Title: Fundamental groups of low-coregularity Calabi-Yau type pairs.
Abstract: Our object of study will be the orbifold fundamental groups of the smooth locus of Calabi-Yau type pairs. The absolute coregularity of a pair, is a measure of combinatorial complexity of a variety. We will show that the fundamental groups of klt pairs with low-coregularity and arbitrary dimension behave similarly to the fundamental groups of low-dimensional Calabi-Yau pairs. More explicitly, we will show that klt pairs with coregularity 0 have finite fundamental groups, and klt pairs with coregularity 1 or 2 have virtually abelian fundamental groups of rank at most 2 or 4. The indices of the abelian subgroups cannot be uniformly bound, even if we control the dimension. In this direction, we will end with a statement that gives effective bounds on the index of solvable groups in the case of pairs of fixed dimension and low-coregularity. This talk is based on joint work with Lukas Braun.
Abstract: I will report on joint work with Valentino Tosatti and Simion Filip in which we show by example that for a pseudo-effective divisor D and ample A, the volume function vol(D+tA) for small values of t can exhibit various pathological behaviors.
Abstract: Esnault-Viehweg (resp. S. Ishii) proved that two-dimensional klt (resp. lc) singularities are stable under small deformations. Unfortunately, an analogous statement fails in higher dimensions, because the generic fiber is not necessarily Q-Gorenstein if the special fiber is klt. In this talk, I present a generalization of the results of Esnault-Viehweg and Ishii under the assumption that the generic fiber is Q-Gorenstein (but the total space is not necessarily Q-Gorenstein). This talk is based on joint work with Kenta Sato.
Title: Irrationality of degenerations of Fano varieties
Abstract:In this talk, I will introduce a recent result about bounding degrees of irrationality of degenerations of klt Fano varieties of arbitrary dimensions. This proves the generically bounded case of a conjecture proposed by C. Birkar and K. Loginov for log Fano fibrations of dimensions greater than three. Our approach depends on a method to modify the klt Fano fibration to a toroidal morphism of toroidal embeddings with bounded general fibres. This is a joint work with Prof. C. Birkar.
Title: Fano 4-folds with b_2 > 12 are products of surfaces.
Abstract: Let X be a smooth, complex Fano 4-fold, and b_2 its second Betti number. We will discuss the following result: if b_2 > 12, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f : X -> Y such that the image S of the exceptional divisor is a surface, together with my previous work on Fano 4-folds. In particular, given f : X -> Y as above, under suitable assumptions we show that S is a smooth del Pezzo surface with -K_S given by the restriction of -K_Y.
Title: Polarized endomorphisms of Fano varieties with complements.
Abstract: An endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to a qA, for some integer q>1. Examples of these endomorphisms are naturally found in toric varieties and abelian varieties. Indeed, it is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. In this talk, I will restrict to the case when X is a Fano type variety admitting a 1-complement, meaning that there exists an effective divisor B such that (X,B) is log Calabi-Yau, and K_X + B ~ 0. I will prove that if (X,B) has a polarized endomorphism that preserves the complement structure, then (X,B) is a finite quotient of a toric log Calabi-Yau pair. This is joint work with Joaquin Moraga and Wern Yeong.
Title: Birational complexity of log Calabi-Yau pairs.
Abstract: The complexity of a log Calabi-Yau pair is a rather simple invariant that measures how close the pair is from being toric.
The birational complexity, as the name suggests, is a birational variant of the aforementioned invariant.
The birational complexity measures how close is a given log Calabi-Yau pair from having a toric birational model.
In this talk, I will introduce this new invariant, and show how it reflects on the topology of dual complexes
and the existence of conic bundles on birational models of a log Calabi-Yau pair.
Abstract: We study equivariant geometry and rationality of moduli spaces of points on the projective line, for twists associated with
permutations of the points. (joint with Yuri Tschinkel and Zhijia Zhang)
Title: Umemura quadric fibrations and maximal subgroups of Cremona groups.
