Selected Research Papers
Higher Dimensional Geometry
KSB Moduli Space
Boundedness
• On the birational automorphisms of varieties of general type (with Christopher Hacon and James McKernan). Annals of Math. 177 (2013), Issue 3, 1077-1111.
This is the first paper of our project which aims to study the boundedness of volumes for singular pairs with arbitrary dimension. In particular, we prove the following result: the order of the automorphism group of a variety of general type X is bounded by N · vol(KX) where N is a constant depending only on the dimension of X. This previously was only known for surfaces. The first step of the proof is to show the birational boundedness of the underlying space for global quotient pairs with volumes bounded above. The remaining parts of the proof work for general log canonical pairs, whose coefficients are in a fixed DCC set.
• ACC for log canonical thresholds (with Christopher Hacon and James McKernan) Annals of Math. 180 (2014), Issue 2, 523-571.
In this paper, we continue our investigation on the boundedness behavior of volumes of singular pairs. We verify Kollár’s following conjecture: if we fix a positive integer n and a DCC set A, then the set of volumes of n-dimensional log canonical pairs (X,∆) with coefficients of ∆ in A satisfies DCC. In the course of proving Kollár’s conjecture, we also prove the ACC conjecture for numerically trivial pairs as part of our induction. Another part of the induction is Shokurov’s ACC conjecture on log canonical thresholds which was known to follow from ACC for numerically trivial pairs.
• Boundedness of moduli of varieties of general type (with Christopher Hacon and James McKernan). J. Euro. Math. Soc. 20 (2018), Issue. 4, 865-901.
We complete the proof of the boundedness of KSB moduli spaces.
Log canonical MMP
• Existence of log canonical closures (with Christopher Hacon). Invent. Math. 192 (2013), Issue 1, 161-195.
We prove the following result on the existence of good models: Let (X, ∆)/U be a proper dlt pair such that over a nonempty open set Uº ⊂ U , the base change (Xº, ∆º) of (X, ∆) has a good model, and any lc center of (X, ∆) has its image meeting Uº, then (X, ∆) has a good model. The result seems to be technical, but it indeed has many interesting consequences including the existence of lc flips, the properness of the moduli spaces of stable varieties the existence of lc closure and so on.
CM polarization
• Nonexistence of asymptotic GIT compactification (with Xiaowei Wang). Duke Math. J. 163 (2014), Issue 12, 2217-2241.
We show that if a KSBA limit of a family of smooth canonically polarized varieties is not asymptotic GIT Chow semistable then this family does not yield any asymptotic GIT Chow semistable limit. This answered a question in the preface of the book ’Geometric Invariant Theory’. In fact, we show that the semi-stable filling minimizes the GIT height and the KSBA limit minimizes the Donaldson-Futaki invariant. Since the latter is the limit of the formers, if there is a asymptotic GIT Chow semistable limit, then it must be identical to the KSBA limit.
K-stability of Fano varieties
K-stability and moduli
• Special test configurations and K-stability of Fano varieties (with Chi Li). Annals of Math. 180 (2014), Issue 1, 197-232.
Let X be a Fano manifold. For any given test configuration of (X,−rKX), we make a few modifications rooted in the minimal model program to simplify the test configuration. We show that the Donaldson-Futaki invariant is always non-increasing along the process. This implies that, when X is Fano, to test K-(semi)stability, we only need to test on the special test configurations. We also show by a counter-example that the ‘right’ definition of K-(poly)stability should only involve normal test configurations.
• On the proper moduli spaces of smooth Kähler-Einstein Fano varieties (with Chi Li and Xiaowei Wang). Duke Math. J. 168 (2019), Issue 08, 1387-1459.
We prove there is a proper algebraic space which is the good moduli space parametrizing all smoothable Fano varieties with Kähler-Einstein metric.
• K-stability of cubic threefolds (with Y. Liu). To appear in Duke Math. J.
We show that the GIT moduli space of cubic threefolds is identical to the K-moduli space. This is achieved by an estimate of the volume of three dimensional klt singularities.
• Uniqueness of K-polystable degenerations of Fano varieties (with H. Blum). To appear in Annals of Math.
We prove any two K-semistable degenerations of a family of Fano varieties are S-equivalent. This verifies that a K-moduli space, if exists, is separated. It is one of the (six) steps to give a purely algebraic construction of the moduli space parametrizing K-polystable Fano varieties. This is achieved by an extensive application of the valuation criterions developed by K. Fujita, C. Li and others, and then use minimal model program techniques to obtain finite generations under suitable conditions.
