Jihao Liu (刘济豪) — Peking University

Algebraic geometer at the School of Mathematical Sciences, Peking University, and a member of the Beijing International Center for Mathematical Research (BICMR). Mostly thinking about birational geometry — the minimal model program, generalized pairs, foliations, singularities, Fano varieties and K-stability, boundedness — and how explicit one can make any of it. From time to time I also work on problems in other parts of mathematics.

Before returning to Beijing in 2024 I spent three years at Northwestern as a Boas Assistant Professor, working with Yuchen Liu. My Ph.D. (Utah, 2021) was supervised by Christopher Hacon; before that, my undergraduate years at PKU were spent under Gang Tian.

Email:

liujihao [at] math.pku.edu.cn

Office:

Beijing International Center for Mathematical Research, Peking University, Jingchunyuan 78402
北京大学北京国际数学研究中心镜春园 78402

CV:

cv.pdf (updated 13 October 2026)

Google Scholar:

profile (cited by 675)

Research

Open a topic to see my papers on it.

Minimal model program (23 papers)

Proof of Serrano’s conjecture. Note.
A klt generalized pair with infinitely generated canonical ring (with Yanze Wang).
Existence of the minimal model program for log canonical generalized pairs (with Zhengyu Hu).
Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures (with Zheng Xu).
Effective termination of general type MMPs in dimension at most five (with Jingjun Han, Ziquan Zhuang).
Boundedness in general type MMP and fivefold effective termination (with Jingjun Han, Lu Qi, Ziquan Zhuang).
Sarkisov program for algebraically integrable and threefold foliations (with Yifei Chen, Yanze Wang). Int. Math. Res. Not. 2026, no. 6, rnag045.
Non-vanishing implies numerical dimension one abundance (with Zheng Xu).
On finite generation and boundedness of adjoint foliated structures (with Paolo Cascini, Jingjun Han, Fanjun Meng, Calum Spicer, Roberto Svaldi, Lingyao Xie).
A generalized non-vanishing theorem on surfaces (with Lingyao Xie). Pure Appl. Math. Q. (special volume in honor of Caucher Birkar), 22 (2026), no. 1, 235–244.
Flop between algebraically integrable foliations on potentially klt varieties (with Yifei Chen, Yanze Wang). Int. J. Math. 36 (2025), no. 11, 2550035.
Minimal model program for algebraically integrable adjoint foliated structures (with Paolo Cascini, Jingjun Han, Fanjun Meng, Calum Spicer, Roberto Svaldi, Lingyao Xie).
Minimal model program for algebraically integrable foliations on klt varieties (with Fanjun Meng, Lingyao Xie). Compos. Math. 161 (2025), no. 12, 3213–3276.
On the equivalence between the effective adjunction conjectures of Prokhorov–Shokurov and of Li (with Jingjun Han, Qingyuan Xue). Algebra Number Theory 19 (2025), no. 11, 2261–2279.
Minimal model program for algebraically integrable foliations and generalized pairs (with Guodu Chen, Jingjun Han, Lingyao Xie).
On effective log Iitaka fibrations and existence of complements (with Guodu Chen, Jingjun Han). Int. Math. Res. Not. (2024), no. 10, 8329–8349.
Semi-ampleness of NQC generalized log canonical pairs (with Lingyao Xie). Adv. Math. 427 (2023), 109126.
On termination of flips and exceptionally non-canonical singularities (with Jingjun Han). Geom. Topol. 29 (2025), no. 1, 399–441.
Uniform rational polytopes for Iitaka dimensions (with Guodu Chen, Jingjun Han). In: Higher Dimensional Algebraic Geometry: A Volume in Honor of V. V. Shokurov (C. D. Hacon, C. Xu eds.), London Math. Soc. Lecture Note Series, Cambridge University Press (2025), 43–68.
Relative Nakayama–Zariski decomposition and minimal models of generalized pairs (with Lingyao Xie). Peking Math. J. 8 (2025), no. 2, 299–349.
On generalized lc pairs with b-log abundant nef part (with an Appendix by J. Han) (with Junpeng Jiao, Lingyao Xie). Front. Math. 21 (2026), no. 4, 769–799.
Existence of flips for generalized lc pairs (with Christopher D. Hacon). Camb. J. Math. 11 (2023), no. 4, 795–828.
Sarkisov program for generalized pairs. Osaka J. Math. 58 (2021), no. 4, 899–920.

