Abstracts of the School Lectures

David Eisenbud: Homological Algebra over Complete Intersections

I'll begin by introducing some of the classical theory about modules over complete intersection, and lead up to current work and open problems.
Topics: Matrix Factorizations and periodic resolutions, Cohen-Macaulay modules, Knörrer periodicity, Ulrich modules, singularity category, support varieties, higher matrix factorizations and layered resolutions.


June Huh: Hard Lefschetz theorems and Hodge-Riemann relations in geometry, algebra, and combinatorics

I will give a broad overview of the Hard Lefschetz theorems and the Hodge-Riemann relations in the theory of polytopes, complex manifolds, reflection groups, algebraic and tropical varieties, in a down-to-earth way. Several applications to the elementary combinatorics of graphs and matroids will be introduced.


Peter Scholze: Integral p-adic Hodge theory

We will discuss some recent results in integral p-adic Hodge theory relating various p-adic cohomologies including crystalline, de Rham and etale cohomology. For the most part, we will discuss some background on these individual theories, and towards the end their relations, which leads to a mysterious q-deformation of de Rham cohomology.


Duco van Straten: Nodal Hypersurfaces

In this series of lectures I will discuss various aspects of hypersurfaces with ordinary double points as singularities. Starting with a review of classical examples, I will explain how Varchenko's spectral bound arises from the asymptotic Hodge structure of isolated hypersurface singungularities.

Abstracts of the Conference Talks

Florin Ambro: Curves with ordinary singularities

I will discuss the classification of projective curves with ordinary singularities (i. e. seminormal), in a way parallel to the classification of projective curves with no singularities.


Yves André: On the status of motives in Algebraic Geometry: from useful fiction to instrument

Grothendieck's dream of motives could be grasped as a "categorification" of the (Grothendieck) ring of algebraic varieties with cut-and-paste relations. In the last two decades, its epistemological status has shifted from a useful fiction (the "philosophy" of motives) to a full-fledged, versatile, instrument of Algebraic Geometry. In this talk, we shall focus on how this instrument serves to elucidate algebraic relations between integrals of rational functions, and related transcendence problems which date back to Newton and Leibniz.


Ana Caraiani: Shimura varieties, torsion classes, and Galois representations

In this talk, I will describe joint work with Peter Scholze on torsion in the cohomology of certain unitary Shimura varieties. I will explain how the theory of perfectoid spaces and p-adic Hodge theory give new insights into the geometry and cohomology of Shimura varieties. I will also mention applications of these results to elliptic curves over imaginary quadratic fields, joint with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne.


Ivan Cheltsov: Finite collineation groups and birationally rigidity

I will report on recent work with Shramov on equivariant birational geometry of ℙ³ .


Aldo Conca: Reduced hyperplane sections of determinantal varieties and Lovasz-Saks-Schrijver ideals associated to graphs

To each graph G and  every positive integer d one associates an ideal, the Lovasz-Saks-Schrijver ideal of the graph, defining the variety of the orthogonal representation of G  in the real space of dimension d. We will discuss algebraic properties of the ideal and its connections with the study of reduced hyperplane sections of determinantal varieties. This is a joint work with Volkmar Welker.


Alessio Corti: Volume preserving birational maps

I report on recent work with Araujo and Massarenti on the structure of the group of birational maps of ℙ³ to itself, preserving a quartic surface with rational double points.


Alexandru Dimca: Rational cuspidal curves vs. free plane curves

A rational cuspidal curve is a complex curve homeomorphic to a projective line. A free plane curve is a curve whose Jacobian ideal is saturated. In our talk we discuss relations between these two classes of curves which, in spite of a very different definition, are closely related.


David Eisenbud: Equations of some degenerate K3 surfaces

It is known that it is possible to construct degenerate K3 surfaces in projective space as certain unions of two rational normal scrolls, or as double structures on a rational normal scroll (``K3 carpets''). It turns out that the generators and Gröbner bases of the ideals of these objects are easy to describe explicitly, and that there is an interesting inductive structure. I’ll describe the objects in question and their ideals; in a later talk Frank Schreyer will explain what we know about their free resolutions.


Gavril Farkas: Moduli of K3 surfaces via cubic 4-folds

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera g<21, with the exception of the case g=14. Using Hassett's identification between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. This is joint work with Verra.


Sergey Galkin: Beautiful formalae, gauged linear sigma models and problems of semi-classical algebraic geometry

I will explain how old intsruments of diophantine and non-commutative geometry together with simple geometric constructions and few new ideas about pre-calculus of algebraic varieties, gauged linear sigma models (equivariant matrix factorizations) and dg-categories could help to attack some classical and semi-classical problems of algebraic geometry, such as rationality questions, derived equivalences, relations between moduli spaces, study of points over rational and finite fields. In particular I will tell why variety of lines on a cubic fourfold melts to a Hilbert scheme of two points on a non-commutative deformation of a K3 surface.


Milena Hering: Frobenius splittings of toric varieties and unimodularity

Many nice properties of toric varieties can be deduced from the fact that they admit a splitting of the Frobenius morphism. Varieties whose product with themselves admits a splitting compatible with the diagonal exhibit even nicer properties. For example, every ample line bundle on such a variety is normally generated. In this talk I will give an overview over the consequences of the existence of such splittings and discuss new combinatorial criteria for toric varieties to be diagonally split.


