arXiv subjects

Runjie Hu

Publications and source records attributed to Runjie Hu.

11 recordsLinked to original sources

arXiv2026

Covering Projective Height Balls by Subspaces in Rigid Adelic Spaces

Let $E$ be an $n$-dimensional rigid adelic space over a number field $K$. We study the minimum number $g_E(R)$ of proper $K$-subspaces needed to cover the projective height ball of radius $R$, together with the maximum cardinality $h_E(R)$ of a subset in linear general position. We show that, once $R$ is sufficiently large compared with the last Roy--Thunder minimum of $E$, both quantities have order $\Psi_E(R)^{[K:\mathbb Q]}$, where $\Psi_E(R)$ is an explicit expression in the Roy--Thunder minima. The comparison constants are effectively computable and uniform in $E$. For the standard adelic space $K^n$, this gives order $R^{[K:\mathbb Q]n/(n-1)}$.

math.NT↗

arXiv2026

$\mathbb{Q}_p$-Homotopy Types and Applications to Topology and Algebraic Geometry

We develop a $\mathbb{Q}_p$-homotopy theory for $p$-complete spaces. To a $p$-complete space $X$, we associate a commutative differential graded algebra over $\mathbb{Q}_p$ by rectifying the $E_\infty$-algebra $S^*(X;\widehat{\mathbb{Z}}_p)\otimes_{\widehat{\mathbb{Z}}_p} \mathbb{Q}_p$ of singular cochains. For nilpotent $p$-complete finite type spaces, we prove that the minimal model of this algebra recovers the $\mathbb{Q}_p$-homotopy groups and Whitehead products, in direct analogy with Sullivan's rational homotopy theory. We also prove that, for a non-simply-connected $p$-complete space, the Lie algebra dual to its $1$-minimal model is the Lie algebra of the continuous Mal'cev $\mathbb{Q}_p$-completion of the fundamental group. We apply the $\mathbb{Q}_p$-homotopy theory to several questions in topology and algebraic geometry, including finite realization problems for $p$-complete spaces, finiteness properties of \'etale homotopy types, formality of smooth proper varieties, Galois representations on \'etale homotopy groups, and constraints on \'etale fundamental groups.

math.AT↗

arXiv2026

Continuous Mal'cev Qp-Completion of Pro-p Groups

We give three explicit constructions of the continuous Mal'cev Qp-completion of a topologically finitely generated pro-p group, using Tannakian formalism, Hopf algebras and p-adic analytic groups. We study properties of the continuous Mal'cev Qp-completion via these explicit constructions.

math.GR↗

arXiv2026

Non-liftable varieties via etale cohomology rings

We construct a smooth projective variety in positive characteristic whose $\mathbb{Q}_{\ell}$-coefficient etale cohomology ring is not the scalar extension of any graded $\mathbb{Q}$-algebra, providing an example of a new type of obstruction to characteristic zero liftability.

math.AG↗

arXiv2025

A necessary condition for liftings of positive characteristic varieties with finite fundamental groups

In this paper, we introduce a necessary condition for the existence of characteristic zero liftings of certain smooth, proper varieties in positive characteristic, using etale homotopy theory and Wall's finiteness obstruction. For a variety with finite etale fundamental group pi, we define a notion of mod-l finite dominatedness based on the F_l-chain complex of the universal cover of its l-profinite etale homotopy type. We prove that such a variety X can be lifted to characteristic zero only if the above chain complex of X is quasi-isomorphic to a bounded complex of finitely generated projective F_l[pi]-modules. To prove this result, we extend Wall's discussions of finiteness obstructions to l-profinite complete spaces with finite fundamental group.

math.AT↗

arXiv2025

Formal Manifold Structures on Positive Characteristic Varieties

In his 1970 ICM report, Sullivan proposes the program of l-adic formalization of the concept of manifolds. In this program, he claims that smooth positive characteristic varieties should carry l-adic formal manifold structures. He also claims the existence of an abelianized Galois symmetry on l-adic formal manifold structures. This paper carries out this program, establishes the claims for certain varieties, and relates the abelianized Galois symmetry on l-adic formal manifold structures to the Galois symmetry of varieties. Meanwhile, we prove that a simply-connected variety is l-adic homotopic equivalent to a simply-connected finite CW complex if and only if the l-profinite completion of its etale homotopy type admits an l-local lifting.

math.AT↗

arXiv2023

Galois Symmetry of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on Topological Manifold Structures of Varieties

We propose a definition of the profinite normal structure set for the set of all manifolds in a fixed profinite homotopy type. Using this framework, we prove that the Galois action of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on the underlying topological manifold structures of smooth, complete, simply-connected complex varieties defined over $\overline{\mathbb{Q}}$ of dimension at least $3$ factors through the abelianization of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Moreover, this abelian action extends canonically to the entire profinite normal structure set. This result provides an answer to the question by Sullivan in the case of topological manifold structures of simply-connected varieties.

