University College London, 15/10/2014
“Non-reductive geometric invariant theory and applications in algebraic, symplectic and hyperkähler geometry.”
Mumford’s geometric invariant theory (GIT) provides a method for constructing (projective completions of) quotient varieties for linear actions of complex reductive groups on affine and projective varieties, and has many applications (for example in the construction of moduli spaces in algebraic geometry). The aim of this talk is to discuss an extension of Mumford’s GIT to actions of linear algebraic groups which are not necessarily reductive. Its applications include the construction known as symplectic implosion (due to Guillemin, Jeffrey and Sjamaar) and more recently an analogous construction in hyperkähler geometry.
“ Analytic construction of dihedral ALF gravitational instantons.”
We deal in this talk with an analytic constructions of ALF (Asymptotically Locally Flat) gravitational instantons, or 4-dimensional complete hyperkähler manifolds with cubic volume growth. We give the construction of dihedral instantons, previously produced by Cherkis-Hitchin and Cherkis-Kapustin, with very different methods. In particular, we will see how resolving a complex Monge-Ampère equation, given for general ALF manifolds, allows us, on our examples, to correct a simple prototype in order to get the desired hyperkähler metric.
“Curves and cycles on K3 surfaces.”
Rational curves on Calabi-Yau manifolds and in particular on K3 surfaces have attracted a great deal of attention. In this talk I will introduce a class of curves, so called constant cycle curves, that behave in many respects like rational curves but include other curves as well. The interest in these curves comes from the study of Chow groups and I will survey the main features and problems surrounding those.