Timetable and abstracts

The lunches will be served between 12:30 and 14:00, and coffee breaks are scheduled between 10:30-11:00 and 15:30-16:00

Abstracts:

• Fano 3-folds with 1-dimensional K-moduli (Erroxe Etxabarri Alberdi)
  We give an introduction to K-stability and the motivation behind it. We will see how to study and completely describe all one-dimensional components of the K-moduli of smooth Fano 3-folds. And we will finish giving some specific examples for family 3.12. This result is in collaboration with Abban, Cheltsov, Denisova, Kaloghiros, Jiao, Martinez-Garcia and Papazachariou.

• Flags on Fano 3-fold hypersurfaces (Livia Campo)
  The existence of Kaehler-Einstein metrics on Fano 3-folds can be determined by studying lower bounds of stability thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. This is deeply intertwined with the geometry of the hypersurfaces: in fact, birational rigidity and superrigidity play a crucial role. The superrigid case had been attacked by Kim-Okada-Won. In this talk I will touch upon the K-stability of strictly rigid Fano hypersurfaces via Abban-Zhuang Theory. This is a joint work with Takuzo Okada.

• Fano 4-folds with rational fibrations onto threefolds (Cinzia Casagrande)
  Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We will first discuss the following result: if rho(X)>12, then X is a product of del Pezzo surfaces; if rho(X)=12, then X has a rational contraction X-->Y where Y has a dimension 3. A rational contraction is a rational map that factors as a sequence of flips followed by a surjective morphism with connected fibers; we will see an explicit example of such setting.  

Then we will discuss the geometric properties of Fano 4-folds X having a rational contraction onto a threefold; this is a joint project with Saverio Secci. We show that, if X has a rational contraction onto a threefold and is not a product of surfaces, then rho(X) is at most 9. This improves the result above, by showing that also Fano 4-folds with rho(X)=12 are products of surfaces. 

The geometric description when X has a rational contraction onto a threefold also allows to construct new examples of Fano 4-folds with large Picard number.

• The effective cone conjecture, and the nef cones of the (many) minimal models of Schoen Calabi—Yau threefolds (Cécile Gachet)
  In 1988, Schoen constructs Calabi-Yau threefolds as fiber products of rational elliptic surfaces. Subsequently, Namikawa (1991) studies birational geometric properties of Schoen threefolds, shows in particular that they have finitely many non-isomorphic minimal models, and counts the exact number of those minimal models (about 56 \times 10^{19}). In 1993, Grassi and Morrison note that the newly stated cone conjecture — predicting the shape of nef cones of Calabi-Yau manifolds in terms of their automorphism groups — is satisfied by Schoen threefolds.
  In this talk, we report on joint work with H.-Y. Lin, I. Stenger, and L. Wang. We explained how to prove the movable cone conjecture for the general Schoen threefold through the lens of an auxiliary “effective cone conjecture” introduced for the occasion. From there, we derive general and explicit finiteness results for the nef cones of the minimal models of the Schoen threefolds. We conclude by noting that all minimal models of a Schoen threefold X, besides X itself, have two maximal fibrations to smooth del Pezzo surfaces, and explain how the shape of their nef cones can be read from those two fibrations.

• Applications of Laurent inversion to K-moduli (Liana Heuberger)

I will discuss how to use Laurent inversion, a technique coming from mirror symmetry which constructs toric embeddings, to study the local structure of the K-moduli space of a K-polystable toric Fano variety. More specifically, starting from a given toric Fano 3-fold X of anticanonical volume 28 and Picard rank 4, and combining a local study of its singularities with the global deformation provided by Laurent inversion, we are able to conclude that the K-moduli space is rational around X. This is joint work with Andrea Petracci.

• The generalized Franchetta conjecture for K3 surfaces (Robert Laterveer)
  O’Grady has formulated an intriguing conjecture concerning the Chow group of zero-cycles of K3 surfaces; according to this conjecture, the « generically defined » cycles should be proportional to the Beauville-Voisin distinguished zero-cycle. I will introduce this conjecture, and report on known results. In particular I will show how one can prove the conjecture for K3 surfaces of genus 14, using the relation with certain famous Fano varieties, i.e. cubic fourfolds. This is based on joint work with Lie Fu.

