• Algebraic Geometry
• Workshop: Birational geometry and foliations

• Schedule
• Participants

I will explain a description of the birational geometry of hyperkähler varieties deformation equivalent to Hilbert schemes of K3 surfaces. The description is based on using wall-crossing for the case of moduli spaces of sheaves on a K3 surface, and deformation theory for rational curves. This is based on joint work with Macri, and with Hassett and Tschinkel.

We show that if D is a smooth non-uniruled divisor in a projective irreducible hyperkähler manifold X, its characteristic foliation is not algebraic. This extends in this case a former result of Hwang-Viehweg which assumed D to be of general type (but X could be an Abelian variety). As a consequence, we show that if D is nef and K_D is semi-ample (which is then the case if dim(X)=4), then D is semi-ample on X. This is a joint work with E. Amerik.

We show that if (X,B) is a two dimensional Kawamata log terminal pair defined over an algebraically closed field of characteristic p, and p is sufficiently large, depending only on the coefficients of B, then (X,B) is also strongly F-regular. Joint work with Y. Gongyo and K. Schwede.

I will state some interconnected conjectures on (a) the algebraic geometry and moduli spaces, and (b) mirror symmetry, for orbifolds del Pezzo surfaces. I will present some of the evidence. This is joint work in progress with many people and students of the PRAGMATIC school held last Summer in Catania.

A rational curve C in a nonsingular variety X is standard if under the normalization f: P¹ → C, the vector bundle f^*T(X) decomposes as O(2) + O(1)^p + O^q for some nonnegative integers satisfying p+q = dim X-1. For a Fano manifold X of Picard number 1, a general rational curve of minimal degree through a general point is standard. It has been asked whether all rational curves of minimal degree through a general point are standard. In a joint work with Hosung Kim, we find a negative example to this question.

Let d >2 be an odd integer and let f(x₁, ..., x_n, x_n+1), n > d-1, be a weighted homogeneous polynomial of degree 2d with respect to the weights wt(x₁) = ... = wt(x_n) = 1 and wt(x_n+1) = 2. Let X^f be a Veronese double cone of dimension n associated to a general choice of f. This is an n-dimensional Fano manifold of Picard number 1 with index n+2-d. We study the variety of minimal rational tangents at a general point x of X^f, the projective variety defined as the union of the tangent directions to rational curves of minimal degree through x. We show that the normalization morphism of the variety of minimal rational tangents is not an immersion if 2d < n+1, which implies that some rational curves of minimal degree through x are not standard.

This talk surveys recent results on the singularities of the Minimal Model Program and discusses applications to the study of varieties with trivial canonical class. The first part of the talk discusses an infinitesimal version of the classical decomposition theorem for varieties with vanishing first Chern class, and its (conjectural) consequences for the structure theory of varieties with vanishing Kodaira dimension. Secondly, we compare the étale fundamental group of a klt variety with that of its smooth locus. As first major application, we show that any flat holomorphic bundle, defined on the smooth part of a projective klt variety is algebraic and extends across the singularities. This allows to generalise a famous theorem of Yau, which states that any Ricci-flat Kähler manifold with vanishing second Chern class is an étale quotient of a torus. This is joint work with Daniel Greb and Thomas Peternell.

This is joint work with Karl Schwede. We prove that being a DB pair is deformation invariant and that the relationship of the notions of rational and Du Bois pairs resembles that of canonical and log canonical varieties. In particular if a member of a family has Du Bois singularities, then the total space of the family has rational singularities near the given fiber.

The volume of a divisor is an important invariant measuring the "positivity" of its numerical class. I will discuss an analogous construction for cycles of arbitrary codimension.

I will explain a construction of a family of 8-dimensional projective complex symplectic manifolds starting from the moduli space of twisted cubics on a general cubic fourfold. The relation to of a -surface is still open. This is a joint work with Manfred Lehn, Christoph Sorger and Duco van Straten.

We show that supersingular surfaces are related by purely inseparable isogenies. As an application, we deduce that they are unirational, which confirms conjectures of Artin, Rudakov, Shafarevich, and Shioda. The main ingredient in the proof is to use the formal Brauer group of a Jacobian elliptically fibered surface to construct a family of "moving torsors" under this fibration that eventually relates supersingular surfaces of different Artin invariants by purely inseparable isogenies. If time permits, we will show how these "moving torsors" exhibit the moduli space of rigidified supersingular crystals as an iterated projective bundle over a finite field.

We give a geometric description of toric varieties using notions from birational geometry. The proof involves using the Cox ring. This is joint work with Morgan Brown, Roberto Svaldi and Runpu Zong.

The log discrepancy function and the log canonical centers play an important role in birational geometry. I will discuss an analogue of these notions in the context of degenerations of Calabi -Yau varieties. This is based on joint work with Johannes Nicaise.

I will report on a joint work with François Charles, in which we study families of rational curves on certain irreducible holomorphic symplectic varieties. In particular, we prove that projective holomorphic symplectic fourfolds of -type contain uniruled divisors and rationally connected lagrangian surfaces.

I will describe recent progress in joint work with Andreas Höring on the minimal model program for 3-dimensional Kähler varieties.

Shokurov and Reid proved that a Fano 3-fold with canonical Gorenstein singularities has a Du Val elephant, that is, a member of the anticanonical linear system with only Du Val singularities. The classification of Fano 3-folds is based on this fact. However, for a Fano 3-fold with non-Gorenstein terminal singularities, the anticanonical system does not contain such a member in general. Altinok-Brown-Reid conjectured that, if the anticanonical system is non-empty, a -Fano 3-fold can be deformed to one with a Du Val elephant. In this talk, I will explain how to deform an elephant with isolated singularities to a Du Val elephant.

I will discuss recent work of Bhargav Bhatt, myself and Shunsuke Takagi relating several open problems, and building on previous work of Mircea Mustata and Vasudevan Srinivas. The open problems follow. First: whether a smooth complex variety is ordinary after reduction to characteristic for infinitely many . Second: whether multiplier ideals reduce to test ideals for infinitely many (regardless of coefficients). Third: whether complex varieties with Du Bois singularities have -injective singularities after reduction to infinitely many .

I will explain how free rational curves give rise to a foliation. This is interesting in positive characteristic and gives the difference between freely rationally connectedness and separably rationally connectedness.