arXiv subjects
Publications and source records attributed to Xiaojin Lin.
We prove that a nonconstant polynomial $f\in\mathbb{C}[x_1,x_2,x_3]$ is cylindrical (that is, $f\in\mathbb{C}[x_1,x_2]$ after an invertible linear change of coordinates) if and only if its bordered Hessian determinant vanishes identically. The same conclusion holds over $\mathbb{R}$. We also give explicit counterexamples to this criterion in every dimension $n\ge4$.
Classical hyperbolic uniformization identifies every hyperbolic log-orbi curve with a compactified quotient of the upper half-plane by a cofinite Fuchsian lattice. The lattice is unique up to conjugacy. We reconstruct it intrinsically. For each hyperbolic log-orbi curve C we construct a canonical maximal principal PSL2-Higgs object. Etale-locally it comes from the standard square-root SL2-model. The central mu2 ambiguity disappears after passage to PSL2. Using vector tame non-abelian Hodge theory and regular-singular Riemann--Hilbert as input we assemble the required principal realizations Tannakianly. Parahoric structures encode the orbifold and cusp data on the coarse curve. After choosing a base point and conjugating the Betti realization is represented by a discrete faithful finite-covolume representation whose image is the uniformizing lattice. Compatibility with finite etale pullback makes the lattice construction a quasi-inverse to the compactified quotient functor. Thus classical uniformization is recast as an intrinsic Tannakian reconstruction theorem. We also identify finite etale covers with finite continuous sets for the profinite completion of the reconstructed lattice. After fixing a separable closure and the resulting geometric generic point we recover the absolute Galois group of the function field of C as the inverse limit of the based etale fundamental groups of orbifold models over C.
We introduce the concept of parabolic bases to establish a localized framework for parabolic bundles and parabolic $\lambda$-connections. Building on this foundation, we propose a novel method for constructing the parabolic non-abelian Hodge correspondence in positive characteristic, extending the work originally developed by Krishnamoorthy and Sheng for algebraic curves. Additionally, we investigate the rank $2$ parabolic Higgs-de Rham flow operator and present a modified version of the Sun-Yang-Zuo algorithm, specifically adapted to the parabolic setting.
We establish a torsion theorem to the effect that the unique zero of the Kodaira-Spencer map attached to a certain quasi-semistable family of complex projective varieties over the complex projective line is the image of a torsion point of an elliptic curve under the natural projection. The proof is a mod $p$ argument and requires a density one set of primes. There are three essential ingredients in the proof: a solution to the conjecture of Sun-Yang-Zuo, which constitutes the principal part of the paper, Pink's theorem, and Higgs periodicity theorem.