In characteristic zero, birational maps between smooth projective varieties factorize through a sequence of blow-ups and blow-downs along smooth centers. While these centers are not unique in general (leading to e.g. new explanations as to why most Cremona groups are not simple), we show that for maps between complex projective threefolds, the factorization centers are always unique up to stable birational equivalence. Joint work in progress with E. Shinder.)

In this talk, I report recent progress on syzygies of secant varieties of smooth projective curves. First, we extend Green's $(2g+1+p)$-theorem to secant varieties of smooth projective curves. This confirms Sidman-Vermeire's conjecture. This part is joint work with Lawrence Ein and Wenbo Niu. Next, we show a generalization of the gonality conjecture on syzygies of smooth projective curves to their secant varieties. More precisely, we prove that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of curves of large degree. This answers a question of Ein. This part is joint work with Junho Choe and Sijong Kwak. Our results show that there is a "matryoshka structure" among secant varieties of smooth projective curves.

In 1992, Koll'{a}r-Miyaoka-Mori proved that the set of smooth Fano varieties with fixed dimension is bounded. A generalization of this theorem to the singular case was conjectured (BAB conjecture), and recently solved by Birkar. However, little is known in positive characteristic even in dimension three. In this talk, I will show that a family of globally F-split Fano threefolds (with some additional assumptions) is bounded.

In this talk, we will introduce the structure theorem of projective klt varieties with certain positive tangent sheaves. More precisely, if the tangent sheaf is almost nef or positively curved, after taking a quasi-etale cover, we can take a well-defined MRC fibration onto an Abelian variety. Furthermore, we will introduce how the tangent sheaf is related to the structure of projective klt varieties. This is based on joint work with Shin-ichi Matsumura (Tohoku University) and Guolei Zhong (IBS-CCG) (arXiv:2309.09489).

Symmetric varieties are normal equivarient open embeddings of (algebraic) symmetric homogeneous spaces $G/H$, where $G$ is a simply-connected reductive algebraic group and $H$ is the subgroup consisting of elements fixed by an algebraic group involution of $G$. We can consider a symmetric homogeneous space $G/H$ as a kind of complexification of a Riemannian symmetric space. For example, the symmetric homogeneous space $SL(m, \mathbb{C})/S(GL(r, \mathbb{C}) \times GL(m-r, \mathbb{C}))$ of type $AIII(r, m)$ is an open orbit for the diagonal action of $SL(m, \mathbb{C})$ on the product of complex Grassmannians $Gr(r, m)$ and $Gr(m-r, m)$. As we have a combinatorial criterion for K-stability of smooth Fano spherical varieties obtained by Delcroix in terms of algebraic moment polytopes, one can ask the following questions: (1) Is the wonderful compactification of a symmetric homogeneous space K-polystable? (2) Which of the blow-ups of wonderful compactifications of symmetric homogeneous spaces along the (unique) closed orbit are K-polystable? In this talk, we answer the questions in the case of the wonderful compactifications of symmetric homogeneous spaces of type $AIII(2, m)$ and their blow-up

Let $X$ be a complex smooth Fano variety whose minimal anticanonical degree of non-free rational curves on $X$ is at least $\dim(X)-2$. We give a classification of extremal contractions of such varieties. As applications, we obtain a classification of Fano fourfolds whose pseudoindex and Picard number are greater than one and study the structure of Fano varieties with nef third exterior power of the tangent bundle.

The cotangent bundle of a complex projective manifold naturally carries a holomorphic symplectic $2$-form. The structure of Lagrangian fibration on these non-compact complex manifolds has not been extensively studied. In this talk, I will initially introduce well-known facts about proper Lagrangian fibrations, relying on the results of Hwang and Oguiso (2009). Subsequently, I will present some examples of non-proper Lagrangian fibrations defined on cotangent bundles and compare their fibration structure with that of proper ones.