Sándor Kovács: Cohomological equivalences

A cohomological equivalence is a proper birational morphism for which the derived push-forwards of the structure sheaf and the dualizing complex of the source are isomorphic to those of the target. In this talk I will discuss cohomological equivalences of Cohen-Macaulay schemes. This allows for studying variants of rational singularities in arbitrary (including mixed) characteristic without resolution of singularities, but it also leads to new results in characteristic zero as well.


Mircea Mustaţă: Invariants of singularities coming from D-module theory

For a reduced hypersurface in a smooth complex algebraic variety, Saito's theory of Hodge modules gives a sequence of invariants, the Hodge ideals, that measure the singularities of the hypersurface. I will discuss these invariants, as well as some generalizations to rational coefficients. This is joint work with Mihnea Popa.


Jérôme Poineau: Berkovich spaces over Z

Although Berkovich spaces usually appear in a non-archimedean setting, their general definition actually allows arbitrary Banach rings as base rings, e.g. Z endowed with the usual absolute value. Over the latter, Berkovich spaces look like fibrations that contain complex analytic spaces as well as p-adic analytic spaces for every prime number p. We will try to show what those spaces look like and describe their main properties both local (local path-connectedness, noetherianity of local rings, etc.) and global (vanishing of higher cohomology of disks, noetherianity of rings of global functions, etc.). If time allows, we will explain that Berkovich spaces over Z may be used to parametrize certain natural families such as Schottky groups and Mumford curves over arbitrary local fields. This last part is joint work with Daniele Turchetti.


Claudiu Raicu: Homological invariants of determinantal thickenings

Given a closed subscheme in affine or projective space which is equipped with a large group of symmetries it is an important problem to determine some of its basic homological invariants (such as sheaf and local cohomology, syzygies etc) and to understand how the symmetries are reflected in the structure of the invariants. Explicit descriptions of these invariants are notoriously difficult to find and often involve combining an array of techniques from different fields. In my talk I will explain how instruments of algebraic geometry and representation theory can be used to shed some light on the structure of some of the most fundamental homological invariants associated to schemes that are supported on determinantal varieties and whose symmetries come from the natural action via row and column operations on the matrix entries.


Bertrand Rémy: Wonderful compactifications of Bruhat-Tits buildings

We will introduce the general theme of compactifying buildings attached to semisimple groups over local fields. In the case of a split group, we can identify equivariantly the maximal Satake-Berkovich compactification of the corresponding Euclidean building with the compactification obtained by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, can be described. This is joint work with Amaury Thuillier and Annette Werner.


Maria Evelina Rossi: An effective characterization of the Gorenstein k-algebras

We present a recent generalization of Macaulay’s Inverse System to higher dimensions. To date a general structure for  Gorenstein k-algebras of any dimension (and codimension)  is not understood. We extend Macaulay's correspondence  characterizing  the submodules of the divided powers ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings with given numerical invariants. Several examples illustrating the results are given.


Frank Olaf Schreyer: Syzygies of some degenerate K3 surface, and Green's conjecture in positive characteristic

I am reporting on joined work with David Eisenbud about resonance phenomena of syzygies of some degenerate K3 surfaces and computation for small genus of their syzygies over arbitrary fields. I will also formulate a version for Green's conjecture which so far has no counter examples by a smooth curve of small genus in positive characteristic.


Misha Verbitsky: Transcendental Hodge algebra

Let M be a projective manifold. Transcendental Hodge lattice of weight p is the smallest rational Hodge substructure in H^p(M) containing H^p,0(M). Transcendental Hodge lattice is a birational invariant of M. I will prove that the direct sum of all transcendental Hodge lattices for M is an algebra, and compute it (using Yu. Zarhin's theorem) explicitly for all hyperkahler manifolds. This result has many geometric consequences. In particular, I will show that any family X of hyperkahler manifolds containing a symplectic torus of dimension m satisfies m ≤ 2^d/2-1, where d is effective dimension of Х.


Annette Werner: Vector bundles on p-adic varieties and parallel transport

In this talk we report on joint work with Christopher Deninger. We develop a theory of étale parallel transport for vector bundles on a smooth, projective p-adic variety X which can be extended to some integral model of X in such a way that the special fiber is a numerically flat bundle.


Jaroslaw Wisniewski: Instruments of algebraic torus action (handout)

I will report on the ongoing project whose task is to use combinatorics arising from algebraic torus action to understand complex contact manifolds. The project is developed in collaboration with Jarosław Buczyński and Andrzej Weber.


Olivier Wittenberg: Zero-cycles on homogeneous spaces of linear groups

The Brauer-Manin obstruction is expected to control the existence and weak approximation properties of rational points on homogeneous spaces of linear algebraic groups over number fields. We establish the zero-cycle variant of this conjecture. The same method also leads to a new proof of Shafarevich's theorem that finite nilpotent groups are Galois groups over any number field. Joint work with Yonatan Harpaz.