Main Courses

• COURSE 1 (second week): Hyperbolicity properties of algebraic varieties

  Jean-Pierre DEMAILLY (Université Grenoble Alpes, France)

  Abstract

  The study of entire holomorphic curves drawn in projective algebraic varieties is intimately related to fascinating questions of value distribution theory and arithmetic geometry. One of the important unsolved questions is the Green-Griffiths-Lang conjecture which stipulates that for every projective variety $X$ of general type over $\mathbb{C}$ there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f\colon \mathbb{C} \to X$ . Such questions can be investigated by studying the rich geometry of jet bundles and the neativity properties of their curvature. We will try to present the main ideas of these techniques, and especially the recent proof by D.Brotbek of a long-standing conjecture of Kobayashi (1970), according to which a generic algebraic hypersurface of $\mathbb{P}^n$ of sufficiently large degree is hyperbolic in the sense of Kobayashi.

• COURSE 2 (first week): Automorphisms of algebraic varieties

  Igor DOLGACHEV (University of Michigan, USA)

  Abstract     Lecture Notes     TS
  • General facts about the scheme of automorphisms of a projective algebraic variety
  • Action on cohomology: cohomologically trivial automorphisms
  • Automorphisms of algebraic curves and surfaces of general type
  • Automorphisms of algebraic surfaces: K3, Enriques and rational surfaces

• COURSE 3 (second week): Prym varieties and Prym maps

  Angela ORTEGA (Humboldt Universität, Germany)

  Abstract     Lecture Notes     TS

  Prym varieties are abelian varieties constructed from coverings of algebraic curves. It is well known that a general principally polarized abelian variety of dimension at most 6 is a Prym variety. For a given g, degree, and ramification degree of a covering over a genus g curve, there is a corresponding Prym map that associates to every covering of this type a polarized abelian variety. We will start with some generalities about Prym varieties; we will present the different strategies to compute the degree of the Prym map (when it is finite), and various cases of Prym maps whose fibres carry a particularly beautiful structure.

• COURSE 4 (second week): Algebraic curves and automorphic forms

  Vincent PILLONI (École normale supérieure de Lyon, France)

  Abstract     Lecture Notes     TS

  It has been conjectured that the zeta function of an algebraic curve defined over a number field has a meromorphic continuation to the complex plane. A strategy to prove this is to show that such curve is "automorphic". In this course, we will explain what this means.

• COURSE 5 (first week): Stable birational invariants and the stable Lüroth problem

  Claire VOISIN (Collège de France, France)

  Abstract

  The lectures will discuss cohomological and Chow-theoretic obstructions to rationality or stable rationality of complex projective varieties. Of course, there are many such geometric obstructions, like the plurigenera, but we will rather focus on the case of rationally connected varieties, where these obvious obstructions are trivial. We will discuss:
     - Unramified cohomology and its link to the integral Hodge conjecture, (this provides very interesting irrationality criteria)
     - Various notions of decompositions of the diagonal
  The study of the (stably birationally invariant) property of having a "decomposition of the diagonal" in the Bloch-Srinivas sense allowed Claire Voisin and subsequently other people to prove that many rationally connected varieties are not stably rational. These works culminated with the proof by Hassett-Pirutka-Tschinkel that rationality is not deformation invariant. Comparing this birational invariant with previously defined invariants (e.g. the Clemens-Griffiths criterion) is also very interesting and has been completely done in dimension 3. This could be the last point developed in the lectures, if time allows.

Mini-Course

• Kähler metrics, connections and curvature

  Pedro MONTERO (Universidad Técnica Federico Santa María , Valparaiso, Chile)

  Abstract     Lecture Notes

  We will recall some basic notions related with Kähler geometry. In particular, we will discuss how to construct the Chern curvature tensor on a given hermitian vector bundle and we will state the Kodaira embedding theorem characterizing ample line bundles.

Panoramic Talks

• About mirror symmetry for K3 surfaces

  Michela Artebani (UdeC, Chile)

  Abstract

  Mirror symmetry is both a phenomenon and a conjecture which predicts the existence of a duality between families of Calabi-Yau manifolds exchanging the complex structure with the symplectic structure. In this talk I will survey different versions of mirror symmetry for two-dimensional Calabi-Yau manifolds, i.e. $K3$ surfaces, in terms of dualities between other geometric objects: lattices (lattice mirror symmetry), matrices and groups (Berglund-Hubsch-Krawitz mirror symmetry) and polytopes (Batyrev mirror symmetry and a duality defined by Comparin, Guilbot and myself in arXiv:1501.05681). I will present known results and open questions about the relations between these symmetries. This talk is partially inspired to "Mirror symmetry and $K3$ surfaces" by Kazushi Ueda arXiv:1407.1566.