Local theory
• Stability of Valuations and Kollár Components (with Chi Li). To appear in J. Euro. Math. Soc..
We show that for a klt singularity, a divisorial valuation is the minimizer of the normalized volume function if and only if it is a K-semistable Kollár Component, and in this case its volume is strictly smaller than the volume of other divisorial valuations.
• Stability of Valuations: Higher Rank (with Chi Li). Peking Math. J. 1 (2018) no. 1, 1-79.
We continue to study the stability of valuations for a klt singularity and consider the case when the minimizer of the normalized volume function is a quasi-monomial valuation with possibly higher rank. Under the assumption that the associated graded ring is finitely generated, we show that such a valuation is a minimizer if and only if it is klt and K-semistability, and furthermore such a valuation is unique. As a corollary, this implies that the K-semistable intermediate cone W of the metric tangent cone C of a singularity appearing on a GH limit of KE Fano manifolds is uniquely determined by the algebraic structure of the singularity, as conjectured by Donaldson-Sun.
• Algebraicity of the Metric Tangent Cones and Equivariant K-stability (with Chi Li and Xiaowei Wang)
Combined with previous work, we complete the proof of the algebraicity of the metric tangent cone of a singularity on GH limit of KE Fano manifolds. This is by showing that any K-semistable Fano cone has a unique K-polystable degeneration.
Singularities and Degenerations
Topology
• Finiteness of algebraic fundamental groups. Compos. Math. 150 (2014), Issue 03, 409-414.
Applying the local-to-global induction, especially by considering the Kollár component, we prove that the algebraic fundamental group of a klt singularity is finite.
• The dual complex of singularities (with Tommaso de Fernex, János Kollár). Higher Dimensional Algebraic Geometry in honour of Professor Yujiro Kawamata’s sixtieth birthday, 103-129, Adv. Stud. in Pure Math 74, Math. Soc. Japan, Tokyo, 2017.
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-Gorenstein 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.
• The dual complex of Calabi-Yau pairs (with János Kollár). Invent. Math. 205 (2016), Issue 3, 527-557.
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, 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.
Degeneration and Berkovich space
• The essential skeleton of a degeneration of algebraic varieties (with Johannes Nicaise), Amer. J. Math. 138 (2016), No. 6, 1645-1667.
Positive Characteristic MMP
• On the three dimensional minimal model program in positive characteristic (with Christopher Hacon). J. Amer. Math. Soc. 28 (2015) no. 3, 711-744.
We show that for a 3-fold extremal dlt flipping contraction defined over an algebraically closed field of characteristic p > 5, such that the coefficients of the boundary are in the standard set {1 − 1/n | n ∈ N}, then the flip exists. As a consequence, we prove the existence of minimal models for any projective ℚ-factorial terminal variety X with pseudo-effective canon- ical class. We obtain this result using the original Shokurov’s argument in characteristic 0, but then replace many characteristic 0 techniques by the recent characteristic p tools developed in the F-singularity theory. Especially, we replace Kawamata-Viehweg vanishing theorem by only looking at the image of absolute Frobenius. Then we prove in the case we consider, it is the same as the complete linear system.
• On base point freeness in positive characteristic (with Paolo Cascini and Hiromu Tanaka). Ann. Sci. École Norm. Sup. 48 (2015) no. 5, 1239-1272.
Along the idea of Angehrn-Siu, we investigate a new approach in positive characteristic of cutting subschemes, from which we can extend sections of the line bundle of an adjoint form. As a result, we prove for any F-regular pair (X,B) and an ample divisor A, the ℚ-divisor KX+B+A is ample if it is strictly nef and is big if it is nef and of maximal nef dimension. These are corollaries of standard base point free and cone theorems. Then we also achieve new results in 3-dimensional MMP in positive characteristic.
• Nonvanishing for threefolds in characteristic p > 5 (with Lei Zhang). Duke Math. J. 168 (2019), Issue 07, 1269-1301.
Combing Miyaoka’s original proof in characteristic 0 and the tools developed in the recent years to study vector bundles in positive characteristics, especially Langer’s works, we prove nonvanshing for terminal threefolds.
Rationally Connected Varieties
Deform rational curves
• Strong rational connectedness of surfaces, J. Reine Angew. Math. 665 (2012), 189-205.
We prove the smooth locus of a log del Pezzo surface, whose rational connectedness is proved by Keel-McKernan, is indeed strongly rationally connected. This confirms a conjecture due to Hassett and Tschinkel.