Searcharxiv⌕ Search

arXiv subjects

Jyh-Haur Teh

Publications and source records attributed to Jyh-Haur Teh.

16 recordsLinked to original sources
arXiv2026

Semi-topological Galois cohomology and Weierstrass realizability

Semi-topological Galois theory associates a canonical finite splitting covering to a monic Weierstrass polynomial. The inverse limit of the corresponding deck groups defines the absolute semi-topological Galois group, $\PiST(X,x)$. This paper develops a cohomology theory for $\PiST(X,x)$ with discrete torsion coefficients, establishing its fundamental properties and canonical comparison maps to singular cohomology. A Lyndon-Hochschild-Serre spectral sequence is used to yield an obstruction theory for semi-topological embedding problems. We prove several structural and vanishing results, including ST-fullness for free fundamental groups and triviality for finite fundamental groups. As applications, we provide a criterion for lifting finite projective monodromy to linear monodromy, formulate the $π_1$-detectable Weierstrass realizability conjecture for divisor classes and show that this conjecture is true for abelian varieties, smooth complex projective curves and ruled surfaces over positive-genus curves.

math.AT↗
arXiv2026

Simplicial Cheeger-Simons models and simplicial higher abelian gauge theory

A pair $(K,K')$ consisting of a smooth triangulation $K$ of a compact smooth oriented Riemannian manifold $M$ and a sufficiently fine subdivision $K'$ determines a finite-dimensional Cheeger--Simons model $\mathscr{CS}(K,K')$ built from Whitney-type data on the induced curvilinear complexes. Its associated differential character groups $\Diff^{\bullet}(\mathscr{CS}(K,K'))$ provide a simplicial, finite-dimensional counterpart of the Cheeger--Simons differential characters $\widehat H^{\bullet}(M)$. We prove that every smooth triangulation admits a subdivision $K'$ for which $(K,K')$ is a Cheeger--Simons triangulation in this sense. Under a uniform fullness (shape-regularity) hypothesis, we show that the natural discretization/extension maps between $\widehat H^{k}(M)$ and $\Diff^{k}(\mathscr{CS}(K,K'))$ approximate the identity in a Sobolev-dual seminorm as $\mesh(K')\to 0$. For closed $M$, we further identify $\widehat H^{k}(M)$ canonically with the inverse limit of $\Diff^{k}(\mathscr{CS}(K,K'))$ over refinements. As an application, we formulate a simplicial higher abelian gauge theory whose gauge-invariant configuration space is $\Diff^{p}(\mathscr{CS}(K,K'))$, and we prove that the resulting simplicial (regularized) partition function converges, in the refining limit, to the corresponding smooth regularized partition function of Kelnhofer.

math-ph↗
arXiv2026

Topological fundamental groups of locally finite infinite configuration spaces and infinite braids

We study the topological fundamental groups of the locally finite infinite ordered configuration space \(Conf^{lf}_\infty(\C)\) in the plane and the homotopy quotient of $Conf^{lf}_\infty$ by the canonical action of the infinite permutation group $\Aut(\N)$: \[ H^{lf}(\infty):=π_1^{\mathrm{top}}(Conf^{lf}_\infty(\C),\widetilde{\N}), \qquad B^{lf}(\infty):=π_1^{\mathrm{top}}\!\bigl(Conf^{lf}_\infty(\C)\!/\!/\Aut(\N),[e_0,\widetilde{\N}]\bigr). \] We prove that \(H^{lf}(\infty)\) and \(B^{lf}(\infty)\) are non-discrete and complete topological groups. A main structural theorem identifies \(H^{lf}(\infty)\) with a canonical locally finite inverse-limit model built from finite pure braid groups, and we construct a complete left-invariant ultrametric compatible with the quotient topology from the loop space of $\Conf$. The direct limit of finite pure braid groups admits a dense embedding into \(H^{lf}(\infty)\), and we show that \(H^{lf}(\infty)\) is the Ra\uıkov completion of this subgroup. Moreover, the direct limit of finite braid groups embeds into \(B^{lf}(\infty)\) and is dense in the finitary subgroup \(B^{lf}_{\mathrm{fin}}(\infty)\subseteq B^{lf}(\infty)\).

