arXiv subjects

Xianyu Hu

Publications and source records attributed to Xianyu Hu.

6 recordsLinked to original sources

arXiv2026

Coherent sheaves in logarithmic geometry

This paper introduces an abelian category of logarithmic coherent sheaves that arranges coherent sheaves across all expansions and root stacks of a simple normal crossing degeneration. Formally, logarithmic coherent sheaves are coherent sheaves in the full logarithmic \'etale topology. We develop a suite of tools that reduces the evaluation of the basic functors of homological algebra to the conventional calculation on a computable logarithmic alteration. A second paper will establish good properties of the associated logarithmic derived category. We thus offer a unified perspective on logarithmic moduli spaces of coherent sheaves: The logarithmic Quot spaces motivated by Maulik and Ranganathan's logarithmic Donaldson--Thomas theory, the logarithmic Picard group constructed by Molcho and Wise, and moduli spaces of logarithmic parabolic sheaves as developed by Borne, Talpo, and Vistoli. In establishing the connection with logarithmic Picard groups, we offer a new interpretation of chip firing as the combinatorial shadow to a logarithmic version of S-equivalence.

math.AG↗

arXiv2025

Grothendieck topologies with logarithmic modifications

Many concepts in log geometry are invariant under log blow-ups. To formalize this invariance, we introduce the m-open, m-\'etale, m-smooth, m-fppf, and m-fpqc topologies for fs log schemes. These refine the standard topologies from scheme theory by treating abstract log modifications as covers. For example, the m-\'etale topology is a subtopology of full log \'etale topology, characterized by a stronger lifting property than for log \'etale maps. Along the way, we identify and correct an error in the definition of the full log \'etale topology. We also prove a global integralization theorem by logarithmic blow-ups and use it to describe the m-open topos as a limit over log blow-ups. Finally, we characterize the sheaves on the m-type sites and connect the m-open site to Kato's valuative space.

math.AG↗

arXiv2024

Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components

Let $Y$ be a cubic threefold with a non-Eckardt type involution $τ$. Our first main result is that the $τ$-equivariant category of the Kuznetsov component $\mathcal{K}u_{\mathbb{Z}_2}(Y)$ determines the isomorphism class of $Y$ for general $(Y,τ)$. We shall prove this categorical Torelli theorem via three approaches: a noncommutative Hodge theoretical one (using a generalization of the intermediate Jacobian construction due to Alexander Perry), a Bridgeland moduli theoretical one (using equivariant stability conditions), and a Chow theoretical one (using some techniques in [kuznetsovnonclodedfield2021]).The remaining part of the paper is devoted to proving an equivariant infinitesimal categorical Torelli for non-Eckardt cubic threefolds $(Y,τ)$. To accomplish it, we prove a compatibility theorem on the algebra structures of the Hochschild cohomology of the bounded derived category $D^b(X)$ of a smooth projective variety $X$ and on the Hochschild cohomology of a semi-orthogonal component of $D^b(X)$. Another key ingredient is a generalization of a result in [macri2009infinitesima] which shows that the twisted Hochschild-Kostant-Rosenberg isomorphism is compatible with the actions on the Hochschild cohomology and on the singular cohomology induced by an automorphism of $X$. In appendix, we prove an equivariant categorical Torelli theorem for arbitrary cubic threefold with a geometric involution under a natural assumption.

math.AG↗

arXiv2024

Autoequivalences of Blow-Ups of Minimal Surfaces

Let X be the blow-up of the projective plane in a finite set of very general points. We deduce from the work of Uehara that X has only standard autoequivalences, no nontrivial Fourier-Mukai partners, and admits no spherical objects. If X is the blow-up of the projective plane in 9 very general points, we provide an alternate and direct proof of the corresponding statement. Further, we show that the same result holds if X is a blow-up of finitely many points in a minimal surface of nonnegative Kodaira dimension which contains no (-2)-curves. Independently, we characterize spherical objects on blow-ups of minimal surfaces of positive Kodaira dimension.

math.AG↗

arXiv2023

Equivariant Kuznetsov Components of Certain Cubic Fourfolds

Let $M$ denote a specific cubic fourfold that accommodates a group action by $\mathbb{Z}/3\mathbb{Z}$. Through utilization of derived Mckay correspondence, we present a new proof establishing the identification of the equivariant Kuznetsov component in the equivariant derived category of $M$ with the derived category of certain abelian surface. This surface naturally emerges from the defining equation of the cubic fourfold $M$.

math.AG↗

arXiv2019

Group-theoretic remarks on Goldbach's conjecture

The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees $2n$ and $2n-1$ to characterize this conjecture, and present its six group-theoretic versions; and further prove that this conjecture is true for $p+1$ and $p-1$ whenever $p \geq 11$ is a prime number.

math.GR↗