arXiv subjects

Ruxuan Zhang

Publications and source records attributed to Ruxuan Zhang.

9 recordsLinked to original sources

arXiv2026

Bloch's conjecture for equivalences between twisted abelian surfaces and applications

The Beauville--Voisin conjecture predicts a canonical descending filtration on the Chow group of zero-cycles of a hyperkähler variety, opposite to the conjectural Bloch--Beilinson filtration. A basic test for such filtrations is a Bloch-type principle: the action on zero-cycles should be governed by the action on the holomorphic symplectic form. While this principle has been verified in several cases of hyperkähler varieties of $\mathrm{K3}^{[n]}$-type, the $\mathrm{Kum}_n$-type case remains much less understood. In this paper, we study this problem through twisted abelian surfaces and their associated $\mathrm{Kum}_n$-type varieties. We first construct a natural action of autoequivalences of twisted abelian surfaces on the Albanese kernel and prove Bloch's conjecture for all (anti-)symplectic autoequivalences. As an application, we prove the corresponding Bloch conjecture for symplectic birational automorphisms of twisted modular $\mathrm{Kum}_n$-type varieties; in particular, this applies to those admitting a birational Lagrangian fibration. Finally, we introduce and study a Shen--Yin--Zhao type filtration on twisted modular varieties and compare it with Voisin's filtration in the sixfold case. We also establish the anti-symplectic Bloch conjecture for twisted modular $\mathrm{Kum}_3$-type varieties.

math.AG↗

arXiv2025

A twisted derived category of hyper-Kähler varieties of $K3^{[n]}$-type

We conjecture that a natural twisted derived category of any hyper-Kähler variety of $K3^{[n]}$-type is controlled by its Markman-Mukai lattice. We prove the conjecture under numerical constraints, and our proof relies heavily on Markman's projectively hyperholomorphic bundle and a recently proven twisted version of the D-equivalence conjecture. In particular, we prove that any two fine moduli spaces of stable sheaves on a $K3$ surface are derived equivalent if they have the same dimension.

math.AG↗

arXiv2025

The period-index problem for hyper-Kähler varieties via hyperholomorphic bundles

We prove new bounds for the period-index problem for hyper-Kähler varieties of $K3^{[n]}$-type using projectively hyperholomorphic bundles constructed by Markman. We show that $\mathrm{dim}(X)$ is a bound for any $X$ of $K3^{[n]}$-type. We also show that $\frac{1}{2}\mathrm{dim}(X)$ is a bound for most Brauer classes when the Picard rank of $X$ is at least two, providing evidence for a conjecture of Huybrechts.

math.AG↗

arXiv2025

Filtrations on the derived category of twisted K3 surfaces

We introduce and study the Shen-Yin-Zhao filtration on derived categories of twisted K3 surfaces. A main contribution is the construction of a twisted Beauville-Voisin class $\mathfrak{o}_{\mathscr{X}} \in \operatorname{CH}_0(X)$ that extends fundamental results of O'Grady and Shen-Yin-Zhao \cite{OG13, SYZ20} to twisted settings. This class enables: 1. A derived equivalence-invariant filtration $\mathbf{S}_\bullet(\mathrm{D}^{(1)}(\mathscr{X}))$ preserved under Fourier-Mukai transforms, 2. A birational invariant filtration $\mathbf{S}^{\mathrm{SYZ}}_\bullet \operatorname{CH}_0$ on Bridgeland moduli spaces. We prove $\mathbf{S}^{\mathrm{SYZ}}_\bullet \operatorname{CH}_0$ coincides with Voisin's filtration $\mathbf{S}^{\mathrm{BV}}_\bullet\operatorname{CH}_0$ (Theorem 1.4), providing a canonical candidate for the conjectural Beauville-Voisin filtration. Applications include Bloch's conjecture for (anti)-symplectic automorphisms and existence of algebraically coisotropic subvarieties.

math.AG↗

arXiv2025

The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles

We show that birational hyper-Kähler varieties of $K3^{[n]}$-type are derived equivalent, establishing the D-equivalence conjecture in these cases. The Fourier-Mukai kernels of our derived equivalences are constructed from projectively hyperholomorphic bundles, following ideas of Markman. Our method also proves a stronger version of the D-equivalence conjecture for hyper-Kähler varieties of $K3^{[n]}$-type with Brauer classes.