• Irregular threefolds with nef anticanonical divisors in positive characteristic (Zhan Li)

Abstract: The positivity of anticanonical divisors imposes strong constraints on the geometry of algebraic varieties. Over the complex numbers, it has been proven that the Albanese morphism of varieties with nef anticanonical divisors are isotrivial fibrations. Similar results have been established in positive characteristic, provided that the fibers have good singularities. In this talk, I will present our recent results on the structures of Albanese morphisms for varieties with nef anticanonical divisors, given these morphisms have relative dimension 1 and the anticanonical divisors are relatively ample. This is a joint work with Tongji Gao and Lei Zhang.

• C^*-actions and birational geometry (Gianluca Occhetta)
  In this talk I will explain how some actions of C^* on smooth projective varieties are related to birational transformations between the sink and the source of the action, discussing examples and related problems. These results are scattered in joint works with Jarosław Wiśniewski, Luis E. Solá Conde and Eleonora A. Romano.

• Fano fourfolds with large anticanonical base locus (Saverio Andrea Secci)
  The anticanonical class is probably one of the most natural objects associated with a Fano variety, since it "embodies" many geometric aspects of the ambient space. In this talk I will focus on Fano fourfolds with large anticanonical base locus, i.e. of dimension two, and will present the known examples. The aim is to discuss a new result on the anticanonical system of Fano fourfolds: if the base locus is a normal irreducible surface, then all of its members are singular. Joint work with Andreas Höring.

• Chow quotients of projective varieties by torus actions (Luis Solá Conde)
  Given an action of the 1-dimensional torus on a projective variety, the parameter space of invariant 1-cycles in the variety is called the Chow quotient of the action. In this talk I will discuss the relation of the Chow quotient with the corresponding GIT quotients: in particular, I will show how the Chow quotient can be constructed upon the GIT ones, under some particular assumptions. I will also discuss conditions for the smoothness of the Chow quotient, and present some examples in which it is singular. The contents of this talk belong to a joint project with G. Occhetta, E. Romano and J. Wiśniewski.

• K-stability of Fano weighted hypersurfaces and applications (Luca Tasin)
  I will report on recent progress about determining the K-stability of Fano hypersurfaces in weighted projective space. As an application, I will talk about the existence of Sasaki-Einstein metrics on odd dimensional spheres, a classical problem in differential geometry. 

• Two elephants (almost) imply LeBrun-Salamon conjecture (Jarosław Wiśniewski)
  The celebrated LeBrun-Salamon conjecture about positive Riemannian manifolds with quaternion-Kahler holonomy can equivalently be stated in terms of contact Fano manifolds equipped with a contact structure and the Kahler-Einstein metric. Recently, the work of Buczyński, Weber, Romano, Occhetta, Sola Conde and myself allowed to achieve notable progress on the conjecture in the latter setting. Moreover, building upon our work, Śmiech observed that a slightly stronger version of Kawamata's effective nonvanishing conjecture for smooth Fano manifolds almost implies the LeBrun-Salamon conjecture. In my talk, I will present some aspects of those results.

• Compact Kähler Threefolds with Nef Anticanonical Line Bundles (Xiaojun Wu)

Abstract: This presentation elucidates recent collaborative research conducted with Shin-ichi Matsumura, focusing on the characterization of compact Kähler threefolds with nef anticanonical line bundles. The endeavor to classify compact Kähler manifolds, particularly those exhibiting positive tangent bundles or anticanonical line bundles, constitutes a significant pursuit within mathematical discourse. While notable progress has been achieved, particularly through the seminal work of Cao and Höring in delineating the classification of projective manifolds with nef anticanonical line bundles, the analogous task in the compact Kähler setting presents formidable challenges. In dimension three, based on Kähler MMP developped by Höring-Peternell, a classification can be given using analytic tools like Segre currents, Q-conic bundle, orbifold vector bundle, etc.

• Cones of divisors on some examples of varieties with nef anticanonical bundle (Zhixin Xie)
  Complex projective varieties with nef anticanonical bundle appear as natural generalisations of Fano varieties. However, the nef cone of this class of varieties is usually not rational polyhedral. In this talk, we will first discuss the nef cone of some surfaces with nef anticanonical bundle. Then we will focus on an example of threefold X obtained by blowing up eight very general points of P^3. We describe explicitly the nef cone and the cone of effective divisors on X. Moreover, we show that a certain Weyl group acts with a rational polyhedral fundamental domain on the effective movable cone of X. This is joint work with Isabel Stenger.