In this talk, I will survey aspects of Brill-Noether theory for moduli spaces of sheaves on surfaces. I will first discuss the cohomology of the general stable sheaf on surfaces such as Hirzebruch, K3 and abelian surfaces. I will then describe some recent results on the cohomology jumping loci. This talk is based on joint work with Jack Huizenga, Howard Nuer, Neelarnab Raha and Kota Yoshioka.

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $K$ of characteristic $0$ and $H$ an ample divisor on $X$. Consider $M_{H}\ = M\ \{ X,H\}\ (r;c_{1},...,c_{s}\ )\ $the moduli space of $H$-stable rank-r vector bundles on $X$ with fixed Chern classes $c_{i}(E): = c_{i}$ for $i = 1,...,s: = \text{min}\{ r,n\}$. One way to study the geometry of the moduli space $M_{H}$ is by considering its subvarieties. In particular, one can consider the subvarieties called Brill-Noether loci, whose points are the stable vector bundles having at least $k$ independent sections. In this talk, we will focus the attention on the non-emptiness of Brill-Noether locus in the case when $X$ is a ruled surface and a ruled 3-fold.

In this talk, we will introduce the $3$-dimensional Noether inequality and completely classify the canonical threefolds on the Noether line with $p_{g} \geq 5$ by studying their moduli spaces. For every such moduli space, we establish an explicit stratification, estimate the number of its irreducible components and prove the dimension formula. A new and unexpected phenomenon is that the number of irreducible components grows linearly with the geometric genus, while the moduli space of canonical surfaces on the Noether line with any prescribed geometric genus has at most two irreducible components of the same dimension. This is a joint work with S.Coughlan, R.Pignatelli and T.Zhang.

The family of plane cubic curves is parametrized by the projective space of dimension nine. The assignment to a plane cubic curve its Hessian cubic is a $3 - 1$ map. The Hessian cubic comes with a non-trivial $2$-torsion divisor class, and it allows one to represent the Hessian as the symmetric determinant of a net of cubics. This defines a birational map from the projective space to the Grassmannian $G(3,6)$. We use the notion of net logarithmic sheaves to describe this picture and resolve the birational map. This is a joint work with Simone Marchesi and Joan Pons-Llopis.

We show that the $Q$-Fano index of a canonical weak Fano 3-fold is at most $66$. This upper bound is optimal and gives an affirmative answer to a conjecture of Chengxi Wang in dimension $3$. During the proof, we establish a new Riemann\--Roch formula for canonical 3-folds and provide a detailed study of non-isolated singularities on canonical Fano 3-folds, concerning both their local and global properties. Our proof also involves a Kawamata\--Miyaoka type inequality and geometry of foliations of rank $2$ on canonical Fano 3-folds. This is a joint work with Haidong Liu.

Some invariants such as Picard numbers, degrees, Fano indices, play important roles in the classification of Fano threefolds with mild singularities. In this talk, I will show that the optimal upper bound of degrees of canonical Fano threefolds with Picard number $1$ is $72$. This is a joint work with Chen Jiang and Jie Liu.

Let X be a very general hypersurface of degree d in the projective $(n + 1)$-space with $n \geq 3$, and $f:X \rightarrow Y$ a non-birational surjective morphism to a normal projective variety $Y$ . We first prove that $Y$ is rationally connected. Next, we prove that $Y$ is a Fano variety provided that either $Y$ has only canonical singularities or $\text{deg}\ f\ \geq \ C$ for some constant $C\ = \ C(n,\ d)$ depending only on $n$ and $d$. Furthermore, we show that if the degree of $f\ $is prime and large then $Y\ $is isomorphic to the projective n-space. This is joint work with Yongnam Lee and De-Qi Zhang.