• Exotic algebraic affine spaces and spheres : topology, geometry, algebra and beyond

  Adrien Dubouloz (U. Bourgogne, France)

  Abstract

  The notion of "exotic structure" first appeared in differential topology, in the work of Milnor on equivalence classes of differentiable structures on smooth manifolds homeomorphic to spheres. It entered algebraic geometry later on after a landmark result of Ramanujam which provides a surprising topological characterization of the complex affine plane $\mathbb{A}^2_{\mathbb{C}}$ as an algebraic variety, reminiscent to the characterization of euclidean spaces due to Whitehead. In this talk, after a quick review of the historical background and motivation from differential topology, I will present a selection of results and open problems on algebraic exotic affine spaces and spheres, and explain on examples some of the "standard" tools, combining geometric and algebraic methods, to study these spaces as well as some recent new perspectives. .

• Homogeneous spaces, weak approximation and the inverse Galois problem

  Giancarlo Lucchini (UChile, Chile)

  Abstract

  The inverse Galois problem asks whether a given finite group can appear as a Galois group over a given field. In the case of number fields, this question is still wide open. Nevertheless, this problem and more powerful generalizations can actually be studied using methods from algebraic geometry. In this talk, we will present the notion of weak approximation for varieties over a number field. We will study it in the particular case of homogeneous spaces and find the link between this property and the inverse Galois problem. Time permitting, we will state the main results in this area so far.

• Moduli of points and lines

  Jenia Tevelev (UMass Amherst, USA)

  Abstract

  Moduli space of n distinct points in $\mathbb{P}^1$ admits many GIT compactifications which depend on the choice of stability (allowed “degenerate configurations”). All these compactifications are dominated by the master space, the Grothendieck—Knudsen moduli space of stable rational curves. For the projective plane, the picture is further muddled by projective duality, which identifies arrangements of $n$ lines in the projective plane and arrangements of n points in the dual projective plane. We ask: how does projective duality behave under degenerations? Kapranov introduced the compactified moduli space of line arrangements, later studied by Lafforgue, Keel-Tevelev, and others. In a much less known work, compactified moduli space of point arrangements was introduced by Gerrizen and Piwek. In joint work with Luca Scaffler, we investigate how these two moduli spaces are related.

• The savage geography of surfaces of general type

  Giancarlo Urzúa (UC, Chile)

  Abstract

  The problem of geography asks about the existence of complex surfaces of general type for a given pair of Chern numbers $(c_1^2, c_2)$. Although this problem is almost solved, there are still some pairs where we do not have an example. If, in addition, we fix the (topological) fundamental group, then we get a much harder question, which is wide open in general. To approach to that question, we consider instead Chern slopes $c_1^2/c_2$, which must live essentially in the allowed interval $[1/5,3]$ (by the Noether inequality, and the Bogomolov-Miyaoka-Yau inequality). In this talk, I will survey what is known for various groups, presenting old and new results, and several open questions that go beyond this particular geographical question. These may include geography of rigid surfaces, canonically polarized surfaces, and Brody hyperbolic surfaces.

Young Researchers Talks

• On product identities and the Chow ring of holomorphic symplectic manifolds

  Ignacio Barros (Northestern U., USA)

  Abstract

  We propose a series of basic conjectural identities in the Chow rings of hyperkähler varieties of $K3$ type that generalizes in higher dimensions a set of key properties of cycles on a $K3$ surface. We prove that they hold for the Hilbert scheme of n points on a $K3$ surface. The talk is based on joint work with Laure Flapan, Alina Marian, and Rob Silversmith.

• Non-symplectic automorphisms of $K3$ surfaces

  Paola Comparin (UFRO, Chile)

  Abstract

  Given a $K3$ surface $X$ and an automorphism of $X$, it is called symplectic if the action on the 2-holomorphic form is trivial and non-symplectic otherwise. Automorphisms of $K3$ surfaces have been widely studied in the last years; the aim of the talk is presenting general results in the study of $K3$ surfaces admitting a non-symplectic automorphism of a given order and showing how one can approach the problem of classification and some results in this direction.