math.AT↗
arXiv2025

The asphericity of locally finite infinite configuration spaces and Weierstrass entire coverings

Let $Conf^{lf}_{\infty}(\C)$ and $C^{lf}_{\infty}(\C)$ denote the locally finite infinite ordered and unordered configuration spaces of the complex plane. We prove that both $Conf^{lf}_{\infty}(\C)$ and $C^{lf}_{\infty}(\C)$ are aspherical. We further obtain a locally finite analogue of the braid exact sequence, \[ 1\longrightarrow H^{lf}(\infty)\longrightarrow B^{lf}(\infty)\longrightarrow \Aut(\N)\longrightarrow 1, \] where $H^{lf}(\infty)=π_1(Conf^{lf}_{\infty}(\C))$ and $B^{lf}(\infty)=π_1(Conf^{lf}_{\infty}(\C)//\Aut(\N))$, the fundamental group of the homotopy quotient of $Conf^{lf}_{\infty}(\C)$ by $\Aut(\N)$. Building on this, we classify connected countably infinite--sheeted covering spaces and give a criterion for when such a covering can be realized from the zero set of a family of entire functions $F:X\times\C\to\C$. In particular, if $π_1(X)$ is free and $H^2(X;\Z)=0$, then every countably infinite--sheeted covering space over $X$ is realizable.

math.AT↗
arXiv2020

Real rectifiable currents, holomorphic chains and algebraic cycles

We study some fundamental properties of real rectifiable currents and give a generalization of King's theorem in characterizing currents defined by positive real holomorphic chains. Our proof uses Siu's semicontinuity theorem and largely simplifies King's proof. A consequence of this result is a sufficient condition for the Hodge conjecture.

math.DG↗
arXiv2019

Bott-Chern homology, Bott-Chern differential cohomology and the Hodge conjecture

We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bott-Chern classes for holomorphic vector bundles in this differential cohomology. These refined Bott-Chern classes transform naturally to standard Chern classes, Bott-Chern classes and Cheeger-Simons' refined Chern classes.

math.CV↗
arXiv2016

Aeppli-Bott-Chern cohomology and Deligne cohomology from a viewpoint of Harvey-Lawson's spark complex

By comparing Deligne complex and Aeppli-Bott-Chern complex, we construct a differential cohomology $\widehat{H}^*(X, *, *)$ that plays the role of Harvey-Lawson spark group $\widehat{H}^*(X, *)$, and a cohomology $H^*_{ABC}(X; \Z(*, *))$ that plays the role of Deligne cohomology $H^*_{\mathcal{D}}(X; \Z(*))$ for every complex manifold $X$. They fit in the short exact sequence $$ 0\rightarrow H^{k+1}_{ABC}(X; \Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{δ_1}{\rightarrow} Z^{k+1}_I(X, p, q) \rightarrow 0$$ and $\widehat{H}^{\bullet}(X, \bullet, \bullet)$ possess ring structure and refined Chern classes, acted by the complex conjugation, and if some primitive cohomology groups of $X$ vanish, there is a Lefschetz isomorphism. Furthermore, the ring structure of $H^{\bullet}_{ABC}(X; \Z(\bullet, \bullet))$ inherited from $\widehat{H}^{\bullet}(X, \bullet, \bullet)$ is compatible with the one of the analytic Deligne cohomology $H^{\bullet}(X; \Z(\bullet))$. We compute $\widehat{H}^*(X, *, *)$ for $X$ the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.

math.DG↗
arXiv2016

$E_1$-degeneration and $d'd''$-lemma

For a double complex $(A, d', d'')$, we show that if it satisfies the $d'd''$-lemma and the spectral sequence $\{E^{p, q}_r\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d'd''$-lemma.

