Searcharxiv⌕ Search

arXiv subjects

Yirui Xiong

Publications and source records attributed to Yirui Xiong.

5 recordsLinked to original sources
arXiv2026

Echoes of phantoms on rational surfaces

We construct the first countably infinite family of new universal phantom categories on a smooth rational surface, namely the blow-up of the complex projective plane at ten points in general position. Moreover, these phantom categories are pairwise non-equivalent, are not equivalent to any previously known phantom category on a smooth rational surface, and arise as the orthogonal complements of non-full exceptional collections of line bundles of maximal length. As an application, we show that all of these phantom categories admit bounded $t$-structures.

math.AG↗
arXiv2025

A new phantom on a rational surface

We construct a universal phantom subcategory on the blow-up of the complex projective plane at 11 general points. This phantom subcategory is the orthogonal complement of a non-full exceptional collection of line bundles of maximal length. It provides a new counterexample to a conjecture of Kuznetsov and to a conjecture of Orlov. The first counterexample was constructed by Krah [Invent. Math. {\bf 235} (2024), 1009--1018]. As an application, we construct a new co-connective DG-algebra whose derived category is a phantom. We also show in Appendix B that every smooth projective surface with an effective smooth anti-canonical divisor has no phantom subcategories, e.g. weak del Pezzo surfaces.

math.AG↗
arXiv2025

Cohen-Macaulay modules and the Bondal-Orlov conjecture

Most of the known examples of derived categories of small resolutions arise as the derived category of the endormorphism algebra of tilting bundles or complexes. Given two resolutions connected by a flop, if the strict transform of a tilting bundle is again tilting, then the derived categories of the two resolutions are equivalent, thereby proving the Bondal-Orlov conjecture in this setup. Unfortunately, it is difficult to produce tilting bundles that are compatible with flops. In this article, we introduce the notion of CM-degree of locally-free sheaves on resolutions and use them to construct tilting generators. In particular, we show that if there exists a relative very ample line bundle on the resolution with CM degree equal to the dimension of the exceptional locus, then the generator bundles constructed by Van den Bergh and Toda-Uehara are also tilting bundles. The advantage of our approach is that the CM degree is preserved under strict transform. As a consequence we prove the Bondal-Orlov conjecture in certain cases of small resolutions.

math.AG↗
arXiv2022

T-structures on a local 4-Calabi-Yau variety

Let $X$ denote the total space of cotangent bundle of projective plane. This is a non-compact Calabi-Yau $4$-fold (also called local Calabi-Yau variety in physics literature). The aim of this paper is to use tilting objects to characterize a large class of t-structures in the relevant Calabi-Yau category and try to calculate the combinatorics of them.

math.RA↗