The cycle class of the supersingular locus of principally polarized abelian varieties

The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge bundle over $A_g$ has its Chern classes in the Chow ring of $A_g$ with rational coefficients. We show that over the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so in the best possible way: despite the highly singular nature of $A_g^*$ they are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements in its bivariant Chow ring. This is in contrast to the situation in the analytic topology, where these Chern classes have canonical lifts to the complex cohomology of the minimal compactification as Goresky-Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology.

This note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

We study the Chow ring with rational coefficients of the moduli space F 2 \mathcal{F}_{2} of quasi-polarized K3 surfaces of degree 2. We find generators, relations, and calculate the Chow Betti numbers. The highest nonvanishing Chow group is A 17 ⁢ ( F 2 ) ≅ Q \mathsf{A}^{17}(\mathcal{F}_{2})\cong{\mathbb{Q}} . We prove that the Chow ring consists of tautological classes and is isomorphic to the even cohomology. The Chow ring is not generated by divisors and does not satisfy duality with respect to the pairing into A 17 ⁢ ( F 2 ) \mathsf{A}^{17}(\mathcal{F}_{2}) . The kernel of the pairing is a 1-dimensional subspace of A 9 ⁢ ( F 2 ) \mathsf{A}^{9}(\mathcal{F}_{2}) which we calculate explicitly. In the appendix, we revisit Kirwan–Lee’s calculation of the Poincaré polynomial of F 2 \mathcal{F}_{2} .

The Chow ring of the classifying space [formula omitted

We determine the Chow ring (with Q-coecients) of M6 by showing that all Chow classes are tautological. (In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective.) We stratify the moduli space into locally closed strata which are group quotients, and use a theorem of Vistoli to show that their Chow rings are generated by Chern classes of natural vector bundles. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus six case relies in addition on the particularly beautiful Brill{Noether theory in this case, and in particular on a rank ve vector bundle \relativizing a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the rank six vector bundle of quadrics cutting out the canonical curve.

Let U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, andclr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

For geometrical triangulated motives with rational coefficients over a ground field of characteristic zero which is embeddable into \({\mathbb{C}}\) , Huber (J. Algebraic Geom. 9(4):755–799, 2000; J. Algebraic Geom. 13(1):195–207, 2004) has constructed a realization functor with values in the category of mixed realization of Huber (Mixed motives and their realization in derived categories. Lecture Notes in Mathematics, vol. 1604. Springer, Berlin, 1995). In this sequel to Ivorra (Doc Math 12:607–671, 2007), we prove that the l-adic realization functor obtained in Theorem 4.3 of Ivorra (Doc Math 12:607–671, 2007) is the same up to a canonical isomorphism as the l-adic component of A. Huber’s construction. In this way (Ivorra in Doc Math 12:607–671, 2007) might be viewed as an integral generalization to all noetherian separed schemes of the work (Huber in J. Algebraic Geom. 9(4):755–799, 2000; J. Algebraic Geom. 13(1):195–207, 2004) as far as the l-adic setting is concerned. We also prove a comparison theorem with the classical l-adic cycle class map over a perfect field using a naive motivic cycle class map.

Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-G\"ottsche, we define the orbifold motive (or Chen-Ruan motive) of the quotient stack $[M/G]$ as an algebra object in the category of Chow motives. Inspired by Ruan, one can formulate a motivic version of his Cohomological HyperK\"ahler Resolution Conjecture. We prove this motivic version, as well as its K-theoretic analogue conjectured by Jarvis-Kaufmann-Kimura, in two situations related to an abelian surface $A$ and a positive integer $n$. Case (A) concerns Hilbert schemes of points of $A$ : the Chow motive of $A^{[n]}$ is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack $[A^{n}/\mathfrak{S}_{n}]$. Case (B) for generalized Kummer varieties : the Chow motive of the generalized Kummer variety $K_n(A)$ is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack $[A_{0}^{n+1}/\mathfrak {S}_{n+1}]$, where $A_{0}^{n+1}$ is the kernel abelian variety of the summation map $A^{n+1}\to A$. As a byproduct, we prove the original Cohomological HyperK\"ahler Resolution Conjecture for generalized Kummer varieties. As an application, we provide multiplicative Chow-K\"unneth decompositions for Hilbert schemes of abelian surfaces and for generalized Kummer varieties. In particular, we have a multiplicative direct sum decomposition of their Chow rings with rational coefficients, which is expected to be the splitting of the conjectural Bloch-Beilinson-Murre filtration. The existence of such a splitting for holomorphic symplectic varieties is conjectured by Beauville.

We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree k has dimension at most k-1. Building on the work of Pirola, we show that very general abelian varieties of dimension g have (covering) gonality at least f(g), where f(g) grows like log g. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties A of dimension g, e.g., if g≥2k-1, the set of divisors D∈ Pic 0 (A) such that D k =0 in CH k (A) is at most countable.

We study the product structure on the Chow ring (with rational coefficients) of a cubic hypersurface in projective space and prove that the image of the product map is as small as possible.

We introduce and study a filtration on the representation ring $R(G)$ of an affine algebraic group $G$ over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with the other known filtrations on $R(G)$ and show that all three define on $R(G)$ the same topology. For any $n\geq 1$, we compute the Chow filtration on $R(G)$ for the special orthogonal group $G:=O^+(2n+1)$. In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of $G$ over any field of characteristic $\ne 2$ is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety $X$ such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of $X$.

The B P ∗ BP^* -module structure of B P ∗ ( B G ) BP^*(BG) for extraspecial 2 2 -groups is studied using transfer and Chern classes. These give rise to p p -torsion elements in the kernel of the cycle map from the Chow ring to ordinary cohomology first obtained by Totaro.

An Alexander self-dual complex gives rise to a compactification of $${{\cal M}_{0,n}}$$ , called an ASD compactification, which is a smooth algebraic variety. ASD compactifications include (but are not exhausted by) the polygon spaces, or the configuration spaces of flexible polygons. We present an explicit description of the Chow rings of ASD compactifications. We study the analogs of Kontsevich’s tautological bundles, compute their Chern classes, compute top intersections of the Chern classes, and derive a recursion for the intersection numbers.

This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincaré's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.

The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First, we prove a Beauville–Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilbert square of a K3 surface, we show that the action of the anti-symplectic involution on the Chow group of zero-cycles commutes with the Fourier decomposition of Shen–Vial.