Abstract: Maximal connected algebraic subgroups of groups of birational transformations Bir(X) of a projective variety X appear as automorphism groups of Mori fiber spaces birational to X. When X = P^n, for n = 2, 3, we have a classification of such groups and the
models they act on, while in dimension 4 and on, this is an open problem. In this talk, after explaining the main machinery and known results, we will explore the equivariant geometry of a class of quadric fibrations over P^1. As a result, we will see how their automorphism
groups gives us infinite families of maximal subgroups of Bir(P^n), for any n at least 3. This is joint work with Enrica Floris.
Title:Rationality questions for Fano schemes of intersections of two quadrics.
Abstract:We study rationality questions for the Fano schemes of non-maximal linear spaces on a smooth complete intersection X of two quadrics, especially over non-closed fields. We start by showing that they are all geometrically rational. We then ask their rationality over k and analyze in details the case of second maximal linear spaces. In particular, we generalize results of Hassett-Tschinkel and Benoist-Wittenberg when X has odd dimension, and extend work of Hassett-Kollár-Tschinkel when X has even dimension and k = R. This is joint work with Lena Ji.
Abstract: Recall that an algebraic variety is rational if it is birational to a projective space. In the past decade it was established that several classes of complex varieties are not (stably) rational: this includes very general hypersurfaces in a certain degree range, cyclic covers, and other varieties.
These results were obtained using the specialization method: to summarize, X is not stably rational if it degenerates to a well chosen mildly singular "reference" variety with some particular nontrivial invariants.
Constructing such reference varieties could be a difficult task in general. In this talk, we will give an overview of the recent progress and we will discuss one specific example of a reference variety, namely, a fibration in cubic surfaces: we will describe a general formula for the invariant coming from the unramified Brauer group.
Abstract: We give a classification of smooth Fano varieties X with Picard number greater than one when the pseudoindex of X is at least dimX-2 or equal to (dimX+1)/2.
Title: Moduli space of Fano threefolds and complete intersection curves.
Abstract: As the last step to the Calabi Problem, we are asked to find all the Kaehler-Einstein limits of each deformation family of Fano varieties. In this talk, I will illustrate the application of the moduli continuity method in conjunction with approaches like wall-crossing and moduli of K3 surfaces to explicitly describe the K-moduli space of a specific deformation family of Fano threefold. This is a recent work joint with Yuchen Liu.
Title: On the Albanese fibration for singular varieties with nef anticanonical divisor.
Abstract: To every normal projective variety X one can associate an Abelian variety Alb(X), called the Albanese variety of X, and a morphism alb:X->Alb(X), called the Albanese morphism. In many situations, the geometry of X can be investigated through alb. In particular, if X has klt singularities (e.g., it is smooth) and -K_X is nef, then alb is a surjective analytically locally trivial fibration. In this talk, we investigate what happens if we relax the condition on the singularities of X. First, we show that alb need not be surjective if the singularities are arbitrary. Then, we show that alb is still surjective and flat if X has log canonical singularities. To conclude, we exhibit an example of log canonical log Calabi-Yau pair whose Albanese morphism is not isotrivial, not even birationally. Time permitting, we will discuss implications of this example for the Beauville-Bogomolov decomposition of log canonical pairs. This talk is based on current work in progress with F. Bernasconi, Zs. Patakfalvi, and N. Tsakanikas.
Title: Abundance theorem for minimal projective varieties satisfying Miyaoka's equality.
Abstract: In this talk, I would like to discuss the abundance conjecture for klt projective varieties satisfying the 'Miyaoka-type equality.' Initially, I will review a previous study (which is joint work with Masataka Iwai), where we established the abundance conjecture for minimal smooth projective varieties with vanishing second Chern class. These varieties satisfy the Miyaoka-type equality. Building on this fact, I will show that our previous study can be extended to varieties with Miyaoka-type equality. This talk is based on joint work with Masataka Iwai (Osaka) and Niklas Muller (Duisburg-Essen).
Title: Recent Results on Minimal Exponent for LCI Subvarieties.
