个人简历

Position
I am a Professor in Beijing International Center of Mathematics Research.
2013
Feb. -Jul. I'm organizing a reading seminar on the first 5 Chapters on Lazarsfeld's book 'Positivity in algebraic geometry I'. Contact me if you are interested to attend.
Jul. 1-12 Jason Starr is giving a series of lectures on 'rationally connected varieties'.
Jul. 14 Tong Zhang (University of Alberta): Geography of varieties of large Albanese dimension.
Aug. 21 Zhiyu Tian (California Institute of Technology): Weak Approximation for Cubic hypersurfaces.
Sep. - Dec. There is a joint seminar between PKU, THU and Chinese Academia of Science which aims to understand Mirror symmetry for log Calabi-Yau. See the program.
Oct. 8-17 Claire Voisin (CNRS): Algebraic cycles and Hodge structures.
Nov. 27-Dec. 2 Benjamin Bakker (NYU): Holomorphic Symplectic Manifolds.
2014
May 27-29 Masayuki Kawakita (RIMS): Minimal log discrepancies and generic limits.
May 28 Yi Zhu (University of Utah): A-1 connectedness varieties.
Jul. 21-23 Lie Fu (ENS de Paris): Intersection theory of projective hyperkahler varieties.
Jul. 21-25 Valery Alexeev (University of Georgia): Compact moduli spaces of weighted hyperplane arrangements.
2015
Mar.-Jun. Evgeny Mayanskiy and I are organizing a reading seminar on deformation theory, see the webpage.
Mar.-Jun. Baohua Fu, Eduard Looijenga and I are organizing a reading seminar on geometry of cubic threefolds and fourfolds.
Apr.1-15 Olivier Benoist (CNRS) : Integral decomposition of the diagonal and application.
Apr. 29 Yujiro Kawamata (University of Tokyo): Derived McKay Correspondence for finite abelian group quotients.
May 10 Beijing Algebraic Geometry Colloquium: First meeting. Speaker: Lawrence Ein and Hélène Esnault.
May 14 Chen Jiang (University of Tokyo): Boundedness of anti-canonical volumes of singular log Fano threefolds.
May 14 Tong Zhang (University of Alberta): Relative Clifford’s Theorem for fibered varieties by Curves.
Jun. 8-14 Morgan Brown (University of Michigan): A mini-course on Berkovich spaces.
Jul. 10-13 Zhiyuan Li (Stanford University): Special cycles on Shimura varieties.
Jul. 11-14 Zhiyu Tian (CNRS and Université Joseph-Fourier-Grenoble): Varieties with large fundamental group.
Nov. 21 Beijing Algebraic Geometry Colloquium: Third meeting.
2016
Apr. 9 Beijing Algebraic Geometry Colloquium: Fifth meeting.
Jul. 4-15 Reading group on Stable Rationality and Chow Decomposition.
2017
Mar. 13-18 Tianyuan Advanced Seminar on the Moduli Spaces in Algebraic Geometry.
May 5-7 BICMR-Tokyo Algebraic Workshop.
Aug. 28-Sep. 1 Boundedness, Stability and Fano varieties.
Teaching
2013 Fall: Algebraic Geometry.
2015 Spring: Algebraic Geometry II.
2015 Fall: Algebraic Geometry II.
2016 Spring: Topics on Core Mathematical Subjects I.
2017 Spring: Introduction to Algebraic Geometry.
2017 Spring: Topics on Core Mathematical Subjects I.
2017 Fall: Abstract Algebra (抽象代数).

研究领域

研究方向： Algebraic Geometry"Research Interests
Birational Geometry:
1. Geometric and Arithmetic theory of Rationally Connected Varieties.
2. Minimal Model Program and Classification of varieties.
3. Stability.
4. Topology and Geometry of Singularities."