• Birational geometry of blow-ups of projective spaces along points and lines

  Zhuang He (Northeastern U., USA)

  Abstract

  In this talk we study the birational geometry of the blow-up $X$ of $\mathbb{P}^3$ at 6 points in very general position and the 15 lines joining the 6 points. We construct an infinite-order pseudo-automorphism $\phi$ on $X$, induced by the complete linear system of a divisor of degree 13. The effective cone of $X$ has infinitely many extremal rays and hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section which is a Jacobian Kummer surface $S$ of Picard number 17. The restriction of $\phi$ to $S$ is one of the 192 infinite-order automorphisms constructed by Keum. In general we show the blow-up of $\mathbb{P}^n$ ($n\geq 3$) at $(n+3)$ very general points and certain $9$ lines through them is not Mori Dream, with infinitely many extremal effective divisors. As an application, for $n\geq 7$, the blow-up of the moduli space of stable $n$-pointed rational curves $\overline{M}_{0,n}$ at a very general point has infinitely many extremal effective divisors. This is a joint work with Lei Yang.

• The Hodge conjecture for moduli spaces of stable sheaves over a nodal curve

  Inder Kaur (IMPA, Brazil)

  Abstract

  The Hodge conjecture is known for the Jacobian variety of a general, smooth, projective curve. Balaji-King-Newstead used this to prove the conjecture for the moduli space of rank $2$ stable sheaves with fixed odd degree determinant over a general, smooth, projective curve of genus $g \geq 2$. In this talk I will discuss an analogous result when the underlying curve is general, irreducible nodal and show why techniques from the smooth case fail. This is joint work with A. Dan.

• Quotients of products of curves with big cotangent bundle

  Juliana Restrepo (U. Grenoble-Alpes, France)

  Abstract

  The study of the positivity of the cotangent bundle of projective algebraic varieties has attracted a lot of attention due to its many geometric implications. In this talk, we are interested in quotients of products of curves of genus $\geq 2$ by the action of a finite group. These varieties have been subject of recent intensive work, among others, since they have been a fruitful method to construct interesting surfaces of general type. We will present some criteria to prove the bigness of the cotangent bundle for some of these varieties. These criteria will apply, in particular, to the case of product-quotient surfaces of general type with geometric genus, irregularity and second Segre number all equal to zero and to the case of the symmetric products of curves

• Segre Invariant and rank 2 Vector Bundles on $\mathbb{P}^2$

  Leonardo Roa (CIMAT, Mexico)

  Abstract

  In this talk, we define the Segre invariant for rank 2 vector bundles on surfaces and give a stratification of the moduli space $M_{\mathbb{P}^2}(2; c_1, c_2)$ of stable vector bundles of rank 2 and Chern classes $c_1$ and $c_2$ on $\mathbb{P}^2$. For a vector bundle $E$ of rank 2 on $\mathbb{P}^2$, the Segre invariant is defined as the minimum of the differences between the slope of $E$ and the slope of all line subbundles of $E$. This invariant defines a semicontinuous function on the families of vector bundles on $\mathbb{P}^2$. Thus, the Segre invariant gives a stratification of the moduli space $M_{\mathbb{P}^2}(2; c_1, c_2)$ into locally closed subvarieties $M_{\mathbb{P}^2}(2; c_1, c_2,s)$. We study the stratification, determine conditions under which the different strata are non-empty and compute their dimensions. As a consequence of this stratification, we give results related to the Brill-Noether problem. This is joint work with H. Torres-Lopez and A. Zamora.

• Bott vanishing using GIT and derived categories

  Sebastián Torres (UMass Amherst, USA)

  Abstract

  A smooth projective variety is said to satisfy Bott vanishing if $\Omega^j \otimes L$ has no higher cohomology for every $j$ and every ample line bundle $L$. This is a very restrictive property, and there are few examples known to satisfy it. Among them are toric varieties and recently $\bar{M_{0,5}}$ (shown by Totaro). I will present a new class of examples obtained as smooth GIT quotients of $(\mathbb{P}^1)^n$. For this, I will need to use the work by Teleman and Halpern-Leistner about the derived category of a GIT quotient, and explain how this allows us, in some cases, to compute cohomologies directly in an ambient quotient stack.

• Periods of algebraic cycles and variational Hodge conjecture

  Roberto Villaflor (IMPA, Brazil)

  Abstract

  In order to compute periods of algebraic cycles inside even dimensional smooth degree d hypersurfaces of the projective space, we restrict ourselves to cycles supported in a complete intersection subvariety. When the description of the complete intersection is explicit, we can compute its periods, and furthermore its cohomological class in primitive de Rham cohomology. As an application, we prove that the locus of general hypersurfaces containing two linear cycles whose intersection is of dimension less than $\frac{n}{2}-\frac{d}{d-2}$, corresponds to the Hodge locus of any integral combination of such linear cycles. This proves variational Hodge conjecture for those algebraic cycles.