ICTP Mathematics Seminar - Thursday 15 November, at 14:30 hrs. (Zakarias Sjöström Dyrefelt)

[ICTP Math Section]

MATHEMATICS SEMINARS 2018


Thursday 15 November, at 14:30 hrs.

Zakarias Sjöström Dyrefelt
(ICTP)


Title:

Stability notions in Kähler geometry


Abstract:

In an effort to find a higher dimensional analogue of the classical 
uniformization theorem in complex geometry, it was suggested by E. 
Calabi that one should study existence of "canonical metrics" on compact 
Kähler manifolds. These are the so called extremal metrics, 
characterized by the fact that they minimize a certain functional on the 
space of Kähler metrics. Many natural and important classes of metrics 
fall into this category, such as the Kähler-Einstein metrics, that play 
a prominent role also in physics. Kähler-Einstein metrics were proven to 
always exist on Calabi-Yau manifolds and on manifolds of general type. 
In the case of Fano manifolds however, it has been long known that they 
do not always exist, and it was conjectured by S-T. Yau, S. Donaldson 
and G. Tian that existence should be linked to stability of the 
underlying manifold. In the case of Kähler-Einstein metrics on Fano 
manifolds this conjecture was confirmed in 2012 by Chen-Donaldson-Sun 
and Tian independently, who showed that existence of Kähler Einstein 
metrics is equivalent to a stability condition in algebraic geometry 
called K-stability.

Generalizing the above picture further we may ask about existence of a 
larger class of extremal metrics, namely the constant scalar curvature 
metrics. In this talk I will present an analytic/pluripotential approach 
to stability notions that obstruct existence of such metrics on 
arbitrary (not necessarily projective) compact Kähler manifolds. I will 
explain how the methods used allow us to view classical algebraic 
stability notions in a new light, and gives a framework for comparing 
certain stability notions to each other. Finally, I will discuss how the 
pluripotential approach is related to recent developments regarding 
Tian's properness conjecture and generalized Yau-Tian-Donaldson conjectures.



VENUE: Stasi Seminar Room (ICTP Leonardo da Vinci Building, first floor)

-- 
Monica Petronela Ancuta
for Ms Mabilo
Secretariat of the ICTP Mathematics Section
math at ictp.it

Abdus Salam International Centre for Theoretical Physics
Strada Costiera 11
34151 Trieste
Italy

Tel.: +39 040 2240 318