Fano's inequality is a mistake

具有典范奇点的$\u211a$-Fano簇的Kawamata--Miyaoka型不等式

Kawamata--Miyaoka type inequality for $\mathbb{Q}$-Fano varieties with canonical singularities

针对余维二光滑、皮卡数为1的$n$维正规$\u211a$-因子典范奇点射影$\u211a$-Fano簇，证明其满足Kawamata--Miyaoka型不等式，并对带终端奇点的三维簇得到更强结果。

Generalized Bogomolov-Gieseker type inequalities on Fano 3-folds

We modify the conjectural Bogomolov-Gieseker type inequality introduced by Bayer, Macri and Toda to construct a family of geometric Bridgeland stability conditions on smooth projective 3-folds. We give an equivalent conjecture which needs to check these inequalities for a small class of tilt stable objects. We extend some of the techniques in Li's work for Fano 3-folds of Picard rank one to establish our modified Bogomolov-Gieseker type inequality conjecture for general Fano 3-folds.

On Q-Fano threefolds of Fano index 2

We show that, for a Q-Fano threefold X of Fano index 2, the inequality dim |-1/2K_X| <= 4 holds with a single well understood family of varieties having dim |-1/2K_X| = 4.

Q-Fano threefolds of index $7$

We show that, for a $\mathbb Q$-Fano threefold $X$ of Fano index 7, the inequality $\dim |-K_X| \ge 15$ implies that $X$ is isomorphic to one of the following varieties $\mathbb P (1^2,2,3)$, $X_6 \subset \mathbb P (1,2^2,3,5)$ or $X_6 \subset \mathbb P (1,2,3^2,4)$.

法诺簇的宫冈-丘不等式与δ不变量

The Miyaoka-Yau inequality and the delta invariant for Fano varieties I -klt Fano case-

研究针对\(n\)维klt法诺簇建立宫冈 - 丘不等式，还受他人启发在孤子情形下证明其等变形式，用等变陈类和加权δ不变量取代相关量，得出允许凯勒 - 里奇孤子的光滑法诺流形的等变宫冈 - 丘不等式。

具有nef反典范除子的终端三维簇的Miyaoka型不等式

Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors

本文研究带nef反典范除子的终端射影3-簇的Miyaoka型不等式，给出陈类乘积的下界及紧性结果，通过部分分类与例子完成证明，还拓展了终端弱Fano簇的相关非零性与Miyaoka–Kawamata型不等式研究。

$cA_1$ 点的 $2n^2$ 不等式及其在双有理刚性中的应用

$2 n^2$-inequality for $cA_1$ points and applications to birational rigidity

本文将光滑点的 $4n^2$ 不等式推广为 $cA_1$ 端点奇点的 $2n^2$ 不等式，并据此证明多类 Fano 三维簇与 del Pezzo 纤维化的双有理刚性。

Fano流形Kähler度量空间上的反向Hölder不等式与有效开性

Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness

本文在Fano流形Kähler度量空间上建立反向Hölder不等式，利用隐藏对数凹性证明并推广至对数终端奇点Fano簇，进而得到测地线速度不等式、有效开性结果及与Igusa zeta函数的联系。

论文《Weyl群与极小模型之间的双有理变换》的补遗/勘误

Addendum/Erratum to the paper "Weyl groups and Birational transformations among minimal models"

本文为1995年关于Fano三维簇与Weyl群论文的补遗和勘误，补入Mori–Mukai分类中遗漏的第88种形变类型，并修正相关错误与计算。

Birational geometry of singular Fano double spaces of index two

In this paper we describe the birational geometry of Fano double spaces $V\stackrelσ{\to}{\mathbb P}^{M+1}$ of index 2 and dimension $\geqslant 8$ with at mostquadratic singularities of rank $\geqslant 8$, satisfying certain additional conditions of general position: we prove that these varieties have no structures of a rationally connected fibre space over a base of dimension $\geqslant 2$, that every birational map $χ\colon V\dashrightarrow V'$ onto the total space of a Mori fibre space $V'/{\mathbb P}^1$ induces an isomorphism $V^+\cong V'$ of the blow up $V^+$ of the variety $V$ along $σ^{…

Arakelov inequalities and the uniformization of certain rigid Shimura varieties

Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1), coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the difference of the slope of E(1,0) and the one of E(0,1) is is smaller than or equal to the degree of the canonical sheaf. We prove a similar inequality in the higher dimensional case. If the latter is an equality, as well as the Bogomolov inequality for E(1,0) or for E(0,1), one hopes that Y is a Shimura variety, and V…