本文首次开展双有理映射分解中心的研究，并针对完美域上的曲面完成了相关研究。我们证明：对于完美域k上光滑射影曲面X的任意双有理自同构φ: X ⇢ X，在φ的每一个弱分解中，胀开中心与胀缩中心都是同构的。这一结论表明，无法基于曲面的双有理自同构构造0维簇的非平凡L-等价关系。该结论同时意味着，有理性中心对每个有理曲面X都是良定义的，即存在一个X内蕴的0维簇，它在X的任意有理性构造中都会被胀开。
英文摘要：
We initiate the study of factorization centers of birational maps, and complete it for surfaces over a perfect field in this article. We prove that for every birational automorphism $ϕ: X \dashrightarrow X$ of a smooth projective surface $X$ over a perfect field $k$, the blowup centers are isomorphic to the blowdown centers in every weak factorization of $ϕ$. This implies that nontrivial L-equivalences of $0$-dimensional varieties cannot be constructed based on birational automorphisms of a surface. It also implies that rationality centers are well-defined for every rational surface $X$, namely there exists a $0$-dimensional variety intrinsic to $X$, which is blown up in any rationality construction of $X$.
We give sufficient cohomological criteria for the classes of given varieties over a field $k$ to be algebraically independent in the Grothendieck ring of varieties over $k$ and construct some examples.
We prove that abelian varieties of small dimension over discrete valuated, stricty henselian ground fields with perfect residue class field obtain semistable reduction after a tamely ramified extension of the ground field. Using this result we obtain perfectness results for Grothendieck's pairing.
We use Bott-Samelson resolutions of Schubert varieties in Grassmannians along with equiariant localization techniques to show that the factorial Schur functions and the factorial Grothendieck polynomials represent Schubert classes in equivariant cohomology and equivariant K-theory respectively.
We give an explicit stratification of the punctual Hilbert schemes of $n$ points of $\mathbb{A}^{m+1}$ with respect to $m$-dimensional partitions in the Grothendieck group of varieties. As an application, we calculate the classes of the punctual Hilbert schemes of $n$ points of $\mathbb{A}^3$ and the classes of the Hilbert schemes of $n$ points of $\mathbb{A}^3$ in the Grothendieck of varieties for $n\leq 5$.
It is shown that if the generalized Hodge conjecture, or some weaker form of it, holds for a Calabi-Yau variety then it holds for any Calabi-Yau variety birationally equivalent to it. The key idea is to construct suitable homomorphisms between the Grothendieck group of varieties and the Grothendieck group of the exact category of filtered Hodge structures. This along with motivic integration yields the first statement.
This is a small note meant to be published in a Conference Proceedings. We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by a finite group. Part of it reproduces the content of a letter dated September 27, 2008 addressed to Johannes Nicaise
For a scheme S with a good action of a finite abelian group G having enough roots of unity we show that the quotient map on the G-equivariant Grothendieck ring of varieties over S is well defined with image in the Grothendieck ring of varieties over S/G in the tame case, and in the modified Grothendieck ring in the wild case. To prove this we use a result on the class of the quotient of a vector space by a quasi-linear action in the Grothendieck ring of varieties due to Esnault and Viehweg, which we also generalize to the case of wild actions. As an application we deduce that the quotient of t…
We construct a presentation for the Grothendieck group of Deligne-Mumford stacks over a field of characteristic zero. The generators for this presentation are smooth, proper Deligne-Mumford stacks and the relations are expressed in terms of stacky blow-ups. In the process we prove a version of the weak factorization theorem for Deligne-Mumford stacks.
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 $λ$-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 t…
We prove that for a normal projective variety $X$ in characteristic 0, and a base-point free ample line bundle $L$ on it, the restriction map of divisor class groups $\Cl(X)\to \Cl(Y)$ is an isomorphism for a general member $Y\in |L|$ provided that $\dim{X}\geq 4$. This is a generalization of the Grothendieck-Lefschetz Theorem, for divisor class groups of singular varieties.
We introduce a Grothendieck group of algebraic stacks (with affine stabilisers) analogous to the Grothendieck group of algebraic varieties. We then identify it with a certain localisation of the Grothendieck group of algebraic varieties. Several invariants of elements in this group are discussed. The most important is an extension of the Euler characteristic (of cohomology with compact support) but in characteristic zero we introduce invariants which are able to distinguish between classes with the same Euler characteristic. These invariants are actually defined on the completed localised Grot…
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 as an exponent of that for the complex affine space of the same dimension. 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.
