Related papers: Infinitesimal Chow Dilogarithm

We construct an algebraic-cycle based model for the motivic cohomology on the category of schemes of finite type over a field, where schemes may admit arbitrary singularities and may be non-reduced. We show that our theory is functorial on…

In this article we prove a result comparing rationality of integral algebraic cycles over the function field of a quadric and over the base field. This is an integral version of the result known for coefficients modulo 2. Those results have…

New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…

Let k be an algebraically closed field of characteristic 0, let X=P^1\times A^N and let f be a rational endomorphism of X given by (x,y)--->(g(x), A(x)y), where g is a rational function, while A is an N-by-N matrix with entries in k(x). We…

In this paper certain Chow weight structures on the "big" triangulated motivic categories $DM_R^{eff}\subset DM_R$ are defined in terms of motives of all smooth varieties over the base field. This definition allows studying basic properties…

We give a presentation for the stack of rational curves with at most 1 node as the quotient by GL(3) of an open set in a 6-dimensional irreducible representation. We then use equivariant intersection theory to calculate the integral Chow…

Let $S$ be a smooth del Pezzo surface over a field $k$ of characteristic $\neq 2, 3$. We define an invariant in the Grothendieck-Witt ring $GW(k)$ for "counting" rational curves in a curve class $D$ of fixed positive degree (with respect to…

Let X be a smooth projective toric surface, and H^d(X) the Hilbert scheme parametrising the length d zero-dimensional subschemes of X. We compute the rational Chow ring A^*(H^d(X))\_Q. More precisely, if T is the two-dimensional torus…

A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path…

We introduce two operads which own the set of planar forests as a basis. With its usual product and two other products defined by different types of graftings, the algebra of planar rooted trees H becomes an algebra over these operads. The…

Let $C$ be a smooth projective irreducible curve defined over a finite field $\mathbb{F}_q$ and $K=\mathbb{F}_q(C)$. Let $A\subset K$ be the ring of functions regular outside a fixed place $\infty$ of $K$. Let…

We study the integral Chow ring of the stack $\mathcal{H}_{g,n}$ parametrizing $n$-pointed smooth hyperelliptic curves of genus $g$. We compute the integral Chow ring of $\mathcal{H}_{g,n}$ for $n=1,2$ completely, while for $3\leq…

The logarithmic Chow semistability is a notion of Geometric Invariant Theory for the pair consists of varieties and its divisors. In this paper we introduce a obstruction of semistability for polarized toric manifolds and its toric…

We introduce a new logarithmic structure on the moduli stack of stable curves, admitting logarithmic gluing maps. Using this we define cohomological field theories taking values in the logarithmic Chow cohomology ring, a refinement of the…

Let $C_{p,d}(\mathbb{P}^n)$ denote the Chow variety of effective algebraic $p$-cycles of degree $d$ in complex projective space $\mathbb{P}^n$. In this paper, we compute the rational Lawson homology groups…

We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3 surface and…

By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of…

We prove that for any monoid scheme M over a field with proper multiplication maps from M x M to M, we have a natural PD-structure on the ideal CH_{>0}(M) of CH(M) with regard to the Pontryagin ring structure. Further we investigate to what…

We study the problem of extending a complex structure to a given Lie algebra g, which is firstly defined on an ideal h of g. We consider the next situations: h is either complex or it is totally real. The next question is to equip g with an…

We prove a restriction isomorphism for Chow groups of zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore, we study torsion subgroups of these groups…