math.AG↗

arXiv2024

Bloch's conjecture for (anti-)autoequivalences on K3 surfaces

In this paper, we investigate Bloch's conjecture for autoequivalence on K3 surfaces. We introduce the notion of reflective autoequivalence of twisted K3 surfaces and prove Bloch's conjecture for such autoequivalences, thereby confirming the conjecture for all (anti-)symplectic autoequivalences of K3 surfaces with Picard number at least $3$. The main idea is that we find a Cartan-Dieudonné type decomposition of (anti)-symplectic autoequivalences. Our findings have several interesting consequences. Firstly, we verify Bloch's conjecture for (anti-)symplectic birational automorphisms of Bridgeland moduli space on a K3 surface with Picard number at least $3$. This notably implies that Bloch's conjecture holds for (anti)-symplectic birational automorphisms of finite order ($\neq 2,4$) on arbitrary hyper-Kähler varieties of $K3^{[n]}$-type. Secondly, we extend Huybrechts' work in \cite{Huy12} to twisted K3 surfaces. This extension enables us to affirm Bloch's conjecture for symplectic birational automorphisms on any hyper-Kähler variety of $K3^{[n]}$-type preserving a birational Lagrangian fibration. Finally, we prove the constant cycle property for the fixed loci of anti-symplectic involutions on hyper-Kähler varieties of $K3^{[n]}$-type, provided that $n \leq 2$ or the invariant sublattice has rank greater than $1$.

math.AG↗

arXiv2023

One-cycles on Gushel-Mukai fourfolds and the Beauville-Voisin filtration

We prove that the invariant locus of the involution associated to a general double EPW sextic is a constant surface and introduce a filtration on $\CH_1$ of a Gushel-Mukai fourfold. We verify the sheaf/cycle correspondence for sheaves supported on low degree rational curves, parallel to the cubic fourfolds case.

math.AG↗

arXiv2022

A Self-Adjusting Fusion Representation Learning Model for Unaligned Text-Audio Sequences

Inter-modal interaction plays an indispensable role in multimodal sentiment analysis. Due to different modalities sequences are usually non-alignment, how to integrate relevant information of each modality to learn fusion representations has been one of the central challenges in multimodal learning. In this paper, a Self-Adjusting Fusion Representation Learning Model (SA-FRLM) is proposed to learn robust crossmodal fusion representations directly from the unaligned text and audio sequences. Different from previous works, our model not only makes full use of the interaction between different modalities but also maximizes the protection of the unimodal characteristics. Specifically, we first employ a crossmodal alignment module to project different modalities features to the same dimension. The crossmodal collaboration attention is then adopted to model the inter-modal interaction between text and audio sequences and initialize the fusion representations. After that, as the core unit of the SA-FRLM, the crossmodal adjustment transformer is proposed to protect original unimodal characteristics. It can dynamically adapt the fusion representations by using single modal streams. We evaluate our approach on the public multimodal sentiment analysis datasets CMU-MOSI and CMU-MOSEI. The experiment results show that our model has significantly improved the performance of all the metrics on the unaligned text-audio sequences.

cs.CL↗

arXiv2022

Beauville-Voisin filtrations on zero cycles of moduli space of stable sheaves on K3 surfaces

The Beauville-Voisin conjecture predicts the existence of a filtration on projective hyper-Kähler manifolds opposite to the conjecture Bloch-Beilinson filtration, called the Beauivlle-Voisin filtration. Voisin has introduced a filtration on zero cycles of an arbitrary projective hyper-Kähler manifold. On moduli space of stable objects of a projective K3 surface, there are other candidates constructed by Shen-Yin-Zhao, Barros-Flapan-Marian-Silversmith and more recently by Vial from different point of views. According to the work of Vial, all of them are proved to be equivalent except Voisin's filtration. In this paper, we show that Voisin's filtration is the same as the other filtrations. As an application, we prove a conjecture in Barros-Flapan-Marian-Silversmith's paper.

math.AG↗