**Jeremy Daniel:**Some characteristic classes of flat bundles in complex geometry

On a compact Kähler manifold $X$, any semisimple flat bundle carries a harmonic metric. It can be used to define some characteristic classes of the flat bundle, in the cohomology of $X$. We show that these cohomology classes come from an infinite-dimensional space – constructed with loop groups – which is an analogue of the period domains used in Hodge theory.**Christian Urech:**Simple subgroups of the Cremona group

The Cremona group is the group of birational transformations of the complex projective plane. In 2012 Cantat and Lamy proved that the Cremona group is not simple, which answered a long standing open question. In this talk we will refine their results and show that a simple group can be embedded into the plane Cremona group if and only if it can be embedded into $\mathbb P GL(3,\mathbb C)$. Our techniques also yield new insights into the structure of subgroups consisting of elliptic elements. In particular, we will describe the structure of torsion subgroups and show that the Cremona group satisfies the Tits alternative for arbitrary subgroups; this extends a result by Cantat.**Michael Hoff:**Moduli of lattice polarized $K3$ surfaces via relative canonical resolutions

For a smooth canonically embedded curve $C$ of genus 9 together with a pencil $|L|$ of degree 6 , we study the relative canonical resolution of $C$ in $X$, where $X$ is the scroll swept out by the pencil $|L|$. In general, the relative canonical resolution is built with certain vector bundles on the projective line. We show that in our case the second syzygy bundle in this resolution of C is unbalanced. The proof reveals a new geometric connection between the universal Brill–Noether variety $W_{9,6}^1$ and a moduli space of lattice polarized $K3$ surfaces (for a certain rank 3 lattice). As a by-product we prove the unirationality of this moduli space. This is a joint work with Christian Bopp.**Matthias Wendt:**Chow-Witt rings of Grassmannians and oriented Schubert calculus

Chow-Witt rings are a recent refinement of Chow rings which take into account orientation information using quadratic forms. While Chow rings of varieties can be compared to cohomology of the complex points of the varieties, the Chow-Witt rings compare better to the cohomology of real points. In the talk I will explain the structure of the Chow-Witt rings of Grassmannian varieties over general base fields (which strongly resembles the integral cohomology of real Grassmannian manifolds). This gives rise to an “oriented” version of Schubert calculus which generalizes some results from real Schubert calculus to (almost) arbitrary fields.

# Author Archives: Stefan Kebekus

# Meeting in February 2017

**Oliver Bräunling (Freiburg): $K$-theory of étale p-torsion sheaves and Cartier modules.**I will discuss the so-called “$K$-theory of semi-linear endomorphisms”, going back to Quillen and Grayson. I’ll present some new computations. The idea is to use a positive characteristic version of the Riemann-Hilbert correspondence (à la Emerton-Kisin and Böckle-Pink) to get a handle on the case of semi-linear endomorphisms coming from the Frobenius.**Enrica Floris (Basel) Invariance of plurigenera for foliations on surfaces.**Recently, Brunella and McQuillan proved some of the main results of birational geometry in

the setup of foliations on surfaces. In this talk, we will discuss to which extent the theorem of Invariance of Plurigenera is true for foliations on surfaces. This is a joint work with Paolo Cascini.**Arvid Perego (Nancy): Kählerness of moduli spaces of sheaves over non-projective $K3$ surfaces.**Kählerness of moduli spaces of slope-stable sheaves over $K3$ surfaces is in known only in particular cases: if the base $K3$ surface is projective, the moduli spaces are known to be quasi-projective; if the dimension of the moduli space is 2, then it is a $K3$ surface; in higher dimension, the moduli spaces which parameterize only locally free sheaves are known to be Kähler. In this talk I will expose some recent results

about the remaining cases: in particular, i will show that a moduli space of slope-stable sheaves (whose rank and first Chern class are prime to each other, and where the stability is with respect to a generic polarization) is Kähler if and only if its second Betti number is the sum of its Hodge numbers $h^{2,0}$, $h^{1,1}$ and $h^{0,2}$.**Robert Laterveer (Strasbourg):**On a conjecture of Voisin. Around 1994, Clair Voisin has made a conjecture concerning algebraic 0-cycles on smooth projective complex varieties of geometric genus 1. We will explain this conjecture, and review cases (both old and new) where the conjecture is known.

