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: The intersection cohomology Hodge modules of toric varieties.
Abstract: The intersection cohomology complex IC_X on a toric variety X has been well studied starting with the works of Stanley and Fieseler, and more recently, the works of de Cataldo-Migliorini-Mustata and Saito. However, it has a richer structure as a Hodge module (denoted IC^H_X) in the sense of Saito’s theory, and so we have the graded de Rham complexes gr_k(DR(IC^H_X)), which are complexes of coherent sheaves carrying significant information about X.
In this talk, I will describe the generating function of gr_k(DR(IC^H_X)) and give a precise formula relating it with the stalks of the perverse sheaf IC_X (in particular, this implies that the generating function depends only on the combinatorial data of the toric variety). This is joint work with Hyunsuk Kim.
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.
Title: Minimal Log Discrepancy and Orbifold Curves.
Abstract: We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves. The talk is based on joint work with Zhengyi Zhou.
Title: On the Casas-Alvero conjecture and its proof.
Abstract: Around 2001, motivated by his work on singularities, Casas-Alvero conjectured the following: If a monic univariate polynomial $f(X)$ of degree $n$, over a field of characteristic $0$, has a non-trivial gcd with each of its formal derivatives $f^{(i)}(X)$ for $1\leq i\leq n-1$, then $f(X)$ is a pure power of a monic linear polynomial. In this talk, we will show that over any field of characteristic $p>0$, there are "finitely many" counterexamples to the conjecture, and also sketch a proof of the conjecture over characteristic $0$.
Title: Fano manifolds of Picard rank 2 with simplest contractions.
Abstract: We will classify Fano manifolds of Picard rank 2 whose each elementary contraction is either a projective bundle or a smooth blow up over a projective space. Most part of the talk will be devoted to explain the classification results with the examples. After that, we will sketch the proof of the classification in the case where one contraction is a projective bundle and the other is a smooth blow up.
Abstract: The complexity is an invariant of log pairs that was shown by Brown-McKernan-Svaldi-Zong to characterize toric varieties. More precisely, they showed that toric Calabi-Yau pairs minimize the complexity among all Calabi-Yau pairs. I will describe joint work with Jennifer Li and José Ignacio Yáñez in which we study other geometric consequences of small complexity. For example, we provide a criterion in terms of the complexity for a variety to be cluster type.
Title: Laurent polynomials and deformations of non-isolated Gorenstein toric singularities.
Abstract: We establish a correspondence between one-parameter deformations of an affine Gorenstein toric variety X, defined by a polytope P, and mutations of a Laurent polynomial f, whose Newton polytope is equal to P. If the Newton polytope P of f is two dimensional and there exists a set of mutations of f that mutate P to a smooth polygon, then, we show that the Gorenstein toric variety, defined by P, admits a smoothing. This smoothing is obtained by proving that the corresponding one-parameter deformation families are unobstructed and that the general fiber of this deformation family is smooth.
Title: Inequalities of Miyaoka-type and Uniformisation for Varieties of
intermediate Kodaira Dimension.
Abstract: Let $X$ be a minimal complex projective variety. Over the past years,
many similar inequalities between the Chern classes of $X$ have been
obtained. Moreover, it is known precisely which varieties $X$ can
achieve the equality. However, so far all results in this direction have
focussed on the case where the numerical dimension of $X$ is either very
small or very large. In this talk, I will present analogous inequalities
for varieties of intermediate Kodaira dimension and I will present a
characterisation of those varieties achieving the equality. This talk is
partially based on joint work with Masataka Iwai and Shin-ichi
Matsumura.
Title: King's conjecture, Birational Geometry, and Mirror Symmetry.
Abstract: King conjectured that the category of coherent sheaves on a projective toric variety always admits a full strong exceptional collection of line bundles. While this conjecture has been proven in many cases, it is false in general, with counterexamples by Hille-Perling, Efimov, and Michalek. I will describe a modification of the conjecture which turns out to be true, at the expense of enlarging the category of coherent sheaves using categories associated to other birational models. The argument is inspired by homological mirror symmetry, which I will also spend some time explaining. Based on joint work with Ballard, Berkesch, Brown, Erman, Favero, Hanlon, Heller and Huang.
Title: Kawaguchi-Silverman conjecture for int-amplified endomorphism.
Abstract: Let f be a surjective endomorphism of a smooth projective variety X over a number field. The Kawaguchi-Silverman conjecture asserts that given a closed point x whose forward orbit is Zariski dense in X, the arithmetic degree of f at x coincides with the first dynamical degree of f. In this talk, based on the previous work of Meng-Zhang, I would like to verify this conjecture when f has a dominant topological degree (or equivalently, f is int-amplified). The key step is to show that the pathological case when f has totally invariant ramifications does not occur up to a finite cover. This is based on a joint work with Sheng Meng.
Title: Variation of cones of divisors in a family of Fano type varieties.
Abstract: The study of moduli problems for algebraic varieties often requires the understanding of how geometric properties relate between the total space and its fibers. In this talk, I will begin by presenting a conjecture concerning the relationship between the Fano type property on a Zariski dense subset of fibers and the global Fano type property of the total space, followed by a partial result toward this conjecture. I will then discuss the invariance of Neron-Severi spaces, nef cones, effective cones, movable cones, and Mori chamber decompositions for a family of Fano type varieties after a generic finite base change. Furthermore, I will describe the uniform behavior of the minimal model program in such families. Various forms of these results are known in the literature and play a central role in the study of moduli spaces of varieties. This talk is based on joint work with Sung Rak Choi and Chuyu Zhou.
Title: Computing direct sum decompositions (and some applications).
Abstract: Describing the indecomposable summands of modules or sheaves is a recurring theme in algebraic geometry; for example, Hartshorne's conjecture about low-rank vector bundles on projective space, or describing the summands of the Frobenius pushforward of a coherent sheaf on a projective variety. However, to this point practical algorithms for producing direct sum decompositions of modules have been lacking. In this talk, we will describe two variants of an algorithm for finding indecomposable summands of finitely generated modules over a finitely generated k-algebra R. The first algorithm applies in the (multi)graded case, which enables the computation of indecomposable summands of coherent sheaves on subvarieties of toric varieties (in particular, for varieties embedded in projective space); the second algorithm applies when R is local and k is a finite field, which opens the door to computing decompositions in singularity theory. We will give multiple examples, including of previously unknown behavior of summands of Frobenius pushforwards (including in the non-homogeneous case) and syzygies over Artinian rings.