arXiv subjects

Paulo Lima-Filho

Publications and source records attributed to Paulo Lima-Filho.

13 recordsLinked to original sources

arXiv2024

LULU is syntomic

Let $G$ be a Chevalley group over a field $\Bbbk$. Fix a maximal torus $\mathbb{T}$ in $ G $, along with opposite Borel subgroups $ B $ and $ B^-$ satisfying $ \mathbb{T} = B \cap B^-$, and denote by $ \mathbb{U} := R_u(B) $ and $ \mathbb{L} := R_u(B^-)$ their respective unipotent radicals. We prove that the multiplication map $ μ\colon \mathbb{L} \times \mathbb{U}\times \mathbb{L} \times \mathbb{U} \longrightarrow G $ is syntomic and faithfully flat over any base field $ \Bbbk$.

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arXiv2023

Multiplicative properties of the current transform regulator

This paper utilizes the properties of transforms of currents under equidimensional cycles, as introduced in \cite{MR4498559}, to establish the multiplicative nature of the resulting regulator map, in the derived category. The construction relies on a synthetic presentation of the fundamental triples of currents from \cite{MR4498559}, which exhibits group-like behavior under an extended Eilenberg-Zilber morphism. A key component of the analysis is a character of the permutation Hopf algebra that takes values in the function field of an infinite-dimensional affine space.

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arXiv2023

Explicit Chern Cycles in BGL and KV-theory

Using determinantal schemes, we construct explicit cycles in the higher Chow complex of BGL that represent the universal Chern classes in higher Chow groups. As an application, we use these cycles, along with a canonical \emph{stable moving lemma} for Karoubi-Villamayor $K$-theory, to give a direct construction of the Chern class homomorphisms $c_{p,r}$ from the r-th KV group of a regular k-algebra over a field $ k $, to the higher Chow group $ CH^p(Spec(R),r) $.

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arXiv2019

An algebraic geometric realization of the Chern character

Using symmetrized Grassmannians we give an algebraic geometric presentation, in the level of classifying spaces, of the Chern character and its relation to Chern classes. This allows one to define, for any projective variety $X$, a Chern character map from the "holomorphic K-theory of X to its morphic cohomology (introduced by Friedlander and Lawson). The holomorphic K-theory of $X$, introduced by Lawson, Lima-Filho and Michelsohn and also by Friedlander and Walker, is defined in terms a group-completion of the space of algebraic morphisms from X into BU. It has been further studied by the authors in a companion paper. Holomorphic K-theory sits between algebraic K-theory and topological K-theory in the same way that morphic cohomology sits between motivic cohomology and ordinary cohomology. Our constructions provide a bridge between these two worlds. We also realize Chern classes in the case where X is smooth, and establish a universal relation between the Chern character and the Chern classes. We use classical constructions with algebraic cycles and infinite symmetric products of projective spaces. The latter can be seen as the classifying space for motivic cohomology, and under this perspective our constructions are essentially motivic.

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arXiv2019

Regulator Maps for Higher Chow Groups via Current Transforms

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 involving the Mähler measure of Laurent polynomials.

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arXiv2013

Arithmetic toric varieties

We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K/k is cyclic, and we also study k-forms of surfaces.

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arXiv2009

Holonomy Lie algebras and the LCS formula for subarrangements of A_n

If X is the complement of a hypersurface in P^n, then Kohno showed that the nilpotent completion of the fundamental group is isomorphic to the nilpotent completion of the holonomy Lie algebra of X. When X is the complement of a hyperplane arrangement A, the ranks phi_k of the lower central series quotients of the fundamental group of X are known for isolated examples, and for two special classes: if X is hypersolvable (in which case the quadratic closure of the cohomology ring is Koszul), or if the holonomy Lie algebra decomposes in degree three as a direct product of local components. In this paper, we use the holonomy Lie algebra to obtain a formula for phi_k when A is a subarrangement of A_n. This extends Kohno's result for braid arrangements, and provides an instance of an LCS formula for arrangements which are not decomposable or hypersolvable.