# Meeting in December 2015

**Jian Xiao: Zariski decomposition for curves.**We study Zariski decompositions using Legendre-Fenchel type transforms. In this way, we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves, yielding some fundamental positivity results for curve classes. We also give a refined structure of the movable cone of curves, and compare the volume function for curves with the mobility, showing some relations between enumerative geometry and convex analysis. This talk is based on the joint work with Brian Lehmann.**Junyan Cao: Kodaira dimension of algebraic fiber spaces over surfaces.**Let $f$ be a fibration between two projective manifolds. The Iitaka conjecture states that the Kodaira dimension of total space is not less than the sum of the Kodaira dimension of the generic fiber and that of the base manifold. We prove a log-version (with klt pair) of the conjecture, under the assumption that the base is of dimension 2. The proof relies mainly on a uniformation theorem for compact kähler orbifolds with trivial first Chern class proved by F. Campana, and also the positivity properties of direct images of relative pluri-canonical bundles. This is a joint work with J.-A. Chen.**Damian Brotbek: Complete intersection varieties with ample cotangent bundle.**This is a joint work with Lionel Darondeau. We prove that any smooth projective variety $M$ contains subvarieties with ample cotangent bundle of any dimension less or equal to half the dimension of $M$. We construct such subvarieties as suitable complete intersection varieties.**Shane Kelly: —**

# Meeting in June 2015

**Robert Laterveer: Hard Lefschetz for Chow groups.**Inspired by the conjectural Bloch-Beilinson filtration, we formulate a conjectural hard Lefschetz property for the Chow groups of a smooth projective algebraic variety over the complex numbers. This property can be verified in some special cases – roughly speaking, for varieties for which the self-product has vanishing middle-dimensional Griffiths group, and for varieties that have finite-dimensional motive (in the sense of Kimura). As we will explain, closely related (but slightly different) results can also be deduced from recent work of Charles Vial.**Susanna Zimmermann: An Abelian quotient of the real Cremona group.**The Cremona group of the complex plane contains many normal subgroups, all of which are of infinite index. There is no proper normal subgroup containing elements of degree 1 (not equal to the identity map), 2, 3 and 4. What about the Cremona group of the real plane? I will present an abelian quotient of it, which implies the existence of normal subgroups of index equal to any given power of 2, all of them containing every map of degree 1, 2, 3, and 4.**Giulia Battiston: A Galois descent theory for inseparable field extension.**Let $L/K$ be a Galois separable field extension, then classical Galois descent theory describes algebraic objects over $K$, such as for example $K$-varieties, as being equivalent to algebraic objects over $L$ endowed with a $Gal(L/K)$-action which is $\sigma$-linear. If $L/K$ is not separable, though, such a theory does not apply for the simple reason that the field of $Gal(L/K)$-invariants is strictly bigger than $K$. We will present how this inconvenient can be bypassed using the automorphism group of truncated polynomials over L and hence obtaining a Galois descent theory for inseparable extensions.**Zsolt Patakfalvi: Projectivity of the moduli space of KSBA stable pairs and applications.**KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs are higher dimensional generalizations of (weighted) stable pointed curves. I will present a joint work in progress with Sándor Kovács on proving the projectivity of this moduli space, by showing that certain Hodge-type line bundles are ample on it. I will also mention applications to the subadditivity of logarithmic Kodaira dimension, and to the ampleness of the CM (Chow-Mumford) line bundle.

# Meeting in December 2014

**Damien Megy: The period map of certain families of singular hypersurfaces.**This is a joint work with Philippe Eyssidieux. We consider a natural Deligne-Mumford stack parametrizing degree $d$ hypersurfaces of $\mathcal P^n$ with ADE singularities, and prove an infinitesimal Torelli property along the stacky strata. This construction gives rise to examples of smooth projective varieties with interesting fundamental groups and universal covers. If time permits, I will discuss the Toledo and Shafarevich conjecture for these examples.**Jean-Philippe Furter: Some properties of the group ${\rm GL}_2 ( {\mathbb C}[x_1,\ldots,x_m])$ and some applications to the polynomial automorphism group ${\rm Aut}( {\mathbb A}^n)$ of the affine space ${\mathbb A}^n$.**The groups ${\rm GL}_2 ( {\mathbb C}[x_1,\ldots,x_m])$ and ${\rm Aut}( {\mathbb A}^n)$ can naturally be considered as ind-groups (algebraic groups of infinite dimension). As such, they are endowed with the Zariski topology. We will describe several topological properties of these two groups. In particular, we will give examples of closed subgroups.**Behrouz Taji: Semistable reflexive sheaves over klt spaces.**We prove that any semistable reflexive sheaf over a klt projective variety with vanishing orbiflod Chern classes comes form a linear representation of $\pi_1(X_{\rm{reg}})$. In the smooth setting such a correspondence goes back to the classical results of Narasimhan-Seshadri , Donaldson-Uhlenbeck-Yau and subsequently Simpson. As an application we establish a characterization of finite quotient of Abelian varieties via vanishing of orbifold Chern classes. This is a joint work with Steven Lu.**Paolo Cascini: Toroidal modifications.**I will survey on a conjecture due to Shokurov on the ACC for the set of minimal log discrepancies and I will describe an approach towards this conjecture using toroidal modifications. Joint work with J. McKernan.