Generalized pairs (11 papers)

A klt generalized pair with infinitely generated canonical ring (with Yanze Wang).
Existence of the minimal model program for log canonical generalized pairs (with Zhengyu Hu).
A generalized non-vanishing theorem on surfaces (with Lingyao Xie). Pure Appl. Math. Q. (special volume in honor of Caucher Birkar), 22 (2026), no. 1, 235–244.
Minimal model program for algebraically integrable foliations and generalized pairs (with Guodu Chen, Jingjun Han, Lingyao Xie).
Vanishing theorems for generalized pairs (with Bingyi Chen, Lingyao Xie).
Semi-ampleness of NQC generalized log canonical pairs (with Lingyao Xie). Adv. Math. 427 (2023), 109126.
Relative Nakayama–Zariski decomposition and minimal models of generalized pairs (with Lingyao Xie). Peking Math. J. 8 (2025), no. 2, 299–349.
On generalized lc pairs with b-log abundant nef part (with an Appendix by J. Han) (with Junpeng Jiao, Lingyao Xie). Front. Math. 21 (2026), no. 4, 769–799.
Existence of flips for generalized lc pairs (with Christopher D. Hacon). Camb. J. Math. 11 (2023), no. 4, 795–828.
Accumulation point theorem for generalized log canonical thresholds.
Sarkisov program for generalized pairs. Osaka J. Math. 58 (2021), no. 4, 899–920.

Foliations (17 papers)

Shokurov's global index conjecture for threefold foliations (with Sheng Qin).
Optimal bend-and-break for foliations (with Zeming Sun, Jiedong Jiang).
Birational boundedness of stable families (with Paolo Cascini, Calum Spicer, Roberto Svaldi).
Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures (with Zheng Xu).
Variation of algebraically integrable adjoint foliated structures (with Paolo Cascini, Fanjun Meng, Roberto Svaldi, Lingyao Xie).
Sarkisov program for algebraically integrable and threefold foliations (with Yifei Chen, Yanze Wang). Int. Math. Res. Not. 2026, no. 6, rnag045.
On finite generation and boundedness of adjoint foliated structures (with Paolo Cascini, Jingjun Han, Fanjun Meng, Calum Spicer, Roberto Svaldi, Lingyao Xie).
Flop between algebraically integrable foliations on potentially klt varieties (with Yifei Chen, Yanze Wang). Int. J. Math. 36 (2025), no. 11, 2550035.
Minimal model program for algebraically integrable adjoint foliated structures (with Paolo Cascini, Jingjun Han, Fanjun Meng, Calum Spicer, Roberto Svaldi, Lingyao Xie).
Volume of algebraically integrable foliations and locally stable families (with Jingjun Han, Junpeng Jiao, Mengchu Li). Trans. Amer. Math. Soc. 379 (2026), no. 10, 6809–6832.
Minimal model program for algebraically integrable foliations on klt varieties (with Fanjun Meng, Lingyao Xie). Compos. Math. 161 (2025), no. 12, 3213–3276.
Minimal model program for algebraically integrable foliations and generalized pairs (with Guodu Chen, Jingjun Han, Lingyao Xie).
ACC for lc thresholds for algebraically integrable foliations (with Omprokash Das, Roktim Mascharak). Selecta Math. (N.S.) 32 (2026), no. 4, Paper No. 70, 34 pp.
Uniform rational polytopes of foliated threefolds and the global ACC (with Fanjun Meng, Lingyao Xie). J. Lond. Math. Soc. 109 (2024), no. 6, e12950.
Complements, index theorem, and minimal log discrepancies of foliated surface singularities (with Fanjun Meng, Lingyao Xie). Eur. J. Math. 10 (2024), no. 1, Paper No. 6.
On global ACC for foliated threefolds (with Yujie Luo, Fanjun Meng). Trans. Amer. Math. Soc. 376 (2023), no. 12, 8939–8972.
On invariance of plurigenera for foliated surface pairs.