The Beauville--Bogomolov decomposition states that a compact K\"{a}hler manifold with numerically trivial canonical bundle admits a finite \'{e}tale cover that decomposes into a product of three types of manifolds; a torus, simply-connected Calabi--Yau, and symplectic manifolds. Recently, in positive characteristic, Patakfalvi and Zdanowicz established a weak version of Beauville--Bogomolov decomposition, which claims that a smooth projective F-split variety with trivial canonical bundle admits a finite cover that decomposes into a product of two types of varieties; a torus and a projective variety with mild singularities whose augmented irregularity is zero. Here, although the cover is not necessarily \'{e}tale, it is proved to have a good property in some sense. In this talk, we generalize Patakfalvi and Zdanowicz's theorem to the case when the anti-canonical bundle is numerically equivalent to a semi-ample line bundle. Its characteristic-zero counterpart has been shown by using Ambro's theorem and the abundance theorem for numerically trivial canonical bundles.

In recent years, significant progress has been made in the field of birational geometry for foliations. Notably, the Minimal Model Program (MMP) has been shown to work for foliations on threefolds. In this talk, I will demonstrate that the MMP is applicable to toric foliations as well. Specifically, I will discuss how non-dicritical singularities (and foliated dlt singularities if time permits) are preserved under the MMP. This is a joint work with Chih-Wei Chang.

Over the past decade, the Minimal Model Program (MMP) has been largely established for threefolds over perfect fields of characteristic $>3$. A central conjecture remaining is the abundance conjecture, which predicts that if $(X,B)$ is a projective log canonical threefold pair over an algebraically closed field of characteristic $>3$ and $K_{X}+B$ is nef, then $K_{X}+B$ is semiample. In this talk, I will discuss some recent results towards this conjecture.

It's well-known that there are no nonconstant morphisms from $\mathbb{P}^m$ to $\mathbb{P}^n$ if $m>n$. The same statement with $\mathbb{P}^n$ replaced by general Grassmannian $\mathbb{G}(k,n+1)$ was obtained by Tango. Recently, Naldi and Occhetta extended the result to morphisms $M\rightarrow \mathbb{G}(k,n+1)$ from smooth projective variety $M$, with the concept of effective good divisibility $e.d.(M)$. In this talk, I will talk about morphisms from projective spaces to flag varieties. Moreover, I will give an application of this result on the classification of higher rank uniform bundles on projective spaces. This is a joint work with Peng Ren.

A fine classification of minimal surfaces of general type with $p=q=2$ is still missing and even an effective bound of the degrees of the Albanese morphisms of these surfaces is out of reach. We give a classification theory on these surfaces with $K^2=5$ or $6$ via cohomological rank functions. This is a joint work with Zhi Jiang and Guoyun Zhang.

Due to a conjecture of Bloch, the transcendental lattice of a surface should control its transcendental motive. In particular, the conjecture implies that any symplectic automorphism of a K3 surface acts trivially on $CH_0$. Voisin and Huybrechts have confirmed the conjecture for automorphism of finite order, which forces the Picard number to be large. We showed that any symplectic automorphism of a K3 surface acts trivially on $CH_0$ as long as the Picard number is larger than $2$. We also obtains some results about Bloch's conjecture on HK varieties of $K3^{[n]}$−type. This is a joint work with Zhiyuan Li and Xun Yu.

In 1983, J. Koll\'ar and T. Matsusaka proved that for a big and semi-ample line bundle $L$ over a nonsingular projective variety $X$ of dimension $n$ over $\mathbb{C}$, $\left | h^{0}(X,L^{\otimes r}- \frac{L^{n}}{n!}r^{n}) \right |$ can be controlled by some polynomial of $L^{n}, K_{X} \cdot L^{n-1}$ and the multiples $r$, whose degree is at most $n-1$ with respect to $r$, and whose coefficients only depend on the dimension $n$ of $X$. These uniform controls are generalized by T. Luo to big and nef line bundles, and sharpened by Peter A. Nielsen. When L is ample, involving Hirzebruch-Riemann-Roch formula, we discover that these inequalities can be interpreted as uniform controls of intersection numbers of Chern classes of $X$ and powers of $L$, by polynomials of $L^{n}$ and $K_{X} \cdot L^{n-1}$. Our results recover the results of Rong Du and Hao Sun in [DS22]. The proof of our results relies on Demailly's results on Fujita's very ampleness conjecture, and inequalities of intersection numbers of Chern classes of nef vector bundles and powers of ample line bundles.