The geometric theory of uniruled projective manifolds introduced by Hwang-Mok revolves around the notion of the variety of minimal rational tangents (VMRT), i.e., the collection at a general point of tangents to *minimal rational curves*. With an aim to schematize the proof of rigidity under Kähler deformation, I formulated the *recognition problem* on rational homogeneous manifolds $X$ of Picard number $1$ in VMRTs, viz., the question of recognizing a rational homogeneous manifold $X\ = \ G/P$ of Picard number 1 by its VMRT at a *general* point. In works with J. Hong on mappings and Y. Zhang on geometric substructures I formulated a relative version of the recognition problem for *admissible* pairs $(X_{0},X)$ consisting of rational homogeneous manifolds $X_{0} \subset X$ of Picard number $1$, including those defined by marked Dynkin subdiagrams. The pair $(X_{0},X)$ is said to be rigid whenever the inclusion $\mathcal{C}_{0}\left( X_{0} \right) \subset \mathcal{C}_{0}(X)\mathbb{\subset P}T_{0}X$ at a *general* point determines the pair. This problem has at its origin the problem of Schur rigidity in relation to Schubert differential systems first studied by Walters and Bryant in the Hermitian symmetric case. It turns out that, for applications to several complex variables, it is necessary to study the rigidity problem for certain admissible pairs $(X_{0},X)$ of non-subdiagram type. With this background I will introduce the semi-rigidity theorem of Kim-Mok-Seo (2025) characterizing proper holomorphic maps $f:\Omega \rightarrow \Omega'$ between certain classical symmetric domains $\Omega$ and $\Omega'$ of rank $r$ resp. $r'$ satisfying $r \geq 2$ and $r' < 2r - 1$, proving a well-known conjecture confirming *semi-rigidity* of such maps in these cases. Our proof exploits a combination of techniques from several complex variables (Mok-Tsai), CR-geometry (Kim, Kim-Zaitsev) and the geometric theory of uniruled projective manifolds. I will focus on the most difficult case of proper holomorphic maps from a type-III domain (i.e., biholomorphically the Siegel upper half-plane) to a type-I domain, which necessitates the solution of Hwang-Li (2021) of the recognition problem for *symplectic Grassmannians* and a technique of proving rigidity of admissible pairs of *non-subdiagram* type using the theory of sub-VMRT structures, especially the *thickening lemma* applied to the transport of geometric substructures along minimal rational curves.

I will show how to construct the Chow quotient of the complete flag variety of linear subspaces of the three dimensional complex projective space by its maximal torus, proving that it is smooth and a Mori Dream Space, and describing in detail its rich birational geometry. This is a joint work with Lorenzo Barban and Luis Eduardo Solá Conde.

A Q-conic bundle is a contraction $f:\ X\ \rightarrow \ Z$ of a threefold with only terminal singularities to a surface such that the anticanonical divisor $- K_{X}$ is $f$-ample and the relative Picard number $\rho(X/Z)$ equals $1$. In my talk I introduce several important classes of elementary birational transformations that improve the singularities of the underlying variety $X$, as well as, the singularities of linear systems on $X$. These transformations are naturally appear as steps of the Sarkisov program.

In this talk, I will discuss how to apply the local theory of K-stability to obtain some boundedeness results in the Minimal Model Program for general type varieties. Based on joint work with Jingjun Han and Ziquan Zhuang.

A pluricanonical fibration of a compact complex manifold is a meromorphic map defined by a sufficiently large and divisible multiple of its canonical class. For a projective variety, a theorem due to P. Deligne and K. Ueno asserts that the image of the automorphism group of the variety in the automorphism group of the base of its pluricanonical fibration is finite. I will tell about an analog of this result for compact complex manifolds of dimension $N$ and Kodaira dimension $N - 1$. The talk is based on a joint work with K.Loginov.

I will report a preprint joint with Mingshuo Zhou, in which we prove that moduli space of parabolic bundles with a fixed determinant on a generic curve is globally $F$-regular.

Kawaguchi and Silverman proposed a conjecture about arithmetic and dynamical degrees of dominant rational self-maps defined over number fields. There are several approaches to this conjecture. We will give an overview, based on the joint works with Yohsuke Matsuzawa, and with Sheng Meng and Tianle Yang.

Let $X$ be a complex surface admitting a nowhere vanishing holomorphic $2$-form; such a form induces a holomorphic symplectic structure on $X$. In this talk, we consider the case when $X$ is an elliptic surface and study how the symplectic geometry is related to the underlying complex geometry of the elliptic fibration. In particular, when the elliptic fibration is non-isotrivial, we introduce a symplecto-functional invariant which determines its symplectic geometry. Applications will be addressed to isogenies of elliptic K3 surfaces. This is based on a joint work with Jun-Muk Hwang.

Classical Teichmüller space plays a central role in geometry, connecting Riemann surfaces, hyperbolic geometry, and complex analysis. Hitchin gave a modern perspective on this space using the non-abelian Hodge correspondence, which provides a striking interaction between complex geometry and topology via differential equations. In the $p$-adic setting, we propose a parallel program that can be developed through the $p$-adic Simpson correspondence. In this talk, I will describe an approach to constructing a $p$-adic analogue of Teichmüller space by applying this correspondence to certain special Higgs bundles. This $p$-adic Teichmüller space is expected to reveal deep connections with anabelian geometry, a field that studies how algebraic fundamental groups encode arithmetic information. The talk will draw on joint work with Mao Sheng and Jinbang Yang, as well as ongoing collaborations with Xiaotao Sun and Jinbang Yang.