Singularities: minimal log discrepancies, thresholds, ACC (29 papers)

Singular points of rank-one del Pezzo surfaces in characteristic three. Note.
On two questions of Qi on saturated filtrations (with Haofeng Zhang).
A Criteria of Weighted Homogeneity via Logarithmic Vector Fields (with Xiping Zhang). Not AI-generated. Presents the same result as no. 60b; the mathematics is identical.
Criteria of isolated weighted homogeneous hypersurface singularities using Logarithmic vector fields (with Xiping Zhang). AI-generated (human-verified). Presents the same result as no. 60a; the mathematics is identical.
Boundedness of total Cartier indices for rational singularities in families (with Ruicheng Hu, Sheng Qin).
A question on klt type varieties of Han and Jiang.
On a question of Kollár and Kovács.
Dual complexes of qdlt Fano type models and strong complete regularity (with Konstantin Loginov).
Upper bound of minimal log discrepancies.
Classification of threefold enc cDV quotient singularities (with Jingjun Han).
ACC for local volumes (with Jingjun Han, Lu Qi).
Exceptional Fano varieties with small minimal log discrepancy (with Louis Esser, Chengxi Wang). Math. Res. Lett. 33 (2026), no. 1, 161–198.
ACC for lc thresholds for algebraically integrable foliations (with Omprokash Das, Roktim Mascharak). Selecta Math. (N.S.) 32 (2026), no. 4, Paper No. 70, 34 pp.
Uniform rational polytopes of foliated threefolds and the global ACC (with Fanjun Meng, Lingyao Xie). J. Lond. Math. Soc. 109 (2024), no. 6, e12950.
Optimal bounds on surfaces (with Vyacheslav V. Shokurov). To appear in Algebraic Geometry and Physics.
Complements, index theorem, and minimal log discrepancies of foliated surface singularities (with Fanjun Meng, Lingyao Xie). Eur. J. Math. 10 (2024), no. 1, Paper No. 6.
On global ACC for foliated threefolds (with Yujie Luo, Fanjun Meng). Trans. Amer. Math. Soc. 376 (2023), no. 12, 8939–8972.
On termination of flips and exceptionally non-canonical singularities (with Jingjun Han). Geom. Topol. 29 (2025), no. 1, 399–441.
Infinitesimal structure of log canonical thresholds (with Fanjun Meng, Lingyao Xie). Doc. Math. 29 (2024), no. 3, 703–732.
Remark on complements on surfaces. Forum Math. Sigma 11 (2023), e42.
Second largest accumulation point of minimal log discrepancies of threefolds (with Yujie Luo).
ACC for minimal log discrepancies of terminal threefolds (with Jingjun Han, Yujie Luo). Adv. Math. 480 (2025), 110457.
Number of singular points on projective surfaces (with Lingyao Xie). Chin. Ann. Math. Ser. B 46 (2025), no. 5, 713–724.
Divisors computing minimal log discrepancies on lc surfaces (with Lingyao Xie). Math. Proc. Camb. Philos. Soc. 175 (2023), no. 1, 107–128.
An optimal gap of minimal log discrepancies of threefold non-canonical singularities (with Liudan Xiao). J. Pure Appl. Algebra 225 (2021), no. 9, 106674.
Bounded deformations of (ε, δ)-log canonical singularities (with Jingjun Han, Joaquín Moraga). J. Math. Sci. Univ. Tokyo 27 (2020), no. 1, 1–28.
ACC for minimal log discrepancies of exceptional singularities (with Jingjun Han, Vyacheslav V. Shokurov). Peking Math. J. 9 (2026), no. 2, 225–257.
Accumulation point theorem for generalized log canonical thresholds.
Toward the equivalence of the ACC for a-log canonical thresholds and the ACC for minimal log discrepancies. To appear in Algebraic Geometry and Physics.

