimensional algebras, including moduli spaces and quiver grassmannians. Moreover, we show that several already known results can be traced back to the Hilbert scheme construction and Beilinson's equivalence.

On the power structure over the Grothendieck ring of varieties and its applications (original) (raw)

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