James D. Lewis
We give an explicit description of a real regulator from the cohomology of a Milnor complex associated to a projective algebraic manifold, to a certain quotient of Deligne cohomology.
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Milnor K-theory, F-isocrystals and Syntomic Regulators
该研究引入带滤过的F-等晶范畴，构造与对数syntomic上同调的syntomic符号映射相容的Milnor K理论符号映射，为syntomic调节子的相关应用研究提供基础材料。
Building on Kerr, Lewis and Mueller-Stach's work on the rational regulator, we prove the existence of an integral regulator on higher Chow complexes and give an explicit expression. This puts firm ground under some earlier results and speculations on the torsion in higher cycle groups by K-L-M, Petras, and Kerr-Yang.
Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as ``the non-crossing algebra'', which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre $F$ of $W$. In the process we define a new, larger algebra $\widetilde{A}$, which seems to be ``dual'' to the Fomin-Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection bet…
We study a natural Hodge module on the Hilbert scheme of four points on affine three-space, which categorifies the Donaldson--Thomas invariant of the Hilbert scheme. We determine the weight filtration on the Hodge module explicitly in terms of intersection cohomology complexes, and compute the E-polynomial of its cohomology. The computations make essential use of a description of the singularity of the Hilbert scheme as the degeneracy locus of the Pfaffian function.
We discuss several approaches to motivic complexes and explicit constructions of the regulator maps from the motivic complexes to Deligne complexes.
We study Milnor fibers of complexified real line arrangements. We give a new algorithm computing monodromy eigenspaces of the first cohomology. The algorithm is based on the description of minimal CW-complexes homotopic to the complements, and uses the real figure, that is, the adjacency relations of chambers. It enables us to generalize a vanishing result of Libgober, give new upper-bounds and characterize the $A_3$-arrangement in terms of non-triviality of Milnor monodromy.
We show that the cohomology of a rank 1 local system on the complement of a projective hyperplane arrangement can be calculated by the Aomoto complex in certain cases even if the condition on the sum of the residues of connection due to Esnault et al is not satisfied. For this we have to study the localization of Hodge-logarithmic differential forms which are defined by using an embedded resolution of singularities. As an application we can compute certain monodromy eigenspaces of the first Milnor cohomology group of the defining polynomial of the reflection hyperplane arrangement of type $G_{…
We define a regulator map from the weight n polylogarithmic motivic complex to the weight n Deligne complex of an algebraic variety X. The regulator map is constructed explicitly via the classical polylogarithms with some funny combinations of Bernoulli numbers as coefficients. This leads to conjectures on special values of L-functions at s=n which reduce to Zagier's conjecture when X is of dimension zero over Q. The cone of this map, shifted by one, should be called the Arakelov motivic complex of X. Its last cohomology group is identified with a group of codimension n Arakelov cycles on X.
We construct an explicit regulator map from the weigh n Bloch Higher Chow group complexto the weight n Deligne complex of a regular complex projective algebraic variety X. We define the Arakelovweight n motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet. and C.Soule. We relate the Grassmannian n-logarithms (defined as in [G5]) to geometry of the symmetric space for GL_n(C). For n=2 we recover Lobachevsky's formula for the volume of an ideal geodesic tetrahedron via the dilogarithm. Using the relations…
We show how to use equidimensional algebraic correspondences between complex algebraic varieties to construct pull-backs and transforms of certain classes of geometric currents. Using this construction we produce explicit formulas at the level of complexes for a regulator map from the Higher Chow groups of smooth complex quasi-projective algebraic varieties to Deligne-Beilinson cohomology with integral coefficients. A distinct aspect of our approach is the use of Suslin's complex of equidimensional cycles over $ Δ^n $ to compute Bloch's higher Chow groups. We calculate explicit examples involv…
We introduce exponential complexes of sheaves on manifolds. They are resolutions of the (Tate twisted) constant sheaves of the rational numbers, generalising the short exact exponential sequence. There are canonical maps from the exponential complexes to the de Rham complex. Using this, we introduce new complexes calculating the rational Deligne cohomology. We call them exponential Deligne complexes. Their advantage is that, at least at the generic point of a complex variety, one can define Beilinson's regulator map to the rational Deligne cohomology on the level of complexes. Namely, we defin…
We define complexes analogous to Goncharov's complexes for the K-theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K-theory, there is a map from the cohomology of those complexes to the K-theory of the ring. In case the ring is the localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the K-theory with the syntomic regulator. The result can be described in terms of a p-adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on B…
In 1986, Spencer Bloch gave an abstract definition of a (regulator) map from higher Chow groups to Deligne-Beilinson cohomology. This map can be defined on the underlying complexes, and Kerr, Lewis and Müller-Stach gave an explicit description of this map in terms of currents. Using a multiplicative version of the Deligne complex, we give a commutative version of this map.
Isotypical components of the homology of ICIS and images of deformations of map germs
本文提出研究有限群作用单纯复形同调同型分量的方法，并用于ICIS的Milnor纤维，系统分析映射芽形变之像的同调与像Milnor数的推广，修正并证明Houston猜想。
We prove that there is a map from Bloch's cycle complex to Kato's complex of Milnor K-theory, which induces a quasi-isomorphism from étale sheafified cycle complex to the Gersten complex of logarithmic de Rham--Witt sheaves. Next we show that the truncation of Bloch's cycle complex at -3 is quasi-isomorphic to Spiess' dualizing complex.
Torus-enriched Motivic Bruhat Complexes and Maximal Compact Groups
本文在特征零完美域上，针对连通分裂半单单连通群构造环面富集动机胞腔复形，通过多种理论关联与计算，实现了外尔复形与旗复形的插值并揭示高阶微分。
Lotuses as computational architectures
本研究提出可将莲（一类有限可缩单纯复形）作为计算架构，关联到十字构型以逐步计算复约化平面曲线奇点的多项不变量，并给出正特征下的调整方案。
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