Fano varieties and K-stability (10 papers)

K-polystability of nilpotent orbit and Slodowy slice bases. AI-generated, not verified by a human.
Disproof of the Yau–Tian–Donaldson conjecture.
The equality case of Ehrhart’s volume conjecture.
On two questions of Qi on saturated filtrations (with Haofeng Zhang).
K-polystable toric Fano varieties with small alpha invariants (with Ziwen Zhu).
An equivariant fixed-level Demailly identity for Fano manifolds (with Sheng Qin).
Dual complexes of qdlt Fano type models and strong complete regularity (with Konstantin Loginov).
ACC for local volumes (with Jingjun Han, Lu Qi).
Exceptional Fano varieties with small minimal log discrepancy (with Louis Esser, Chengxi Wang). Math. Res. Lett. 33 (2026), no. 1, 161–198.
On explicit bounds of Fano threefolds (with Caucher Birkar). J. Reine Angew. Math. (Crelle's Journal) (2025).

Calabi–Yau varieties and log Calabi–Yau pairs (4 papers)

A counterexample to the odd-dimensional rank bound for abelian p-group actions (with Yanze Wang).
On a conjecture of Esser, Totaro, and Wang. Note.
On a question of Mauri and Moraga.
Dual complexes of qdlt Fano type models and strong complete regularity (with Konstantin Loginov).

Boundedness (12 papers)

Boundedness of complements of log canonical pairs. Note.
Boundedness of total Cartier indices for rational singularities in families (with Ruicheng Hu, Sheng Qin).
Shokurov's global index conjecture for threefold foliations (with Sheng Qin).
Birational boundedness of stable families (with Paolo Cascini, Calum Spicer, Roberto Svaldi).
Boundedness in general type MMP and fivefold effective termination (with Jingjun Han, Lu Qi, Ziquan Zhuang).
On finite generation and boundedness of adjoint foliated structures (with Paolo Cascini, Jingjun Han, Fanjun Meng, Calum Spicer, Roberto Svaldi, Lingyao Xie).
Volume of algebraically integrable foliations and locally stable families (with Jingjun Han, Junpeng Jiao, Mengchu Li). Trans. Amer. Math. Soc. 379 (2026), no. 10, 6809–6832.
On effective log Iitaka fibrations and existence of complements (with Guodu Chen, Jingjun Han). Int. Math. Res. Not. (2024), no. 10, 8329–8349.
Remark on complements on surfaces. Forum Math. Sigma 11 (2023), e42.
Uniform rational polytopes for Iitaka dimensions (with Guodu Chen, Jingjun Han). In: Higher Dimensional Algebraic Geometry: A Volume in Honor of V. V. Shokurov (C. D. Hacon, C. Xu eds.), London Math. Soc. Lecture Note Series, Cambridge University Press (2025), 43–68.
On effective birationality for sub-pairs (with Jingjun Han). Int. J. Math. 34 (2023), no. 6, Article No. 2350029.
Bounded deformations of (ε, δ)-log canonical singularities (with Jingjun Han, Joaquín Moraga). J. Math. Sci. Univ. Tokyo 27 (2020), no. 1, 1–28.

Explicit birational geometry (17 papers)

Fujita freeness on complex projective sevenfolds. AI-generated, not verified by a human.
Fujita freeness in dimension six. AI-generated, not verified by a human.
The minimal volume of normal stable surface pairs with reduced boundary containing a zero-dimensional non-klt center (with Weili Shao).
On a conjecture of Esser, Totaro, and Wang. Note.
Optimal bend-and-break for foliations (with Zeming Sun, Jiedong Jiang).
The minimal volume of stable surfaces of rank one (with Wenfei Liu).
Effective termination of general type MMPs in dimension at most five (with Jingjun Han, Ziquan Zhuang).
Boundedness in general type MMP and fivefold effective termination (with Jingjun Han, Lu Qi, Ziquan Zhuang).
Classification of threefold enc cDV quotient singularities (with Jingjun Han).
Exceptional Fano varieties with small minimal log discrepancy (with Louis Esser, Chengxi Wang). Math. Res. Lett. 33 (2026), no. 1, 161–198.
On explicit bounds of Fano threefolds (with Caucher Birkar). J. Reine Angew. Math. (Crelle's Journal) (2025).
The minimal volume of surfaces of log general type with non-empty non-klt locus (with Wenfei Liu). To appear in Algebr. Geom.
Optimal bounds on surfaces (with Vyacheslav V. Shokurov). To appear in Algebraic Geometry and Physics.
Second largest accumulation point of minimal log discrepancies of threefolds (with Yujie Luo).
On the fixed part of pluricanonical systems for surfaces (with Lingyao Xie). Math. Nachr. 296 (2023), no. 5, 2046–2069.
Number of singular points on projective surfaces (with Lingyao Xie). Chin. Ann. Math. Ser. B 46 (2025), no. 5, 713–724.
An optimal gap of minimal log discrepancies of threefold non-canonical singularities (with Liudan Xiao). J. Pure Appl. Algebra 225 (2021), no. 9, 106674.