In this paper, we discuss the problem of whether the difference $[X]-[Y]$ of the classes of a Fourier--Mukai pair $(X, Y)$ of smooth projective varieties in the Grothendieck ring of varieties is annihilated by some power of the class $\mathbb{L} = [ \mathbb{A}^1 ]$ of the affine line. We give an affirmative answer for Fourier--Mukai pairs of very general K3 surfaces of degree 12. On the other hand, we prove that in each dimension greater than one, there exists an abelian variety such that the difference with its dual is not annihilated by any power of $\mathbb{L}$, thereby giving a negative an…
Let $\operatorname{K}_0(\operatorname{Var}_k)$ denote the Grothendieck ring of $k$-varieties over an algebraically closed field $k$. Larsen and Lunts asked if two $k$-varieties having the same class in $\operatorname{K}_0 (\operatorname{Var}_k)$ are piecewise isomorphic. Gromov asked if a birational self-map of a $k$-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group. Furthermore, if $\math…
We establish a formula for the classes of certain tori in the Grothendieck ring of varieties, in terms of its lambda-structure. More explicitly, we will see that if L* is the torus of invertible elements in the n-dimensional separable k-algebra L, then the class of L* can be expressed as an alternating sum of the images of the spectrum of L under the lambda-operations, multiplied by powers of the Lefschetz class. This formula is suggested from the cohomology of the torus, illustrating a heuristic method that can be used in other situations. To prove the formula will require some rather explici…
The notion of the orbifold Euler characteristic came from physics at the end of 80's. 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 $K_0^{\rm fGr}(Var)$ 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^{\rm fGr}(V…
In each dimension n>2 there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly.
Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck-Katz $p$-Curvature Conjecture for certain locally symmetric varieties, including the moduli space of abelian varieties ${\cal A}_g$ when $g > 1.$
A power structure over a ring is a method to give sense to expressions of the form $(1+a_1t+a_2t^2+\ldots)^m$, where $a_i$, $i=1, 2,\ldots$, and $m$ are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of $λ$- and power structures over some Grothendieck rings of varieties. The main example is for the Grothendieck ring of maps of complex quasi-projective varieties. We describe two natural $λ$-structures on it which lead to the same power structure. We s…
We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any elliptic curve $A$ in any characteristic.
Let X be a projective hypersurface in P_k^n of degree d <= n. In this paper we study the relation between the class [X] in K_0(Var_k) and the existence of k-rational points. Using elementary geometric methods we show, for some particular X, that X(k) is nonempty if and only if [X] is equivalent to 1 modulo the class of the affine line in K_0(Var_k). More precisely we consider the following cases: a union of hyperplanes, a quadric, a cubic hypersurface with a singular k-rational point, and a quartic which is a union of two quadrics one of which being smooth.
We prove, that Hausel's formula for the number of rational points of a Nakajima quiver variety over a finite field also holds in a suitable localization of the Grothendieck ring of varieties. In order to generalize the arithmetic harmonic analysis in his proof we use Grothendieck rings with exponentials as introduced by Cluckers-Loeser and Hrushovski-Kazhdan.
We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several refinements and generalizations of the Feit-Fine formula. Our main application is to motivic Donaldson-Thomas theory.
This note describes the subring of the Grothendieck ring of k-varieties generated by smooth conics; finding many zero divisors. The proof uses only elementary projective geometry.
We define a special type of hypersurface varieties inside $\mathbb{P}_k^{n-1}$ arising from connected planar graphs and then find their equivalence classes inside the Gröthendieck ring of projective varieties. Then we find a characterization for graphs in order to define irreducible hypersurfaces in general.
We prove a conjecture of A. S. Buch concerning the structure constants of the Grothendieck ring of a flag variety with respect to its basis of Schubert structure sheaves. For this, we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities, have alternating signs. Equivalently, the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of Schubert varieties.
We prove that Grothendieck's Hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the Hodge conjecture holds for complex abelian varieties of CM-type. For abelian varieties with no exotic algebraic classes, we prove the Hodge standard conjecture unconditionally.
We introduce a Grothendieck ring of higher Artin stacks generalizing the Grothendieck ring of algebraic varieties. We show that this ring is not trivial by noticing that it factors the invariant "number of rational points over a finite field". We also introduce the notion of "special Artin stacks", which by definition have affine homotopy groups π_{i}, and furthermore unipotent for i>1. Our principal theorem states that the natural inclusion morphism from the Grothendieck ring of varieties to the Grothendieck ring of special Artin stacks is an isomorphism after inverting the class of the affin…
Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group generated by the geometrically reduced varieties over k, modulo the relations of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of varieties makes K_0(V_k) into a ring. We prove that if the characteristic of k is zero, then K_0(V_k) is not a domain.
We discuss the notion of a power structure over a ring and the geometric description of the power structure over the Grothendieck ring of complex quasi-projective varieties and show some examples of applications to generating series of classes of configuration spaces (for example, nested Hilbert schemes of J. Cheah) and wreath product orbifolds.
