Xiaolong Liu

Mathematics · AMSS, Chinese Academy of Sciences

Xiaolong LIU 刘 晓龙 / 劉 曉龍

Basic Information

Curriculum vitae
Xiaolong Liu wearing late Ming dynasty hanfu
Hanfu of the late Ming Dynasty.
01

Research

My research focuses on moduli spaces, derived geometry and enumerative geometry, particularly Donaldson–Thomas theory of Calabi–Yau 4-folds and cohomological Donaldson–Thomas theory.

Preprint 2026

On logarithmic Donaldson-Thomas invariants for local Calabi-Yau $4$-folds

Xiaolong Liu · 62 pages· Submitted

arXiv:2609.17720 PDF
Abstract Hide abstract

In this paper, we study logarithmic Donaldson-Thomas invariants for local log Calabi-Yau $4$-folds with simple normal crossing divisors. Using the family version of shifted Lagrangian classes announced by Khan-Kinjo-Park-Safronov, we construct relative and family logarithmic $\mathsf{DT}_4$-theories for logarithmic Hilbert schemes of curves, and prove a degeneration formula.

For logarithmic Hilbert schemes of points, we construct virtual classes and prove a degeneration formula in Chow groups; in particular, it is independent of shifted Lagrangian classes. Finally, combining our degeneration formula with logarithmic cobordism, we explicitly compute the zero-dimensional logarithmic $\mathsf{DT}_4$-invariants for local surface snc pairs.

Preprint 2025

Donaldson–Thomas invariants of \([\mathbb{C}^4/\mathbb{Z}_r]\)

Xiaolong Liu · 31 pages · Submitted

arXiv:2507.21582 PDF
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We compute the zero-dimensional Donaldson-Thomas invariants of the quotient stack $[\mathbb{C}^4/\mathbb{Z}_r]$, confirming a conjecture of Cao-Kool-Monavari. Our main theorem is established through an orbifold analogue of Cao-Zhao-Zhou's degeneration formula combined with the zero-dimensional Donaldson-Thomas invariants for $\widetilde{\mathbb C^2/\mathbb Z_r}\times\mathbb{C}^2$ and an explicit determination of orientations of Hilbert schemes of points on $[\mathbb{C}^4/\mathbb{Z}_r]$.

02

Talks

2026

  1. Virtual enumerative geometry and Donaldson–Thomas invariants of \([\mathbb{C}^4/\mathbb{Z}_r]\)

    Invited talk

    Shandong University · Jinan, China

  2. Donaldson–Thomas invariants of \([\mathbb{C}^4/\mathbb{Z}_r]\)

    Invited talk

    Beijing Institute of Technology · Beijing, China

2025

  1. Donaldson–Thomas invariants of \([\mathbb{C}^4/\mathbb{Z}_r]\)

    Strings, Geometry and Supersymmetry · Short talk

    Yau Mathematical Sciences Center, Tsinghua University · Beijing, China

  2. Donaldson–Thomas invariants of \([\mathbb{C}^4/\mathbb{Z}_r]\)

    Friday Morning Seminar

    Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS) · Shanghai, China

  3. Donaldson–Thomas invariants of \([\mathbb{C}^4/\mathbb{Z}_r]\)

    Mathematics Inspired by String Theory · Celebrating the 60th birthday of Conan Leung

    Southern University of Science and Technology · Shenzhen, China