# Meeting in May 2014

**Chenyang Xu: Maximal pole of motivic Zeta function.**We prove a conjecture of Veys, which says that the opposite of the log canonical threshold is the only possible pole of maximal order of the motivic zeta function over a field of characteristic zero. If time permits, we will also discuss how to apply our method to study a family of Calabi-Yau varieties and prove properties for the weight function associated with a degeneration.(joint with Johannes Nicaise)**Tomasz Szemberg: The effect of points fattening.**I recall briefly results due to Bocci and Chiantini on the effect of points fattening on the projective plane. Then I will report on some generalizations to other surfaces. The core of the lecture will be devoted to higher dimensional analogies. Results in that part were obtained jointly with Thomas Bauer (Marburg).**Olivier Benoist: Complete families of smooth space curves.**In this talk, we will study complete families of smooth space curves, that is complete subvarieties of the Hilbert scheme of smooth curves in $\mathbb P^3$. On the one hand, we will construct non-trivial examples of such families. On the other hand, we obtain necessary conditions for a complete family of smooth polarized curves to induce a complete family of non-degenerate smooth space curves. Both results rely on the study of the strong semistability of certain vector bundles.**Gianluca Pacienza: Families of rational curves on holomorphic symplectic varieties.**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 (K3)$^{[2]}$-type contain uniruled divisors and rationally connected lagrangian surfaces. I will also mention some applications to the study of Chow groups of such varieties, generalizing analogous results due to Beauville and Voisin on K3 surfaces.

# Meeting in December 2013

**Damian Brotbek: Height inequality for surfaces in an abelian variety.**Given a function field $K$ and a projective variety $X$ over $K$, Vojta conjectured an inequality between the canonical height of an algebraic point on $X$ and the discriminant of that point. In this talk, I will explain how to obtain such a height inequality when $X$ is a generic surface in an abelian threefold. The proof if based on the study of higher order jet spaces. This is a joint work with Carlo Gasbarri.**Clemens Jörder: On the Poincaré lemma on singular spaces.**On a singular normal complex space the cochain complex of sheaves of reflexive differential forms is not a resolution of the sheaf of locally constant functions, since the Poincaré lemma for reflexive differential forms fails in general. I discuss under which conditions the Poincaré lemma is valid. Furthermore I will relate the question of its failure to vanishing theorems of Kodaira-Akizuki-Nakano type.**Sergei Kovalenko: Smooth Non-Homogeneous Gizatullin Surfaces.**Quasi-homogeneous surfaces, or Gizatullin surfaces, are normal affine surfaces such that the complement of the big orbit of the automorphism group is finite. If the action of the automorphism group is transitive, the surface is called homogeneous. Examples of non-homogeneous Gizatullin surfaces were constructed in by the speaker in arXiv:1304.7116, but on more restricted conditions. We show that a similar result holds under less constrained assumptions. Moreover, we exhibit examples of smooth affine surfaces with a non- transitive action of the automorphism group whereas the automorphism group is huge. This means that the automorphism group is not generated by a countable set of algebraic subgroups and that its quotient by the (normal) subgroup, generated by all algebraic subgroups, contains a free group over an uncountable set of generators.**Giovanni Mongarde: Ample cone and negative divisors for Hilbert schemes of points on K3s**. For K3 surfaces, the ample cone is cut out by rational curves of selfintersection -2. In the case of Hilbert schemes of points of K3 surfaces and their deformations, a similar result can be phrased using certain divisors whose top self intersection is negative.