Let $A_K$ be an abelian variety with semistable reduction over a strictly henselian field of positive characteristic with perfect residue class field. We show that there is a close connection between the pairings of Grothendieck, Bester/Bertapelle and Shafarevic. In particular, we show that the pairing of Bester/Bertapelle can be used to describe the $p$-part of Grothendieck's pairing in the semistable reduction case, thus proving a conjecture of Bertapelle.
We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. We furthermore conjecture that the coefficients in our formula have signs which alternate with degree. The proof of our formula involves $K$-theoretic generalizations of several useful cohomological tools, including the Thom-Porteous formula, the Jacobi-Trudi formula, and a Gysin formula of Pragacz.
We show that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. The argument is based on the Pfaffian-Grassmannian double mirror correspondence.
Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between the Grothendieck groups of these categories. In particular, we derive a binomial formula and use it to give explicit expressions for the classes of universal quasi-split tori in the equivariant Grothendieck group of varieties.
The category of effective Grothendieck-Witt-motives $\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ (and Witt-motives $\mathbf{DM}^W_{\mathrm{eff},-}(k)$) by Voevodsky-Suslin method starting with some category of GW-correspondences (and Witt-correspondences) over a perfect field $k$, $char\,k\neq 2$, is defined. The functor $M^{GW}_{eff}\colon Sm_k\to \mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ of Grothendieck-Witt-motives of smooth varieties is computed and it is proved that for any smooth scheme $X$ and homotopy invariant sheave with GW-transfers $F$ $$ Hom_{\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)}(M^{GW}_{ef…
We introduce a new perspective on the $K$-theory of exact categories via the notion of a CGW-category. CGW-categories are a generalization of exact categories that admit a Qullen $Q$-construction, but which also include examples such as finite sets and varieties. By analyzing Quillen's proofs of dévissage and localization we define ACGW-categories, an analogous generalization of abelian categories for which we prove theorems analogous to dévissage and localization. In particular, although the category of varieties is not quite ACGW, the category of reduced schemes of finite type is; applying d…
We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection $X$ of three quadrics in ${\mathbf P}^5$ and the corresponding double cover $Y \to {\mathbf P}^2$ branched over a sextic curve. We show that as soon as the natural Brauer class on $Y$ vanishes, so that $X$ and $Y$ are derived equivalent, …
A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the th…
Let $\cal R$ be either the Grothendieck semiring (semiring with multiplication) of complex algebraic varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We introduce a construction which defines operations of taking powers of series over these (semi)rings. This means that, for a power series $A(t)=1+\sum\limits_{i=1}^\infty A_i t^i$ with the coefficients $A_i$ from $\cal R$ and for $M\in {\cal R}$, there is defined a series $(A(t))^M$ (with coefficients from $\cal R$ as well) so that all the usual properties of the…
In this article, by combining the recent theory of noncommutative motives with the classical theory of motives, we prove that if two quadrics (or, more generally, two involution varieties) have the same Grothendieck class, then they have the same even Clifford algebra and the same signature. As an application, we show in numerous cases (e.g., when the base field is a local or global field) that two quadrics (or, more generally, two involution varieties) have the same Grothendieck class if and only if they are isomorphic.
Lev A. Borisov has shown that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. We improve the final formula by removing a factor.
In this note we discuss a generalization ("extension") of the pre-lambda structure on the Grothendieck ring of quasi-projective varieties and of the corresponding power structure over it to the Grothendieck ring of stacks, discuss some of their properties and give some explicit formulae for the Kapranov zeta function for some stacks.
In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.
Motivated by [Bor] and [Mar], we show the equality $\left([X] - [Y]\right) \cdot [\mathbb{A}^1] = 0$ in the Grothendieck ring of varieties, where $(X, Y)$ is a pair of Calabi-Yau 3-folds cut out from the pair of Grassmannians of type $G_2$.
We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham-Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures of Grothendieck and Kontsevich-Zagier concerning transcendence properties of the torsors of periods of varieties over number fields. We notably establish that the Grothendieck period conjecture holds in degree 1 for products of curves, of abelian varieties, and of K3 surfaces, and that it holds in degre…
We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety $X$ over a complete discretely valued field $K$ with perfect residue field $k$. If $K$ has characteristic zero, we extend the definition to arbitrary $K$-varieties using Bittner's presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties. The motivic Serre invariant can be considered as a measure for the set of unramified points on $X$. Under certain tameness conditions, it admits a cohomological interpretation by means of a trace formula. In the curve case, we use T. Sait…
We consider the "limiting behavior" of *discriminants*, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motiva…
The purpose of the present paper is to generalize Sakai's work on anticanonical models of rational surfaces to varieties of Fano type. We first prove a characterization of Fano type varieties using the singularities of anticanonical models. Secondly, we study the decomposition of the anticanonical map using the $K_X$-minimal model program.
