Yoshihiko Mitsumatsu (Chuo University, Tokyo) will speak in the geometry seminar on Tuesday 24 March 2020 at 14h in the Salle des Profs. There will be two geometry seminars that day. Yoshihiko’s title is Lefschetz fibration on Milnor fibres of certain singularities and his abstract is below. Update 8/03: Yoshihiko’s talk has been cancelled because of travel restrictions due to the ongoing pandemic of Covid-19. 

Recently we found that the Milnor fibre of cusp or simple elliptic singularities of complex 3-variables admits a Lefschetz fibration to a disk with general fibres 2-torus. As Milnor fibres with natural complex structure are Stein manifolds, our Lefschetz fibration is not holomorphic. The construction will be only roughly indicated. Rather I will explain the folllowing applications. This fact implies, for example, that the Lawson type foliations, a codimension 1 foliation which is obtained as a modification of the Milnor fibration of the singularity, admits a leafwise symplectic structures (a regular Poisson structure). Also Arnold’s strange duality between certain mordality 1 singularities has a deep relation with this Lefschetz fibration. This view point gives a smooth decomposition of a K3 surface into two Milnor fibres. This talk is based on a recent joint work with Naohiko Kasuya, Hiroki Kodama, and Atsuhide Mori.

Daniel Waldram (Imperial College) will speak in the geometry seminar on Tuesday 28 April 2020 at 11h am in the Salle des Profs. Daniel’s title and abstract are to be announced.

Daniel Greb (Universität Duisburg-Essen) will speak in the geometry seminar on Tuesday 17 March 2020 at 11h am in the Salle des Profs. Daniel’s title is Projectively flat sheaves and characterisations of finite quotients of projective spaces and his abstract is below.

I will explain how to extend the classical characterisation of projective space among Kähler-Einstein Fano manifolds in terms of a Chern class (in)equality to the class of Fano varieties with Kawamata log terminal singularities. I will spend significant time on discussing the necessary tools, which range from analysis (harmonic metrics / Simpson correspondence) and classical differential geometry to algebraic geometry (local fundamental groups of klt singularities). This is joint work with Stefan Kebekus and Thomas Peternell, and partly with Behrouz Taji.

Mircea Petrache (Pontificia Universidad Catolica, Santiago de Chile) will speak in the geometry seminar on Friday 28 February 2020 at 11h am in the Salle N 08.08 (this is the room behind the elevators in the 8th floor of the NO building). Mircea’s title is Uniform measures and manifolds all of whose curvatures are constant and his abstract is below.

A uniform measure in Euclidean space R^d is a measure with respect to which balls B(x,r) with center x in the support, are assigned mass dependent of r and independent of the choice x. For example any invariant measure with respect to a subgroup of the isometry group of R^d is uniform, and called a homogeneous measure. However we also have a few exotic examples of non-homogeneous uniform measures, such as the volume measure of the “light cone” {x^2+y^2+z^2=w^2} in R^4.This class of measures was first studied by David Preiss as the crucial ingredient of his 1987 proof of the Besicovitch conjecture. The complete classification of uniform measures remains a difficult open problem, even restricted to ambient dimension d=2. I will detail the known classification of 1-dimensional uniform measures in R^d for general d, for which, in joint work with Paul Laurain, we show that they are constituted of disjoint unions of helices or of toric knots, or equivalently, of analytic curves all of whose curvatures are constant.