Abstract: The minimal exponent of a complex hypersurface singularity is a refinement of the log canonical threshold of that hypersurface, which has been shown to relate to recently established classes of ``higher" singularities. Together with Chen, Mustaţa and Olano, we have defined and studied the minimal exponent for local complete intersection subvarieties, and established various analogous properties of this invariant.
Recently, some new properties were shown: the first relates the invariant to the Bernstein-Sato polynomial of the LCI subvariety, which is what one would expect by naively extending the definition from the hypersurface case. Another is a lower bound in terms of numerical data from a resolution which is known for hypersurface singularities but was not yet known for the higher codimension case. I will explain these results and show how the lower bound helps to compute the minimal exponent in a class of examples.
Title: Boundedness of singularities and discreteness of local volumes.
Abstract: The local volume of a klt singularity is an invariant that plays a central role in the local theory of K-stability. By the stable degeneration theorem, every klt singularity has a volume preserving degeneration to a K-semistable log Fano cone singularity. I will talk about a joint work with Chenyang Xu on the boundedness of log Fano cone singularities when the volume is bounded away from zero. This has the consequence that local volumes only accumulate at zero in any given dimension.
Title: Hypersurfaces with large automorphism groups.
Abstract: Let X be an n-dimensional smooth hypersurface of degree d in complex projective space, where n is at least 1 and d is at least 3. We show that the Fermat hypersurface has the largest automorphism group among all X of dimension n and degree d, apart from a handful of exceptional pairs (n, d). For those exceptional cases, we find new bounds. This is joint work with Louis Esser.
Title: Curves on complete intersections and measures of irrationality.
Abstract: Given a projective variety X, it is always covered by curves obtained by taking the intersection with a linear subspace. Towards studying the geometry of X, it is fruitful to ask: which other curves lie on X? More specifically, we consider the existence of curves on X that are simpler than complete intersection curves in terms of some numerical invariant. If X is a general complete intersection of large degrees, we show there are no curves on X of smaller degree, nor are there curves of asymptotically smaller gonality. Verifying a folklore conjecture on the degrees of subvarities of complete intersections as well as a conjecture of we verify a conjecture of Bastianelli--De Poi--Ein--Lazarsfeld--Ullery on measures of irrationality for complete intersections. This work is joint with Nathan Chen and Junyan Zhao.
Title: Minimal metrics and the Abundance conjecture.
Abstract: The Abundance conjecture predicts that on a minimal projective klt pair (X,D), the adjoint divisor K_X+D is semiample. When the Euler-Poincaré characteristic of X does not vanish, I will present a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of K_X+D. I will also give strong evidence that an important class of currents with minimal singularities - supercanonical currents - is central to the completion of the proof of the Abundance conjecture for minimal klt pairs with non-vanishing Euler-Poincaré characteristic.
Title: K-moduli of log del Pezzo pairs and variations of GIT.
Abstract: In recent years K-stability has successfully constructed K-moduli spaces for Fano varieties and log Fano pairs. A natural next direction in this topic is the explicit description of these K-moduli spaces in specific examples. The study of K-moduli of log del Pezzo pairs formed by a del Pezzo surface of degree d and an anti-canonical divisor is of particular importance, as they are the first natural lower-dimensional examples of such explicit descriptions. These moduli spaces naturally depend on one parameter, providing a natural problem in variations of K-moduli spaces, by exhibiting wall-crossing phenomena. In this talk, I will describe these K-moduli spaces for degrees 2, 3, 4, by establishing an isomorphism between the K-moduli spaces and variations of Geometric Invariant Theory compactifications, which generalizes the isomorphisms in the absolute cases established by Odaka--Spotti--Sun and Mabuchi--Mukai. I will also briefly describe the wall-crossing structure for degrees 5,6,7,8,9, that are obtained by computational methods. This is joint work with J. Martinez—Garcia and J. Zhao.