Surfaces (15 papers)

Singular points of rank-one del Pezzo surfaces in characteristic three. Note.
The minimal volume of normal stable surface pairs with reduced boundary containing a zero-dimensional non-klt center (with Weili Shao).
Mori dream Jacobian elliptic surfaces of Kodaira dimension one (with Antonio Laface, Sichen Li).
Examples of infinitely many (-1)-curves with higher genus on elliptic surfaces (with Sichen Li).
An example of a very non-movable effective divisor.
The minimal volume of stable surfaces of rank one (with Wenfei Liu).
A generalized non-vanishing theorem on surfaces (with Lingyao Xie). Pure Appl. Math. Q. (special volume in honor of Caucher Birkar), 22 (2026), no. 1, 235–244.
The minimal volume of surfaces of log general type with non-empty non-klt locus (with Wenfei Liu). To appear in Algebr. Geom.
Optimal bounds on surfaces (with Vyacheslav V. Shokurov). To appear in Algebraic Geometry and Physics.
Complements, index theorem, and minimal log discrepancies of foliated surface singularities (with Fanjun Meng, Lingyao Xie). Eur. J. Math. 10 (2024), no. 1, Paper No. 6.
Remark on complements on surfaces. Forum Math. Sigma 11 (2023), e42.
On the fixed part of pluricanonical systems for surfaces (with Lingyao Xie). Math. Nachr. 296 (2023), no. 5, 2046–2069.
Number of singular points on projective surfaces (with Lingyao Xie). Chin. Ann. Math. Ser. B 46 (2025), no. 5, 713–724.
Divisors computing minimal log discrepancies on lc surfaces (with Lingyao Xie). Math. Proc. Camb. Philos. Soc. 175 (2023), no. 1, 107–128.
On invariance of plurigenera for foliated surface pairs.

General algebraic geometry (6 papers)

Proof of the tropical Hodge conjecture for tropical abelian varieties. AI-generated, not verified by a human.
Syzygy bundles on Picard rank one varieties need not be stable. Note; AI-generated, not verified by a human.
Removing the Torsion-free Hypothesis in a Positivity Theorem on Deligne-Mumford Stacks (with Shengyu Hou).
Twelve common flex lines in a general pencil of cubics (with Yanze Wang).
A slope-unstable bundle on a surface with 1-homogeneous projectivization.
A counterexample to the Popa–Schnell global generation conjecture.

Group theory and operator algebras (2 papers)

Nonhyperlinear groups exist. AI-generated, not verified by a human.
Spectral gap, internality of relative commutants, and non-hyperlinear groups (with Vadim Alekseev, Andreas Thom).

Combinatorics and number theory (2 papers)

Differential geometry (3 papers)

Embedded constant mean curvature tori in hemispheres of Berger spheres (with Xiang Ma).
Disproof of the Yau–Tian–Donaldson conjecture.
An equivariant fixed-level Demailly identity for Fano manifolds (with Sheng Qin).

Danus (1 papers)

Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory (with Guoxiong Gao, Zeming Sun, Bin Wu, Shurui Liu, Jiedong Jiang, Haocheng Ju, Leheng Chen, Ronnie Cheng, Xiping Zhang, Bin Dong).

Currently

Co-organizing the Peking University Algebraic Geometry Seminar with Fumiaki Suzuki, Zhiyu Tian, and Qizheng Yin — talks are also listed on Qizheng's page and on BICMR's site.

About Jihao Liu

Research

Currently