math.AT↗

arXiv2023

A homotopical consequence of branched covers

We prove that the profinite completion of a pseudomanifold is the Artin-Mazur's etale homotopy type construction on its branched covers, which was implicitly conjectured by Sullivan in his MIT note (page 247) around 1970. This is a consequence of the existence of enough $K(\pi,1)$ open dense subspaces in a pseudomanifold.

math.AT↗

arXiv2023

$L$-theory Characteristic Classes

Although the local information of the $L$-spectra is well understood, the problem of whether this local information can be identified with the geometric data for bundles remains open for decades, which was originally raised in the 1960s and 1970s by Sullivan, Brumfiel, Taylor-Williams and others independently. In this paper, we provide an affirmative answer by proving that Levitt-Ranicki's theory of connective $L$-orientations for $TOP$ bundles and spherical fibrations is equivalent to the $2$-local characteristic classes constructed by Brumfiel-Morgan's, Madsen-Milgram's and Morgan-Sullivan's, as well as Sullivan's odd-prime-local real $K$-theory orientation. A key step in our proof involves constructing more geometric homotopy equivalences from the $2$-local quadratic, symmetric and normal connective $L$-spectra to products of Eilenberg-Maclane spectra and those from odd-local quadratic and symmetric connective $L$-spectra to the connective real $K$-spectra. This approach reproves the known local structure of $L$-spectra.

math.AT↗

arXiv2020

Planck Galactic Cold Clumps in Two Regions: the First Quadrant and the Anti-Center Direction Region

Sixty five Planck Galactic cold clumps (PGCCs) from the first quadrant (IQuad) and thirty nine of PGCCs from the Anti-Center direction region (ACent) were observed in $^{12}$CO, $^{13}$CO and C$^{18}$O J=1-0 lines using the PMO 13.7-m telescope. All the targets were detected with all the three lines, except for 12 IQuad and 8 ACent PGCCs without C$^{18}$O detection. Seventy six and 49 velocity components were obtained in IQuad and ACent respectively. One-hundred and forty-six cores were extracted from 76 IQuad clumps and 100 cores from 49 ACent clumps. The average T$_{\mathrm{ex}}$ of IQuad cores and ACent cores are 12.4 K and 12.1 K, respectively. The average line width of $^{13}$CO of IQuad cores and ACent cores are 1.55 km s$^{-1}$ and 1.77 km s$^{-1}$, respectively. Among the detected cores, 24 in IQuad and 13 in ACent have asymmetric line profiles. The small blue excesses, $\sim$0.03 in IQuad and 0.01 in ACent, indicate that the star formation is not active in these PGCC cores. Power-law fittings of core mass function to the high mass end give indexes of -0.57 in IQuad and -1.02 in ACent which are flatter than the slope of initial mass function given by \citeauthor{1955ApJ...121..161S}. The large turnover masses with value of 28 M$_{\odot}$ for IQuad cores and 77 M$_{\odot}$ for ACent cores suggest low star formation efficiencies in PGCCs. The correlation between virial mass and gas mass indicates that most of PGCC cores in both regions are not likely pressure-confined.

astro-ph.GA↗

arXiv2016

Dense gas in molecular cores associated with Planck Galactic cold clumps

We present the first survey of dense gas towards Planck Galactic Cold Clumps (PGCCs). Observations in the J=1-0 transitions of HCO+ and HCN towards 621 molecular cores associated with PGCCs were performed using the Purple Mountain Observatory 13.7-m telescope. Among them, 250 sources have detection, including 230 cores detected in HCO+ and 158 in HCN. Spectra of the J=1-0 transitions from CO, 13CO, and C18O at the centers of the 250 cores were extracted from previous mapping observations to construct a multi-line data set. The significantly low detection rate of asymmetric double-peaked profiles, together with the well consistence among central velocities of CO, HCO+, and HCN spectra, suggests that the CO-selected Planck cores are more quiescent compared to classical star-forming regions. The small difference between line widths of C18O and HCN indicates that the inner regions of CO-selected Planck cores are not more turbulent than the exterior. The velocity-integrated intensities and abundances of HCO+ are positively correlated with those of HCN, suggesting these two species are well coupled and chemically connected. The detected abundances of both HCO+ and HCN are significantly lower than values in other low- to high-mass star-forming regions. The low abundances may be due to beam dilution. On the basis of the inspection of the parameters given in the PGCC catalog, we suggest that there may be about 1 000 PGCC objects having sufficient reservoir of dense gas to form stars.

astro-ph.GA↗