Abstract: Log Calabi-Yau pairs interpolate between Calabi-Yau varieties and Fano varieties, two of the fundamental building blocks of algebraic varieties. The complexity of a log Calabi-Yau pair measures how far it is from being toric, and it’s birational complexity measures how far it is from admitting a birational toric model. Recent work has studied the (birational) complexity of log Calabi-Yau pairs in the presence of restrictions on their index. I will discuss joint work in progress with Fernando Figueroa, in which we provide characterizations of log Calabi-Yau pairs of (birational) complexity zero and arbitrary index.
Title: Moduli of boundary polarized CY surface pairs.
Abstract: While the theories of KSBA stability and K-stability have been successful in constructing compact moduli spaces of canonically polarized varieties and Fano varieties, respectively, the case of K-trivial varieties remains less well understood. I will discuss a new approach to this problem in the case of Calabi-Yau pairs (X,D), where D is ample, in which we consider all semi-log-canonical degenerations. One challenge of this approach is that the set of such degenerations is unbounded. Nevertheless, in the case of surface pairs, we construct a projective moduli space on which the Hodge line bundle is ample. This is based on joint work with Yuchen Liu that builds on previous work with Ascher, Bejleri, DeVleming, Inchiostro, Liu, and Wang.
Title: K-semistability of log Fano cone singularities.
Abstract: K-stability of log Fano cone singularities was introduced by Collins--Sz\’ekelyhidi to serve as a local analog of K-stability of Fano varieties. In the Fano case, the result of Li-Xu states that to test K-stability, it suffices to test the so-called special test configurations. In this talk, I will talk about a local version of this result for log Fano cones. Our method relies on a non-Archimedean characterization of local K-stability.
Abstract: In the context of algebraic geometry, decomposition and inertia groups are special subgroups of the Cremona group which preserve a certain subvariety of $\mathbb{P}^n$ as a set and pointwise, respectively. These groups were and still are classic objects of study in the area, with explicit descriptions in several instances. In the particular case where this fixed subvariety is a hypersurface of degree $n+1$, we have the notion of Calabi-Yau pair which allows us to use new tools to deal with the study of these groups and one of them is the so-called volume preserving Sarkisov Program. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of the decomposition group of a nonsingular plane cubic is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. Regarding now a 3-dimensional context, we give a description of which toric weighted blowups of a point are volume preserving and among them, which ones will initiate a volume preserving Sarkisov link from a Calabi-Yau pair $(\mathbb{P}^3,D)$ of coregularity 2. In this case, $D$ is necessarily an irreducible normal quartic surface having canonical singularities. This last result enhances and extends the recent works of Guerreiro and Araujo, Corti and Massarenti in a log Calabi-Yau geometrical perspective, and it is a possible starting point to study the decomposition group of such quartics.
Title: On the positivity of Frobenius pushforwards on toric varieties.
Abstract: Let L be an invertible sheaf on a smooth projective variety over an algebraically closed field of positive characteristic. By a celebrated result of Kunz, the Frobenius pushforward of L is a locally free sheaf. But what can be said about its positivity (ampleness, nefness, bigness, pseudo-effectiveness)? In this talk, I'll explore answers to this question in the case of toric varieties. I'll show that indeed the positivity of these sheaves reflect the geometry of the underlying variety. This is joint work with Emre Özavci (EPFL).
Abstract: Very general hypersurfaces in complex projective space (of dimension at least 3) are expected to be rational if and only if the degree is at most 2.
T. Okada gave a similar "small degree condition" for weighted hypersurfaces that guarantees rationality and he proved that it is actually equivalent to
rationality for a very general terminal quasismooth hypersurface in dimension 3. In this talk, I'll show that there are rational examples in higher dimensions
that don't satisfy this degree criterion. Along the way, I'll introduce two new rationality constructions for these hypersurfaces, which work over any field.
Title: On the uniformization problem in the log-canonical case.
Abstract: We know since the work of Baily-Borel-Mok that any ball quotient by a lattice (discrete, with finite covolume) admits a structure of quasi-projective variety. The minimal compactifications of these objects are varieties with ample cotangent bundle, obtained by adding a finite number of points at the boundary : the points then give rise to log-canonical singularities. We can naturally ask how to characterize algebraically the varieties obtained in this manner: given a projective variety with ample canonical bundle and log-canonical singularities, when can we exhibit a Zariski open subset uniformizable by the ball? This question aims at generalizing to the lc case several recent results, that mostly deal with the klt or orbifold setting (Greb-Kebekus-Peternell-Taji, Deng, Claudon-Guenancia-Graf...)
