The integral model of a G U ( n − 1 , 1 ) \mathrm {GU}(n-1,1) Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet L L -functions. We also determine the arithmetic volumes of Kudla-Rapoport divisors and relate them to coefficients of Eisenstein series.

We state several questions, and prove some partial results, about the Chow ring $A^{\ast}(X)$ of complete intersections in projective space. For one thing, we prove that if $X$ is a general Calabi–Yau hypersurface, the intersection product $A^{2}(X) \cdot A^{i}(X)$ is one-dimensional, for any $i > 0$. We also show that quintic threefolds have a multiplicative Chow–Künneth (MCK) decomposition. We wonder whether all Calabi–Yau hypersurfaces might have an MCK decomposition, and prove this is the case conditional to a conjecture of Voisin.

We show that the normal points of a cubic hypersurface in projective space have canonical singularities unless the hypersurface is an iterated cone over an elliptic curve. As an application, we give a simple linear algebraic description of all the hyperplane sections of projectivized tangent bundle of projective space, hence describing hyperplane sections of a rational homogeneous manifold of Picard rank 2 2 . This also simplifies and extends recent results of Mazouni-Nagaraj in higher dimensions. We also compute the Chow ring of these hyperplane sections.

Abstract Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the Hodge-Riemann relations of degree zero and one. Motivated by fundamental properties of volume polynomials of Chow rings of simplicial fans, we define a class of multivariate polynomials which we call hereditary polynomials. We give a complete and easily checkable characterization of hereditary Lorentzian polynomials. This characterization is used to give elementary and simple proofs of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a matroid, and the Alexandrov-Fenchel inequalities for convex bodies. We then characterize Chow rings of simplicial fans which satisfy the Hodge-Riemann relations of degree zero and one, and we prove that this property only depends on the support of the fan. Several different characterizations of Lorentzian polynomials on cones are provided.

The Chow Ring of a Sequence of Point and Rational Curve Blow-Ups

The integral Chow rings of the moduli stacks of hyperelliptic Prym pairs I

The Chow ring of the universal Picard stack over the hyperelliptic locus

Correction to: Two Cycle Class Maps on Torsion Cycles

Abstract We prove a formula for the cycle class of the supersingular locus in the Chow ring with rational coefficients of the moduli space of principally polarized abelian varieties of dimension g in characteristic p. This formula determines this class as a monomial in the Chern classes of the Hodge bundle up to a factor that is a polynomial in p. This factor is known for $$g\le 3$$ g ≤ 3 . We also determine the factor for $$g=4$$ g = 4 .

The integral Chow ring of R_2

Abstract The tautological Chow ring of the moduli space $\mathcal{A}_{g}$ A g of principally polarized abelian varieties of dimension $g$ g was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from $\mathcal{A}_{g}$ A g to the moduli space $\mathcal{M}_{g}^{\operatorname{ct}}$ M g ct of genus $g$ g of curves of compact type, we prove that the product class $[\mathcal{A}_{1}\times \mathcal{A}_{5}]\in \mathsf{CH}^{5}( \mathcal{A}_{6})$ [ A 1 × A 5 ] ∈ CH 5 ( A 6 ) is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the tautological ring $\mathsf{R}^{*}(\mathcal{M}_{6}^{\operatorname{ct}})$ R ∗ ( M 6 ct ) in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring $\mathsf{R}^{*}(\mathcal{M}_{6}^{\operatorname{ct}})$ R ∗ ( M 6 ct ) has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of $[\mathcal{A}_{1}\times \mathcal{A}_{5}]$ [ A 1 × A 5 ] . More generally, the Torelli pullback of the difference between $[\mathcal{A}_{1}\times \mathcal{A}_{g-1}]$ [ A 1 × A g − 1 ] and its tautological projection always lies in the Gorenstein kernel of $\mathsf{R}^{*}(\mathcal{M}_{g}^{\operatorname{ct}})$ R ∗ ( M g ct ) . The product map $\mathcal{A}_{1}\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_{g}$ A 1 × A g − 1 → A g is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer’s tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all $g$ g are presented.

