arXiv subjects

Xiaopeng Xia

Publications and source records attributed to Xiaopeng Xia.

5 recordsLinked to original sources

arXiv2026

On the image of Hitchin morphism for some classical groups on algebraic surfaces

In this article, we study the image of the Hitchin morphism for some classical groups over an algebraic surface. The Hitchin morphism is a map from the moduli stack of $G$-Higgs bundles $\mathscr{M}_{X,G}$ to the Hitchin base $\mathscr{A}_{X,G}$, where $X$ is a smooth projective variety. In general, this morphism is not surjective when the dimension of $X$ is greater than one. Chen and Ng{ô} showed that the Hitchin morphism factors through a closed subscheme $\mathscr{B}_{X,G}$ of the Hitchin base, which is called the spectral base. They conjectured that the image of the Hitchin morphism is exactly the spectral base. When $X$ is a smooth projective surface, we prove that this conjecture holds for the special linear algebraic group of odd rank. We also confirm this conjecture for the classical groups ${\rm SL}_n$ and ${\rm Sp}_{2n}$ when $X$ is a product of smooth curves.

math.AG↗

arXiv2025

Mod-$\ell$ monodromy of double covers of $\mathbb{P}^n$ branched along hyperplane arrangements

We determine the mod-$\ell$ geometric monodromy group of the universal family of double covers of projective space branched along hyperplane arrangements in general position.

math.AG↗

arXiv2025

On the normality of commuting scheme for general linear Lie algebra

The commuting scheme $\mathfrak{C}^{d}_{\mathfrak{g}}$ for reductive Lie algebra $\mathfrak{g}$ over an algebraically closed field $\mathbb{K}$ is the subscheme of $\mathfrak{g}^{d}$ defined by quadratic equations, whose $\mathbb{K}$-valued points are $d$-tuples of commuting elements in $\mathfrak{g}$ over $\mathbb{K}$. There is a long-standing conjecture that the commuting scheme $\mathfrak{C}^{d}_{\mathfrak{g}}$ is reduced. Moreover, a higher dimensional analog of Chevalley restriction conjecture was conjectured by Chen-Ngô. We show that the commuting scheme of $\mathfrak{C}^{2}_{\mathfrak{g}l_{n}}$ is Cohen-Macaulay and normal. As a corollary, we prove a 2-dimensional Chevalley restriction theorem for general linear group in positive characteristic.

math.AG↗

arXiv2024

A higher-dimensional Chevalley restriction theorem for classical groups in characteristic p

We establish a theorem concerning the commuting scheme in characteristic p. As a significant application of this theorem, we derive an explicit lower bound for the characteristic p, ensuring the validity of the higher-dimensional Chevalley restriction theorem for classical groups.

math.AG↗

arXiv2023

A higher-dimensional Chevalley restriction theorem for orthogonal groups

We prove a higher-dimensional Chevalley restriction theorem for orthogonal groups, which was conjectured by Chen and Ngô for reductive groups. In characteristic $p>2$, we also prove a weaker statement. In characteristic $0$, the theorem implies that the categorical quotient of a commuting scheme by the diagonal adjoint action of the group is integral and normal. As applications, we deduce some trace identities and a certain multiplicative property of the Pfaffian over an arbitrary commutative algebra.

math.RT↗