I will present such a uniformization theorem, under a stronger assumption on the singularities: as it is usually the case in this theory, the criterion will be stated in terms of the equality case in a certain Miyaoka-Yau inequality. We will see that the strategy is to try to avoid a class of examples obtainable through work of Deligne-Mostow-Siu, Deraux, Stover-Toledo, exploiting the log-canonicity of the singularities. Campana's theory of special varieties will be used quite crucially in arguments.
Title: Group theoretical characterizations of rationality.
Abstract: To a variety X we associate its group of birational transformations Bir(X). In this talk, we will see that the group structure of Bir(X) determines whether X is rational and whether X is ruled. In another direction, I will explain that Borel subgroups of Bir(X) are of derived length <= 2 dim(X) with equality if only if X is rational and the Borel subgroup is standard; this gives a further group theoretical rationality criterion. A necessary ingredient for our proofs are some new results about the structure of families of birational transformations, which build on previous work of Blanc-Furter, Hanamura, and Ramanujam. This is joint work with Regeta and Van Santen.
Title: The Morrison Cone Conjecture under deformation.
Abstract: The Morrison cone conjecture is a fundamental conjecture on the geometry of the nef cone and the movable cone of a Calabi-Yau variety. We prove that if the Morrison Cone Conjecture holds for a smooth Calabi-Yau threefold Y, then it also holds for any smooth deformation of Y.
Title: 2-Gorenstein stable surfaces with volume 1 and Euler characteristic 3.
Abstract: Surfaces of general type with particular invariants and their moduli spaces
have been studied for over a century and the case of K^2 = 1 and p_g = 2
is one of the simplest. Nowadays the Gieseker moduli space of surfaces of
general type is known to admit a natural compactification in the moduli space
of stable surfaces.
I will talk about our ongoing quest to understand this compactification and the
surfaces it parametrises in this supposedly simple case and illustrate the
techniques we use to classify those surfaces in the moduli space for which
$2K_X$ is Cartier.
This is joint work with Stephen Coughlan, Marco Franciosi and Rita Pardini.
Title: Cubic fourfolds with birational Fano varieties of lines.
Abstract: We give four examples of pairs of (conjecturally irrational) cubic fourfolds with birationally equivalent Fano varieties of lines. Two of our examples, which are special families in C_12, provide new examples of cubic fourfolds with equivalent Kuznetsov components, which we show are birational. We will discuss how various types of equivalences for cubic fourfolds are related. This is joint work with Corey Brooke and Sarah Frei, building on our previous work with Xuqiang Qin.
Title: Hyperbolicity of the complement of generic quartic plane curves.
Abstract: A complex algebraic variety is said to be Brody hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. It is conjectured that varieties of (log) general type are hyperbolic outside of a proper subvariety called an exceptional locus. The case of quartic plane curves has drawn interest for decades. In two collaborations (with X. Chen and E. Riedl, and with K. Ascher and A. Turchet), we prove an algebraic version of this conjecture for the complement of a very general quartic plane curve with <=2 components. Moreover, we completely characterise the exceptional locus for the complement of a very general irreducible quartic plane curve, identifying it as the union of its flex and bitangent lines.
Title: Moduli spaces of points in flags of affine spaces and polymatroids.
Abstract: We will discuss two novel moduli spaces of labeled points in flags of affine spaces. The first moduli space parametrizes distinct weighted points, with configurations defined up to translation and scaling. The second moduli space allows points to collide freely, without any notion of equivalence between configurations. We will see that the first moduli space admits a toric compactification, which coincides with the polypermutohedral variety of Crowly-Huh-Larson-Simpson-Wang, while the second one is toric and coincides with the polystellahedral variety of Eur-Larson. This is joint work with P. Gallardo and J.L. Gonzalez.