Kähler-Einstein metrics on toric Fano manifolds and connections to Optimal Transport
Typ
Examensarbete för masterexamen
Program
Engineering mathematics and computational science (MPENM), MSc
Publicerad
2021
Författare
Andreasson, Rolf
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
Abstract
We introduce the problem of finding a Kähler-Einstein metric on a Kähler manifold and
specifically on a Fano manifold. We restrict to the class of toric complex manifolds where
the symmetry can be used to reduce the resulting partial differential equation to a real
equation in Rn. We then introduce the theory of optimal transport, specially adapted to
the application. We present a special transportation problem which is in fact equivalent
to a weak formulation of the Kähler-Einstein equation on toric Fano manifolds. The presentation
is a literature study aimed at presenting the material in a self-contained and
elementary fashion.
We also present a novel variational approach to the existence problem in the language
of optimal transport and equilibrium physics. We show some results towards an existence
result based on this approach.
Finally we exemplify large parts of the theory on complex projective space, an explicit
example of a toric Fano manifold. We also compute the free energy, an invariant we will
introduce, on complex projective space.