Title: Transcendental Minimal Model Program for Projective Varieties.
Abstract: In recent years there has been much progress for the minimal model program for complex projective varieties. It is conjectured that the minimal model program should also hold for compact Kähler varieties. This is known in dimension 3 but open in higher dimensions.
Since Kähler varieties typically admit very few divisors, it is important to work with closed (1,1) forms \beta \in H^{1,1}_{BC}(X). For example, if \omega is a Kähler form one would like to be able to run the K_X+B MMP with scaling of \omega (which is closely related to the corresponding Kähler Ricci flow). In this talk we will discuss (generalized versions of) this MMP which holds for Kähler varieties in low dimensions and for projective varieties in any dimension.
Abstract: In this talk, we will introduce the concept of cluster type varieties which generalize the concept of toric varieties.
Cluster type varieties are compactifications of algebraic tori and the divisor at the boundary has nice singularities.
Then, we will survey some recent developments in the understanding of cluster type varieties.
Throughout the talk, we will discuss some open problems related to these varieties.
The results presented in this talk are work with many mathematicians including Eduardo Alves da Silva, Joshua Enwright,
Fernando Figueroa, Lena Ji, Konstantin Loginov, Artem Vasilkov, and Jose Yañez.
Abstract: The main characters of this talk are two families of algebras:
Cox rings and cluster algebras. Cox rings are graded algebras associated
with algebraic varieties that encode relevant geometric information
about the underlying varieties. Cluster algebras are some families of
algebras of combinatorial nature that have recently proven to be
fundamental tools in many areas of mathematics. In this talk, we will
discuss results that aim to identify the Cox ring of an algebraic
variety Z with a graded cluster algebra, with a focus on the possible
implications that such an identification has for the Cox ring.
Title: Smooth Calabi-Yau varieties with large index and Betti numbers.
Abstract: A normal variety X is called Calabi-Yau if its canonical divisor is Q-linearly equivalent to 0. The index of X is the smallest positive integer m so that mK_X~0. We construct smooth, projective Calabi-Yau varieties in every dimension with doubly exponentially growing index, which we conjecture to be maximal in every dimension. We also construct smooth, projective Calabi-Yau varieties with extreme topological invariants; namely, their Euler characteristics and the sums of their Betti numbers grow doubly exponentially. These are conjecturally extremal in every dimension. The varieties we construct are known in small dimensions but we believe them to be new in general. This work builds off of the singular Calabi-Yau varieties found by Esser, Totaro, and Wang.
Title: Birational geometry of hypersurfaces in products of weighted projective spaces.
Abstract: Mori dream spaces are a class of algebraic varieties that play a significant role in birational geometry, as they exhibit ideal behavior under the minimal model program. This talk explores the birational geometry of hypersurfaces in products of weighted projective spaces, particularly when they are Mori dream spaces. In such cases, we completely determine their birational geometry. Time permitting, we discuss the birational version of the Kawamata-Morrison cone conjecture for terminal, anticanonical hypersurfaces in Gorenstein products of several weighted projective spaces.
Title: A birational description of the minimal exponent.
Abstract: The minimal exponent of a hypersurface is an invariant of singularities introduced by Morihiko Saito via D-module theory, which refines the log canonical threshold. I will give an introduction to this invariant then I will present a joint result with Qianyu Chen giving a birational description of this invariant via twisted sheaves of log differentials.
Abstract: Hironaka's celebrated resolution of singularities in char 0 proceeds by blowing up carefully chosen subvarieties. There is an alternative canonical process (requiring no choices) called the Nash blowup,
and a fundamental question is whether iterated Nash blowups resolve singularities. We show this is not the case in dimensions greater than 3, and in every characteristic, by constructing counterexamples using toric varieties.
This is joint work with Daniel Duarte, Maximiliano Leyton, and Alvaro Liendo.
Title: On the relative cone conjecture for families of hyperkähler manifolds.
Abstract: The Kawamata-Morrison cone conjecture predicts the geometry of the nef cone and the movable cone of a variety with trivial canonical class. In this talk, we will discuss families of varieties with trivial canonical class and vanishing irregularity. We will study the relative nef cone and the relative movable cone of such families, using machinery from the Minimal Model Program. As application, we will show the relative cone conjecture for families whose very general fiber is a projective hyperkähler manifold of one of the known deformation types. This is joint work with Andreas Höring and Gianluca Pacienza.
Title: On the relative cone conjecture for families of hyperkähler manifolds.
Abstract: The minimal exponent of a hypersurface is an invariant of singularities introduced by Morihiko Saito via D-module theory, which refines the log canonical threshold. I will give an introduction to this invariant then I will present a joint result with Qianyu Chen giving a birational description of this invariant via twisted sheaves of log differentials.
Title: Uniformization of log Fano pairs and Equality in the Miyaoka--Yau inequality.
Abstract: At the beginning of the 20th century, it was known that any compact connected, simply connected Riemann surface is biholomorphic to the projective line. Subsequently, several characterizations of complex projective spaces were established. For instance, Siu and Yau stated that projective spaces are the only Kähler manifolds with positive holomorphic bisectional curvature, and Mori proved that they are the only projective manifolds that have an ample tangent bundle. In a different direction, projective spaces are the only Kähler--Einstein manifolds with a positive constant satisfying the equality in the Miyaoka--Yau inequality. This result originating from uniformization theory was generalized in the singular setting by Greb, Kebekus, Peternell and Druel, Guenancia, Paun. More precisely, they characterize singular quotients of P^n(C) by finite groups acting freely in codimension 1. The aim of this talk is to discuss a generalization of Greb--Kebekus--Peternell's result in order to characterize quotients of P^n(C) by any group action.
Title: Uniformization of log Fano pairs and Equality in the Miyaoka--Yau inequality.
Abstract: We introduce the canonical model singularities (CMS) criterion for bigness of the cotangent bundle for surfaces. The CMS-criterion for bigness involves invariants for canonical singularities that we describe and we give formulas for A_n singularities. The CMS-criterion leads to conjectures and some answers about the geography and the possible ratios $c_1^2/c_2$ of surfaces with big cotangent bundle. Two cases are naturally separated: regular and irregular surfaces. For regular surfaces we apply the CMS-criterion to show the existence of deformations of hypersurfaces in $\PP^3$ with big cotangent bundle for degree $d\ge 8$ and give an example of the regular surface with big cotangent bundle with ratio close to 1/5. For irregular surfaces we show that there are examples with ratio as close to 1/5 as possible. If time permits, we talk about ratios below 1/5.
Title: A characterization of uniruled Kaehler manifolds.
Abstract: We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kaehler manifold.
As a consequence, we extend the algebraic integrability criteria of Campana-Paun and of Druel to foliations on compact Kaehler manifolds.
As an application, we prove that a compact Kaehler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.