Abstract: In this talk, I want to relate the graded syzygies of a coherent sheaf on projective space with a family of "Euler" stability conditions. I am promoting a point of view: graded syzygies (especially the minimal free resolution) should be organized into filtrations, which will often be Harder-Narasimhan filtrations for appropriate stability conditions. The picture that emerges is a series of moduli spaces, each constructed by Geometric Invariant Theory, that culminates in the moduli space of Gieseker-stable sheaves.

To a variety over a non-Archimedean field, one can associate its Berkovich analytification, which is a space of semivaluations on the function field of the variety. When working over a discretely-valued field of residue characteristic zero, Mustaţă—Nicaise introduced a canonical subspace of the Berkovich analytification called the essential skeleton; by work of Nicaise—Xu, the essential skeleton is identified with the dual complex of a minimal dlt-model of the variety, when it exists. We will describe how one constructs the essential skeleton of a pair (or, more generally, of a log regular log scheme) and how this compares with canonical skeleta that exist in the toric and trivially-valued settings. This is joint work with Enrica Mazzon.

Autoequivalences of a variety X are natural generalizations of the automorphisms of X. When X is a complex, projective K3 surface, Bridgeland conjectured an alluring description of the group of autoequivalences of X. One part of this conjecture has been established in work of Huybrechts, Macri, and Stellari. In this talk, we explain a new proof of their result. We first review the necessary background on K3 surfaces and derived categories and give a precise definition of the notion of autoequivalences. Then, we construct a natural action of the autoequivalences on the cohomology of X, which is essential in formulating Bridgeland's conjecture. Finally, we demonstrate how passing to the larger class of twisted K3 surfaces helps to prove part of the conjecture.

Recently, Birkar and Zhang introduced the concept of generalized polarized pair. This kind of pairs, which arise naturally from the canonical bundle formula, play a central role in recent developlments, such as the study of the Iitaka fibration and the proof of BAB conjecture. A natural question is whether these generalized pairs satisfy the nice properties of usual log pairs. In this talk, we will discuss invariance of plurigenera and volumes in this new setup. Then, moving to the case of surfaces, we will address the boundedness of these objects.

Let $X$ be a Fano variety defined over a number field. Manin's conjecture predicts an asymptotic formula for the number of rational points on $X$ with bounded height. According to the conjecture, the growth of rational points is governed by the Fujita invariant (or the $a$-constant). In this talk we will use birational geometric methods to prove statements related to Manin's conjecture. In particular, we will study the behavior of the Fujita invariant under pull-back to generically finite covers and prove a statement about geometric consistency of Manin's conjecture.

Fiber spaces play an important role in the minimal model program as the results can be categorized into Mori fiber space, Iitaka fibrations over canonical models and varieties of general type. A natural problem to consider would be, if we started with an algebraic fiber space, how might it behave with respect to the minimal model program. For case of elliptic threefolds, it was shown by Grassi, that minimal models of elliptic threefolds related to log minimal models of the base surface and showed that minimal models, in a sense, has to respect the fiber structure for elliptic threefolds. I will present ideas towards a generalization to the case of elliptic fourfolds and higher dimensions.

Let S be the ring of polynomial functions on the space of n x n complex skew-symmetric matrices. This ring has a natural action of the group GL(n), and the invariant ideals correspond to scheme-theoretic thickenings of the Pfaffian varieties. For each invariant ideal I in S, we compute the modules Ext^i(S/I,S). When the thickening is defined by a power of an ideal of Pfaffians, we use these Ext computations to determine the Castelnuovo-Mumford regularity. This allows us to characterize when powers of ideals of Pfaffians have linear minimal free resolution.

I construct an intersection pairing of cycles modulo relations and the corresponding determinant line bundle. More specifically, I consider the map ${CH^p(X)\times CH^q(X)\to \text{Pic}(S)}$ of Chow groups of a variety $X$ over a base $S$. Here $p+q=d+1$, where $d$ is the relative dimension of the morphism $X\to S$. I treat the Chow groups $CH^p(X)$ as categories with objects cycles of codimension $p$ and morphisms arising from the $Z^p(X,1)$ term in Bloch's complex modulo the image of Tame symbols of $K2$-chains. This pairing coincides with the Knudsen-Mumford determinant line bundle using the structure sheaves of the cycles on $X$.

For codimension $q$ cycles that are algebraically trivial on the generic fiber $X_{\eta}$, I show that the image in $\text{Pic}(S)$ does not depend on the rational equivalence of the codimension $p$ cycles. Nevertheless, when working with numerically trivial divisors and zero cycles, the image does depend on the rational equivalence of the zero cycles. As a part of the proof, I construct an explicit isomorphism ${H^1_{et}(X, \mathbb{Z}/n)\to Hom(H_1(Sus_{\bullet}(X)/n), \mathbb{Z}/n)}$, which in the case of smooth curves is the Weil pairing. Equivalently, this is an isomorphism $H_1(Sus_{\bullet}(X)/n)\to \pi_1^{et,ab}(X)/n$.

Based on the constructed pairing, I construct a line bundle on ${CH^p_{alg}(X)\times CH^q_{alg}(X)}$, which I want to prove is canonically isomorphic to the pull-back via the Abel-Jacobi map of the Poincare line bundle on the Intermediate Jacobians $J^p(X)\times J^q(X)$. I also hope to extend the pairing to algebraic singular (Suslin) homology and in motivic setting.

Let $X$ be a smooth projective variety over $\mathbb{C}$ with an embedding $O_X(1)$, and $D \subset X$ be a simple normal crossing divisor. Suppose $(E, \theta)$ is a Higgs sheaf with trivial \emph{parabolic structure} on $D$ such that 1) $E$ is semistable; 2) $c_i(E) = 0$ for all $i$; 3) $\theta$ is nilpotent. Such a Higgs sheaf arises naturally when taking associate graded sheaf of a polarized Hodge structure with unipotent monodromy at $D$. A result of D.Arapura states that \[ \mathbb{H}^i(X, \text{Gr}_F\text{DR}(V, \nabla)\otimes L) = 0 \] for $i > \dim X$. We will generalize D.Arapura's result via a cyclic cover so that the Higgs sheaf in question could have non-trivial parabolic structure on $D$. Moreover, use the moduli space of Higgs sheaf and upper-semicontinuity of cohomology, we can remove the assumption that $\theta$ is nilpotent.