近期论文

1. Rationally Connected Varieties
(1) (with Amit Hogadi) Degenerations of Rationally Connected Varieties. Trans. Amer. Math. Soc. 361 (2009), no. 7, 3931–3949. We give an affirmative answer to the following question raised by Kolla´r: Do the degenerations of rationally connected varieties defined over a field k (char(k) = 0 and k may not be algebraic closed) always contain a rationally connected subvariety (by definition, it says in particular that the subvariety is absolute irreducible). In an early paper, Koll´ar gave a partial answer to this question, namely, he proved that a degeneration of Fano varieties contains an absolute irreducible subviariety over k. Combining his idea with Hacon-Mckernan’s extension theorem, we first prove the theorem for the cases when the general fibers are Fano varieties or the relative dimension is 0. Then we use the results from the minimal model program (MMP), which is proved by Birkar, Cascini, Hacon and Mckernan, to reduce the general case to these 2 special cases. (2) Notes on π1 of Smooth Loci of Log Del Pezzo Surfaces. Michigan Math. J. 58 (2009), no. 2, 489–515. It is known that the fundamental groups of smooth loci of Log del Pezzo surfaces are finite groups. In this note, based on the classification results of Dolgachev and Iskovskikh on finite subgroups of Cremona groups, we study these finite groups. A short table containing these groups is given. And lots of groups on the table are proved to be fundamental groups.
(3) Strong Rational Connectedness of Surfaces. J. Reine Angew. Math. 665 (2012), 189-205. For a nonproper smooth variety defined over C, it is asked whether the rational connectedness implies the strong rational connectedness. We discuss this question in the case of surfaces. And we prove in lots of situations, including the smooth locus of a log del Pezzo surface, whose rational connectedness is proved by Keel-McKernan, the rationally connected surface is indeed strongly rationally connected. This confirms a conjecture due to Hassett and Tschinkel. The main idea we use here is to investigate the rational curve theory of the canonical smooth separated cover stack of the compactification surface. The natural frame is the n-pointed twisted stable map theory with the target space a Deligne-Mumford stack, which is constructed by Abramovich-Vistoli. Then we study the deformation theory there. (4) Weak Approximation for Low Degree del Pezzo Surfaces. J. Algebraic Geom. 21 (2012), no. 4, 753-767. Applying the main theorem of the above paper, we verify the weak approximation conjecture of Hassett-Tschinkel for del Pezzo surfaces of degree 1 defined over the function field of curve K(C), which can be completed to a ‘generic family’ over C. The approach follows from idea of Hassett-Tschinkel: we get a section with prescribed jets by adding appropriate ‘teeth’ to an arbitrary section and deform it. (5) (with Ja´nos Kolla´r) Fano Varieties with Large Degree Endomorphisms. 2 pages. arXiv:0901.1692 In this unpublished note, we give examples of Fano varieties X with Picard number 1, which have terminal singularities and admit endomorphisms with degree larger than 1. This is a counterexmaple to a conjecture of De-Qi Zhang. However, if we assume X is smooth, the long standing conjecture says that X is indeed Pn.
2. Singularities
(6) Finiteness of algebraic fundamental groups. Compos. Math. 150 (2014), Issue 03, 409-414. Applying the local-to-global induction, especially by considering the Koll´ar component, we prove that the algebraic fundamental group of a klt singularity is finite. (7) (with Tommaso de Fernex, J´anos Kolla´r) The dual complex of singularities. To appear in Proceedings of the conference in honor of Yujiro Kawamatas 60th birthday, Advanced Studies in Pure Mathematics. By investigating when the MMP process does not change the homotopic class of the dual complex of a dlt pair, we prove there is a well defined up to PL homeomorphism topological space which is a representative of the
dual complex of a Q-Cartier isolated singularity. Using the same idea, we also show that the dual complex of a klt singularity as well as the degeneration of rationally connected varieties is contractible. (8) (with Johannes Nicaise) Poles of maximal order of motivic zeta function. Duke Math. J. 165 (2016), Issue 2, 217-243. We prove a conjecture of Veys which says that the only maximal order pole of zeta functions is the opposite of the log canonical threshold. We prove this by running MMP and analyze how the dual complex changes in the process. So we prove a stronger geometric statement which says that the only candidate of pole with maximal order is the opposite of the log canonical threshold. (9) Motivic zeta function via dlt modification. Michigan Math. J. 65 (2016), Issue 1, 89-103. We invent a notion called dlt motivic zeta function using the dlt modification and show it does not depend on the choice of the dlt modification. (10) (with J´anos Kolla´r) The dual complex of Calabi-Yau pairs. Accepted by Invent. Math.. We show that for a Calabi-Yau pair (X,D) the fundamental group of the dual complex of D is a quotient of the fundamental group of the smooth locus of X if dimD(D) ≥ 2, hence its pro-finite completion is finite. This is achieved by running a carefully chosen MMP to change the model such that D supports a large divisor. And then the Lefschetz hyperplane theorem for open varieties connects the fundamental group of the smooth locus of X and the fundamental group of the divisor. (11) (with Zhiyu Tian) Finiteness of fundamental groups. We use stratified Morse theory to refine the argument in the study of (6), and show that the finiteness of the fundamental group of the smooth locus of n−1 dimensional log Fano varieties implies the same thing for the local fundamental group of n-dimensional klt singularities. (12) (with Chi Li) Stability of Valuations and Kolla´r Components We use the construction of Koll´ar component in (6) to study the minimizer of the normalized volume function, especially the case of rational rank 1. We show that if there is K-semistable Kolla´r component, then it is the minimizer, and it’s the only one among all Kolla´r components. This study clarifies the quasi-regular case, when the entire subject of studying the normalized volume function aims at giving a picture similar to Sasakian geometry, but for any klt singularity.