## Dynamic alpha-invariants of del Pezzo surfaces with boundary

##### Abstract

The global log canonical threshold, algebraic counterpart to Tian's alpha-invariant, plays an
important role when studying the geometry of Fano varieties. In particular, Tian showed that
Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In
recent years Donaldson drafted a programme to precisely determine when a smooth Fano variety
X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is
equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's
programme consists on finding a Kahler-Einstein metric with edge singularities of small angle
along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic
version of the alpha-invariant could be used to find such metrics.
The global log canonical threshold measures how anticanonical pairs fail to be log canonical.
In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings.
First we extend Cheltsov's computation of the global log canonical threshold of complex
del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically
closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be
log canonical. In particular, we give a very explicit classifiation of very singular anticanonical
pairs for del Pezzo surfaces of degree smaller or equal than 3. We conjecture under which
circumstances such a classifcation is plausible for an arbitrary Fano variety and derive several
consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo
surfaces of small degree, where the boundary is any smooth elliptic curve C.
Our main result is a computation of the dynamic alpha-invariant on all smooth del Pezzo
surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend
on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge
singularities of angle β along C.