math.AT↗
arXiv2015

Aeppli and Bott-Chern cohomology for bi-generalized Hermitian manifolds and $d'd''$-lemma

We define Aeppli and Bott-Chern cohomology for bi-generalized complex manifolds and show that they are finite dimensional for compact bi-generalized Hermitian manifolds. For totally bounded double complexes $(A, d', d'')$, we show that the validity of $d'd''$-lemma is equivalent to having the same dimension of several cohomology groups. Some calculations of Bott-Chern cohomology groups of some bi-generalized Hermitian manifolds are given.

math.DG↗
arXiv2010

Semi-topological Galois theory and the inverse Galois problem

We enhance the analogy between field extensions and covering spaces by introducing the concept of splitting covering which correspondences to the splitting field in Galois theory. We define semi-topological Galois groups for Weierstrass polynomials and prove the existence of a Galois correspondence. This new tool enables us to study the inverse Galois problem from a new viewpoint.

math.GR↗
arXiv2008

Motivic integration and projective bundle theorem in morphic cohomology

We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycles are $K$-statements. We define stringy functions which enable us to ask stringy Grothendieck standard conjecture and stringy Hodge conjecture. We prove a projective bundle theorem in morphic cohomology for trivial bundles over any normal quasi-projective varieties.

math.AG↗
arXiv2007

Grothendieck standard conjectures, morphic cohomology and Hodge index theorem

Using morphic cohomology, we produce a sequence of conjectures, called morphic conjectures, which terminates at the Grothendieck standard conjecture A. A refinement of Hodge structures is given, and with the assumption of morphic conjectures, we prove a Hodge index theorem. We answer a question of Friedlander and Lawson by assuming the Grothendieck standard conjecture B and prove that the topological filtration from morphic cohomology is equal to the Grothendieck arithmetic filtration for some cases.

math.AG↗
arXiv2007

A homology and cohomology theory for real projective varieties

In this paper we develop homology and cohomology theories which play the same role for real projective varieties that Lawson homology and morphic cohomology play for projective varieties respectively. They have nice properties such as the existence of long exact sequences, the homotopy invariance, the Lawson suspension property, the homotopy property for bundle projection, the splitting principle, the cup product, the slant product and the natural transformations to singular theories. The Friedlander-Lawson moving lemma is used to prove a duality theorem between these two theories. This duality theorem is compatible with the $\Z_2$-Poincaré duality for real projective varieties with connected full real points.

math.AG↗
arXiv2006

Harnack-Thom Theorem for higher cycle groups and Picard varieties

We generalize the Harnack-Thom theorem to relate the ranks of the Lawson homology groups with $\Z_2$-coefficients of a real quasiprojective variety with the ranks of its reduced real Lawson homology groups. In the case of zero-cycle group, we recover the classical Harnack-Thom theorem and generalize the classical version to include real quasiprojective varieties. We use Weil's construction of Picard varieties to construct reduced real Picard groups, and Milnor's construction of universal bundles to construct some weak models of classifying spaces of some cycle groups. These weak models are used to produce long exact sequences of homotopy groups which are the main tool in computing the homotopy groups of some cycle groups of divisors. We obtain some congruences involving the Picard number of a nonsingular real projective variety and the rank of its reduced real Lawson homology groups of divisors.

math.AG↗
arXiv2006

Complexification of real cycles and Lawson suspension theorem

We embed the space of totally real $r$-cycles of a totally real projective variety into the space of complex $r$-cycles by complexification. We provide a proof of the holomorphic taffy argument in the proof of Lawson suspension theorem by using Chow forms and this proof gives us an analogous result for totally real cycle spaces. We use Sturm theorem to derive a criterion for a real polynomial of degree $d$ to have $d$ distinct real roots and use it to prove the openness of some subsets of real divisors. This enables us to prove that the suspension map induces a weak homotopy equivalence between two enlarged spaces of totally real cycle spaces.

math.AG↗