arXiv subjects
Publications and source records attributed to Lingyao Xie.
In this article, we show that flips and divisorial contractions preserve the Kähler condition (for strongly $\mathbb{Q}$-factorial compact Kähler generalized klt pairs with $B+β_X$ big), and we give a criterion for varieties in Fujiki's class $\mathcal C$ to be Kähler. We also prove the existence of small $\mathbb Q$-factorializations for generalized klt pairs and of dlt modifications for generalized pairs.
In this article, we establish the minimal model program for big gklt Kähler pairs, and in particular we prove Tosatti's transcendental base-point-free conjecture.
Using techniques from the theory of foliations, we establish the cone theorem and the contraction theorem for lc generalized pairs in full generality, and meanwhile develop the minimal model program for $\mathbb Q$-factorial foliated dlt algebraically integrable foliations. As an application, we obtain the canonical bundle formula for generalized pairs completely, together with several further consequences, including answering a question of Cascini and Spicer.
Given a canonical algebraically integrable foliation on a klt projective variety, we study the variation of the ample models of the associated adjoint foliated structures with respect to the parameter. When the foliation is of general type, we show the finiteness of ample models if the parameter is sufficiently close to $1$. When the ambient variety is of general type, we show the finiteness of ample models for all parameters. A key ingredient in our proof is the equivalence between the existence of minimal models and the termination of MMP with scaling for algebraically integrable adjoint foliated structures.
We prove the boundedness of complements for Fano type generalized pairs (with the boundary coefficient set $[0,1]$) after Shokurov.
For lc algebraically integrable foliations on klt varieties, we prove the base-point-freeness theorem, the contraction theorem, and the existence of flips. The first result resolves a conjecture of Cascini and Spicer, while the latter two results strengthen a result of Cascini and Spicer by removing their assumption on the termination of flips. Moreover, we prove the existence of the minimal model program for lc algebraically integrable foliations on klt varieties and the existence of good minimal models or Mori fiber spaces for lc algebraically integrable foliations polarized by ample divisors on klt varieties. As a consequence, we show that $\mathbb{Q}$-factorial klt varieties with lc algebraically integrable Fano foliation structures are Mori dream spaces. We also show the existence of a Shokurov-type polytope for lc algebraically integrable foliations.
Let $f:X\to S$ be a projective morphism of normal varieties. Assume $U$ is an open subset of $S$ and $L_U$ is a $\mathbb{Q}$-divisor on $X_U:=X\times_S U$ such that $L_U\equiv_U 0$. We explore when it is possible to extend $L_U$ to a global $\mathbb{Q}$-divisor $L$ on $X$ such that $L\equiv_f 0$. In particular, we show that such $L$ always exists after a (weak) semi-stable reduction when $\dim S=1$. On the other hand, we give an example showing that $L$ may not exist (after any reasonable modification of $f$) if $\dim S\ge 2$, which also gives an $f_U$-nef divisor $M_U$ that cannot extend to an $f$-nef ($\mathbb{Q}$) divisor $M$ for any compactification of $f|_U$, even after replacing $X_U$ with any higher birational model.
We prove the existence of good minimal models for any klt algebraically integrable adjoint foliated structure of general type, and that Fano algebraically integrable adjoint foliated structures with total minimal log discrepancies and parameters bounded away from zero form a bounded family. These results serve as the algebraically integrable foliation analogues of the finite generation of the canonical rings proved by Birkar-Cascini-Hacon-M\textsuperscript{c}Kernan, and the Borisov-Alexeev-Borisov conjecture on the boundedness of Fano varieties proved by Birkar, respectively. As an application, we prove that the ambient variety of any lc Fano algebraically integrable foliation is of Fano type, provided the ambient variety is potentially klt.
We show that the anti-canonical bundle of any $\mathbb Q$-factorial surface is numerically effective if and only if it is pseudo-effective. To prove this, we establish a numerical non-vanishing theorem for surfaces polarized with pseudo-effective divisors. The latter answers a question of C. Fontanari.
For $\mathbb Q$-factorial klt algebraically integrable adjoint foliated structures, we prove the cone theorem, the contraction theorem, and the existence of flips. Therefore, we deduce the existence of the minimal model program for such structures. We also prove the base-point-freeness theorem for such structures of general type and establish an adjunction formula and the existence of $\mathbb Q$-factorial quasi-dlt modifications for algebraically integrable adjoint foliated structures.
We present an extension of several results on pairs and varieties to foliated surface pairs. We prove the boundedness of local complements, the local index theorem, and the uniform boundedness of minimal log discrepancies (mlds), as well as establishing the existence of uniform rational lc polytopes. Furthermore, we address two questions posed by P. Cascini and C. Spicer on foliations, providing negative responses. We also demonstrate that the Grauert-Riemenschneider type vanishing theorem generally fails for lc foliations on surfaces. In addition, we determine the set of minimal log discrepancies for foliated surface pairs with specific coefficients, which leads to the recovery of Y.-A. Chen's proof on the ascending chain condition conjecture for mlds for foliated surfaces.
In this paper, we show the existence of uniform rational lc polytopes for foliations with functional boundaries in dimension $\leq 3$. As an application, we prove the global ACC for foliated threefolds with arbitrary DCC coefficients. We also provide applications on the accumulation points of lc thresholds of foliations in dimension $\leq 3$.
We show that log canonical thresholds of fixed dimension are standardized. More precisely, we show that any sequence of log canonical thresholds in fixed dimension $d$ accumulates in a way which is i) either similar to how standard and hyperstandard sets accumulate, or ii) to log canonical thresholds in dimension $\leq d-2$. This provides an accurate description on the infinitesimal structure of the set of log canonical thresholds. We also discuss similar behaviors of minimal log discrepancies, canonical thresholds, and K-semistable thresholds.
Given an NQC log canonical generalized pair $(X,B+M)$ whose underlying variety $X$ is not necessarily $\mathbb{Q}$-factorial, we show that one may run a $(K_X+B+M)$-MMP with scaling of an ample divisor which terminates, provided that $(X,B+M)$ has a minimal model in a weaker sense or that $K_X+B+M$ is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance, when the boundary contains an ample divisor.
We establish a Kollár-type gluing theory for NQC generalized log canonical pairs and use it to prove semi-ampleness results of NQC generalized pairs. As consequences, we prove the existence of flips for any NQC generalized log canonical pair, and show that NQC generalized log canonical singularities are Du Bois.
We prove some basic properties of the relative Nakayama-Zariski decomposition. We apply them to the study of lc generalized pairs. We prove the existence of log minimal models or Mori fiber spaces for (relative) lc generalized pairs polarized by an ample divisor. This extends a result of Hashizume-Hu to generalized pairs. We also show that, for any lc generalized pair $(X,B+A,{\bf{M}})/Z$ such that $K_X+B+A+{\bf{M}}_X\sim_{\mathbb R,Z}0$ and $B\geq 0,A\geq 0$, $(X,B,{\bf{M}})/Z$ has either a log minimal model or a Mori fiber space. This is an analogue of a result of Birkar/Hacon-Xu and Hashizume in the category of generalized pairs, and is later shown to be crucial to the proof of the existence of lc generalized flips in full generality.
We establish the Kodaira vanishing theorem and the Kawamata-Viehweg vanishing theorem for lc generalized pairs. As a consequence, we provide a new proof of the base-point-freeness theorem for lc generalized pairs. This new approach allows us to prove the contraction theorem for lc generalized pairs without using Kollár's gluing theory.
We show that generalized log canonical thresholds for complex analytic spaces satisfy the ACC and we characterize the accumulation points.