## Smooth exceptional del Pezzo surfaces

##### Abstract

For a Fano variety V with at most Kawamata log terminal (klt) singularities and a finite group
G acting bi-regularly on V , we say that V is G-exceptional (resp., G-weakly-exceptional) if the
log pair (V,∆)
is klt (resp., log canonical) for all G-invariant effective Q-divisors ∆ numerically
equivalent to the anti-canonical divisor of V. Such G-exceptional klt Fano varieties V are
conjectured to lie in finitely many families by Shokurov ([Sho00, Pro01]). The only cases for
which the conjecture is known to hold true are when the dimension of V is one, two, or V is
isomorphic to n-dimensional projective space for some n. For the latter, it can be shown that
G must be primitive—which implies, in particular, that there exist only finitely many such G
(up to conjugation) by a theorem of Jordan ([Pro00]).
Smooth G-weakly-exceptional Fano varieties play an important role in non-rationality
problems in birational geometry. From the work of Demailly (see [CS08, Appendix A]) it
follows that Tian’s αG-invariant for such varieties is no smaller than one, and by a theorem
of Tian such varieties admit G-invariant Kähler-Einstein metrics. Moreover, for a smooth
G-exceptional Fano variety and given any G-invariant Kähler formin the first Chern class, the
Kähler-Ricci iteration converges exponentially fast to the Kähler form associated to a Kähler-
Einsteinmetric in the C∞(V)-topology. The termexceptional is inherited from singularity
theory, to which this study enjoys strong links.
We classify two-dimensional smooth G-exceptional Fano varieties (del Pezzo surfaces)
and provide a partial list of all G-exceptional and G-weakly-exceptional pairs
(S,G), where S
is a smooth del Pezzo surface and G is a finite group of automorphisms of S. Our classification
confirms many conjectures on two-dimensional smooth exceptional Fano varieties.