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arXiv2008

Integral Deligne Cohomology for Real Varieties

We develop an integral version of Deligne cohomology for smooth proper real varieties. For this purpose the role played by singular cohomology in the complex case has to be replaced by ordinary bigraded G-equivariant cohomology, where G=Gal(C/R). This is the G-equivariant counterpart of singular cohomology. We establish the basic properties of the theory and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2.

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arXiv2006

Bigraded Equivariant Cohomology of Real Quadrics

We give a complete description of the bigraded Bredon cohomology ring of smooth projective real quadrics, with coefficients in the constant Mackey functor $ \mathbf{Z} $. These invariants are closely related to the integral motivic cohomology ring, which is not known for these varieties. Some of the results and techniques introduced can be applied to other geometrically cellular real varieties.

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arXiv2005

Algebraic cycles and the classical groups II: Quaternionic cycles

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space P(V), where V carries a real structure, and completely determined their homotopy type. We also extended some functors in K-theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel-Whitney classes. In this sequel, we establish corresponding results in the case where V has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a similarly simple description. The stabilized space of quaternionic algebraic cycles admits a nontrivial infinite loop space structure yielding, in particular, a delooping of the total Pontrjagin class map. This stabilized space is directly related to an extended notion of quaternionic spaces and bundles (KH-theory), in analogy with Atiyah's real spaces and KR-theory, and the characteristic classes that we introduce for these objects are nontrivial. The paper ends with various examples and applications.

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arXiv1999

Holomorphic K-theory, algebraic co-cycles, and loop groups

In this paper we study the "holomorphic K-theory" of a projective variety, which is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory was introduced by Lawson, Lima-Filho and Michelsohn, and also by Friedlander and Walker, and a related theory was considered by Karoubi. Using the Chern character studied by the authors in a companion paper, we give a rational isomorphism between holomorphic $K$-theory and the "morphic cohomology" defined by Lawson and Friedlander. In doing so, we describe a geometric model for rational morphic cohomology groups in terms of algebraic maps from the variety to the "symmetrized loop group" Loops(U(n)/S_n) where the symmetric group S_n acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K-theory by inverting the Bott class, then it is rationally isomorphic to topological K-theory. Finally we produce explicit obstructions to periodicity in holomorphic K-theory, and show that these obstructions vanish for generalized flag manifolds.

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arXiv1999

Algebraic Cycles and the Classical Groups - Part I, Real Cycles

Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or quaternionic algebraic cycles on P(V). In this paper and its sequel the homotopy structure of these cycle groups is determined. They bear a direct relationship to characteristic classes for the classical groups, and functors in K-theory extend directly to these groups. These groups give rise to E-infinity-ring spaces, and the maps extending the K-theory functors are ring maps. The stabilized space of cycles is a product of (Z/2Z)-equivariant Eilenberg-MacLane spaces indexed by the representations R^{n,n} for n > 0. This gives a wide generalization of the results in Boyer, Lawson, Lima-Filho, Mann and Michelsohn on the Segal question. The ring structure on the homotopy groups of these stabilized spaces is explicitly computed. In the real case it is a quotient of a polynomial algebra on two generators corresponding to the first Pontrjagin and first Stiefel-Whitney classes. This yields an interesting total characteristic class for real bundles. It is a mixture of integral and mod 2 classes and has nice multiplicative properties. The class is shown to be the (Z/2Z)-equivariant Chern class on Atiyah's KR-theory.

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arXiv1998

Chow quotients and projective bundle formulas for Euler-Chow series

Given a projective algebraic variety $X$, let $Π_p(X)$ denote the monoid of effective algebraic equivalence classes of effective algebraic cycles on $X$. The $p$-th Euler-Chow series of $X$ is an element in the formal monoid-ring $Z[[Π_p(X)]]$ defined in terms of Euler characteristics of the Chow varieties $\cvpd{p}α{X}$ of $X$, with $α\inΠ_p(X)$. We provide a systematic treatment of such series, and give projective bundle formulas which generalize previous results by B. Lawson and S.S.Yau and Elizondo. The techniques used involve the Chow quotients introduced by Kapranov, and this allows the computation of various examples including some Grassmannians and flag varieties. There are relations between these examples and representation theory, and further results point to interesting connections between Euler-Chow series for certain varieties and the topology of the closure of moduli spaces $M_{0,n+1}$.

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