Paolo Cascini, Calum Spicer
We show that termination of flips for $\mathbb Q$-factorial klt pairs in dimension $r$ implies existence of minimal models for algebraically integrable foliations of rank $r$ with log canonical singularities over a $\mathbb Q$-factorial klt projective variety.
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MMP for smooth co-rank one foliations
该研究证明了一般类型的光滑余秩一叶状结构的极小模型存在性
Counterexamples to the MMP for 1-foliations in positive characteristic
本文证明了正特征下1-叶状曲面对的极小模型纲领（MMP）中锥定理、无基点定理及Mori纤维空间存在性等核心结论不成立，提供了反例。
On the MMP for rank one foliations on threefolds
本文证明 Q-因子射影 klt 三维簇上秩一对数典范叶状配对存在翻转，从而推广了三维秩一叶状结构极小模型的存在性结果。
Recent progress on the Minimal Model Program for foliations
本文综述了复杂变分上叶结构有理几何的最新进展，重点探讨了最小模型程序视角下的奇点、邻接及在曲面和三维翻转存在性中的应用。
Log Canonical Minimal Model Program for corank one foliations on Threefolds
本文研究了三重形上秩为一的叶形的Log Canonical极小模型程序，证明了通过运行$(K_\mcF+\Delta)$-MMP可以得到极小模型或Mori纤维空间，并探讨了极小模型之间的连接及有限数量性质。
MMP for co-rank one foliations on threefolds
该研究针对Q-阶乘射影三维簇上余秩一的F-dlt叶状配对，证明了翻转存在、特殊终止、基点自由定理及对数一般型下极小模型存在，推导了相关修正与奇点性质，验证了数值平凡情形的丰饶性，推进了叶状结构极小模型计划研究。
Sarkisov program for algebraically integrable and threefold foliations
本研究利用伴随叶状结构的极小模型纲领理论，建立了klt簇上代数可积叶状结构的Sarkisov纲领，结合已有结果推广到维数≤3的温和奇点叶状结构，并给出其对数与伴随叶状版本。
Volumes of foliations birationally bounded by algebraically integrable families
研究证明了由代数可积族理性有界所限定的log canonical叶体积集合满足DCC，并建立了弱半稳定映射家族的相对log canonical体积的变形不变性。
Existence of complements for foliations
本文证明了klt环境簇上代数可积log-Fano叶状结构存在$\mathbb{Q}$-补，并研究了其伴随反转与连通性原理。
In his work on birational classification of foliations on projective surfaces, Brunella showed that every regular foliation on a rational surface is algebraically integrable with rational leaves. This led Touzet to conjecture that every regular foliation on a rationally connected manifold is algebraically integrable with rationally connected leaves. Druel proved this conjecture for the case of weak Fano manifolds. In this paper, we extend this result showing that Touzet's conjecture is true for codimension one foliations on threefolds with nef anti-canonical bundle.
In this paper we prove that a regular foliation on a complex weak Fano manifold is algebraically integrable.
ACC for lc thresholds for algebraically integrable foliations
本文研究代数可积叶状结构的 lc 阈值，证明其满足 ACC 及全局 ACC，并给出相关应用。
A basepoint free theorem for algebraically integrable foliations
本文证明了代数可积叶状结构在特定条件下的无基点定理，并给出了 F-dlt 对的收缩定理及 b-半充足猜想特例等相关应用。
We give a characterization theorem for non-degenerated plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree $r$ of a non-degenerated foliation as above provides the minimum number, $r+1$, of points in the projective plane through which infinitely many algebraic leaves of the foliation go.
Minimal model program for algebraically integrable foliations on klt varieties
本文研究了klt变种上的lc代数积分可积foliation的最简模型程序，证明了基点自由定理、收缩定理和翻转的存在性，展示了Q-因子理klt变种的Mori梦空间性质。
Flop between algebraically integrable foliations on potentially klt varieties
该研究证明潜在klt簇上lc代数可积叶状三元组的极小模型可经翻转变换连接，还探讨了三维非代数可积叶状结构极小模型的联系。
Minimal model program for algebraically integrable foliations and generalized pairs
利用叶状结构理论，建立了一般lc广义对的锥定理与收缩定理，并发展了Q-因子叶状dlt代数可积叶状结构的最小模型纲领，从而完全得到了广义对的典范丛公式，并回答了Cascini与Spicer的一个问题。
Foliated Minimal Models and Flops
本文研究叶状结构的极小模型与翻转，证明秩1叶状结构的$K_F$-MMP输出同构、三维簇余秩1叶状结构的$D$-翻转存在性，同时构造例子揭示秩1叶状结构的病态。
Volume of algebraically integrable foliations and locally stable families
本文研究代数可积叶状结构的体积与局部稳定族，证明典范代数可积叶状结构的体积属于仅依赖秩和一般叶体积的离散集，一般型时有正下界，并推导了稳定族相对体积的离散性及对数版本结果。
Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open set of stable points, through which every invariant morphism factors uniquely.
On finite generation and boundedness of adjoint foliated structures
本文证明了一般型klt代数可积伴随叶状结构存在良好极小模型、特定Fano代数可积伴随叶状结构构成有界族，并给出环境簇为Fano型的应用，分别类比典范环有限生成性与Fano簇有界性结果。
Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures
本文在丰裕猜想下建立叶状结构非代数性判据，证明伴随叶状结构的丰裕性及良好极小模型存在性，并解决曲面情形问题。
Variation of algebraically integrable adjoint foliated structures
该研究针对klt射影簇上的典范代数可积叶状结构，分析伴随叶状结构丰富模型随参数的变化，证明了不同一般型条件下的丰富模型有限性，核心依据是极小模型存在性与带标度MMP终止性的等价性。
We will work with codimension one holomorphic foliations over the complex projective space, represented by integrable forms $ω\in H^0(Ω^1_{\PP^n}(e))$. Our main result is that, under suitable hypotheses, the Kupka set of the singular locus of $ω\in H^0(Ω^1_{\PP^3}(e))$, defined algebraically as a scheme, turns out to be arithmetically Cohen-Macaulay. As a consequence, we prove the connectedness of the Kupka set in $\PP^n$, and the splitting of the tangent sheaf of the foliation, provided that it is locally free.
Birational boundedness of stable families
本文证明了固定维数和有界余体积的稳定族的理查德森有界性，通过研究代数可积的叶结构，建立了叶状结构的理查德森几何框架，解决经典纤维化有界性问题。
Let X be a variety over an algebraically closed field, η:Ω^1_X\to L a one-dimensional singular foliation, and C\subseteq X a projective leaf of η. We prove that 2p_a(C)-2=°(L|C)+λ(C)-°(C\cap S) where p_a(C) is the arithmetic genus, where λ(C) is the colength in the dualizing sheaf of the subsheaf generated by the Kähler differentials, and where S is the singular locus of η. We bound λ(C) and °(C\cap S), and then improve and extend some recent results of Campillo, Carnicer, and de la Fuente, and of du Plessis and Wall.
We prove the Cone Theorem for algebraically integrable foliations. As a consequence, we show that termination of flips implies the b-nefness of the moduli part of a log canonical pair with respect to a contraction, generalising the case of lc trivial fibrations.
Holomorphic foliations of degree two and arbitrary dimension
本文研究了二次全纯叶形在任意维度上的分类，证明了其分类结果并探讨了相关泊松结构的几何特性。
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