Xun Lin · 林汛

Research

Research

My research lies in algebraic geometry. I am interested in the geometry of derived categories and GLSM models, Hochschild–Serre algebras of smooth proper dg categories, Hodge theory of algebraic varieties, and Hitchin fibrations.

Current projects concern categorical Torelli problems, categorical Hitchin fibrations, and Torelli problems for hypersurfaces in projective space.

Publications

Published and forthcoming work
  1. On nonexistence of semi-orthogonal decompositions in algebraic geometry.

    International Mathematics Research Notices, 2026, Issue 2.

  2. Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces.

    With Shizhuo Zhang. Mathematische Annalen 391 (2025), 163–177.

  3. Kuznetsov’s Fano threefold conjecture via Hochschild–Serre algebra.

    With Shizhuo Zhang. Mathematische Zeitschrift 308 (2024), article 26. arXiv

  4. Categorical Torelli theorems for Gushel–Mukai threefolds.

    With Augustinas Jacovskis, Zhiyu Liu, and Shizhuo Zhang. Journal of the London Mathematical Society 109 (2024), e12878.

  5. Indecomposability of the bounded derived categories of Brill–Noether varieties.

    With Chenglong Yu. Proceedings of the American Mathematical Society 152 (2024), 1457–1465.

  6. Infinitesimal categorical Torelli theorems for Fano threefolds.

    With Augustinas Jacovskis, Zhiyu Liu, and Shizhuo Zhang. Journal of Pure and Applied Algebra 227 (2023), no. 12, 107418.

  7. Noncommutative Hodge conjecture.

    Journal of Noncommutative Geometry 17 (2023), 1367–1389.

Preprints

Recent manuscripts
  1. Semiorthogonal indecomposability for Hilbert schemes of points on integral locally planar curves.

    With Qingyuan Jiang. arXiv:2608.11608 [math.AG].

  2. Infinitesimal Torelli problems for special Gushel–Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives.

    With Shizhuo Zhang and Zheng Zhang. arXiv:2507.06995 [math.AG].

  3. IVHS via Kuznetsov components and categorical Torelli theorems for weighted hypersurfaces.

    With Jørgen Vold Rennemo and Shizhuo Zhang. arXiv:2408.08266 [math.AG].

  4. Three approaches to categorical Torelli theorem for cubic threefold of non-Eckardt type via equivariant Kuznetsov components.

    With Xianyu Hu, Sebastian Casalaina-Martin, Shizhuo Zhang, and Zheng Zhang. arXiv:2405.20554 [math.AG].

  5. Fully faithful functors, skyscraper sheaves, and birational equivalence.

    With Chunyi Li and Xiaolei Zhao. arXiv:2402.12573 [math.AG].

Works in progress

Ongoing collaborations
  1. Higgs objects over noncommutative varieties.

    With Shizhuo Zhang.

  2. Categorical Torelli theorem for Verra threefolds and fourfolds.

    With Zhiyu Liu and Shizhuo Zhang.

  3. Singular categories: Hodge theory and (birational) categorical Torelli theorem for nodal Fano threefolds.

    With Fei Xie, Daniele Faenzi, and Shizhuo Zhang.