Abstract We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas’s results on an integral version of Grothendieck–Riemann–Roch. If S is smooth quasi-projective of dimension d over a field and $$\pi :X\rightarrow S$$ π : X → S is a g -dimensional abelian scheme, we prove, under very mild assumptions on X / S , that all classical results about Fourier duality, including the existence of a Beauville decomposition, are valid for the Chow ring $$\textrm{CH}(X;\Lambda )$$ CH ( X ; Λ ) with coefficients in the ring $$\Lambda = \mathbb {Z}[1/(2g+d+1)!]$$ Λ = Z [ 1 / ( 2 g + d + 1 ) ! ] . If X admits a polarization $$\theta $$ θ of degree $$\nu (\theta )^2$$ ν ( θ ) 2 we further construct an $$\mathfrak {sl}_2$$ sl 2 -action on $$\textrm{CH}(X;\Lambda _\theta )$$ CH ( X ; Λ θ ) with $$\Lambda _\theta = \Lambda [1/\nu (\theta )]$$ Λ θ = Λ [ 1 / ν ( θ ) ] , and we show that $$\textrm{CH}(X;\Lambda _\theta )$$ CH ( X ; Λ θ ) is a sum of copies of the symmetric powers $$\textrm{Sym}^n(\textrm{St})$$ Sym n ( St ) of the 2-dimensional standard representation, for $$n=0,\ldots ,g$$ n = 0 , … , g . For an abelian variety over an algebraically closed field, we use our results to produce torsion classes in $$\textrm{CH}^i(X;\Lambda _\theta )$$ CH i ( X ; Λ θ ) for every $$i\in \{1,\ldots ,g\}$$ i ∈ { 1 , … , g } .

Consider the canonical morphism from the Chow ring of a smooth variety X to the associated graded ring of the topological filtration on the Grothendieck ring of X . In general, this morphism is not injective. However, Nikita Karpenko conjectured that these two rings are isomorphic for a generically twisted flag variety X of a semisimple group G . The conjecture was first disproved by Nobuaki Yagita for G = Spin ( 2 n + 1 ) with n = 8 , 9 . Later, another counter-example to the conjecture was given by Karpenko and the first author for n = 10 . In this note, we provide an infinite family of counter-examples to Karpenko’s conjecture for any 2 -power integer n greater than 4 . This generalizes Yagita’s counter-example and its modification due to Karpenko for n = 8 .

Abstract We study the Hilbert series and the representations of ${\mathfrak{S}}_{n}$ and $GL_{n}(\mathbb{F}_{q})$ on the (augmented) Chow rings of uniform matroids $U_{r,n}$ and $q$-uniform matroids $U_{r,n}(q)$. The Frobenius series for uniform matroids and their $q$-analogs are computed. As a byproduct, we recover Hameister, Rao, and Simpson’s formula for the Hilbert series of Chow rings of $q$-uniform matroids in terms of permutations and further obtain their augmented counterpart in terms of decorated permutations. We also show that the equivariant Charney–Davis quantity of the (augmented) Chow ring of a matroid is nonnegative (i.e., a genuine representation of a group of automorphisms of the matroid). When the matroid is a uniform matroid and the group is ${\mathfrak{S}}_{n}$, the representation either vanishes or is a Foulkes representation (i.e., a Specht module of a ribbon shape). Specializing to the usual Charney–Davis quantities, we obtain an elegant combinatorial interpretation of Hameister, Rao, and Simpson’s formula for Chow rings of $q$-uniform matroids and its augmented counterpart.

The integral chow ring of the stack of pointed hyperelliptic curves

Straightening laws for Chow rings of matroids

Abstract We give an explicit quadratic Gröbner basis for generalized Chow rings of supersolvable built lattices, with the help of the operadic structure on geometric lattices introduced in a previous article. This shows that the generalized Chow rings associated to minimal building sets of supersolvable lattices are Koszul. As another consequence, we get that the cohomology algebras of the components of the extended modular operad in genus $0$ are Koszul.

The integral Chow ring of weighted blow-ups

We investigate the real cycle class map for singular varieties. We introduce an analog of Borel–Moore homology for algebraic varieties over the real numbers, which is defined via the hypercohomology of the Gersten–Witt complex associated with schemes possessing a dualizing complex. We show that the hypercohomology of this complex is isomorphic to the classical Borel–Moore homology for quasi-projective varieties over the real numbers.

Chow Ring of Horospherical Variety with Picard Number 1