Sergey A. Merkulov
Twistor Theory, Complex Homogeneous Manifolds and G-Structures
Preprint series: ESI preprints

MSC:

53C10 $G$-structures

53C05 Connections, general theory

53C15 General geometric structures on manifolds (almost complex, contact, symplectic, almost product structures, etc.)

Abstract: One of the most useful characteristics of an affine connection on a manifold
$M$ is its (restricted) holonomy group which is defined, up to a conjugation,
as a subgroup of $GL(T_t M)$ consisting of all automorphisms of the tangent
space $T_t M$ at a point $t\in M$ induced by parallel translations along the
$t$-based contractible loops in $M$. Which groups can occur as holonomies of
affine connections? By Hano and Ozeki \cite{HO}, any closed subgroup of a
general linear group can be realised as a holonomy of some affine connection
(which in general has a non-vanishing torsion tensor). The same question, if
posed in the class of {\em torsion-free}\, (non-locally symmetric) affine
connections only, is
not yet answered. According to Berger \cite{Berger}, the list of all possible
irreducibly acting holonomies of such connections is very
restricted. How much is known about this list?
In his seminal paper \cite{Berger}, Berger found a list of groups which
embraces all possible holonomies of torsion-free {\em metric}\, connections,
though his approach provides no method to distinguish which entries
can indeed be realised as holonomies and which are superfluous.
Later much work has been done to refine this list and to prove existence
of Riemannian metrics with special holonomies \cite{Al,Bryant1,Bryant2,S}.
In the same paper Berger presented
also a list of all but a finite number of possible candidates to irreducible
holonomies of "non-metric" torsion-free affine connections. How many holonomies
are missing from this second list is not known, but, as was recently shown
by Bryant~\cite{Br}, the set of missing, or {\em exotic}, holonomies is
non-empty. As usual in the representation theory, in order to get a deeper
understanding of all irreducible real holonomies one should first
try to address a complex version of the problem. The main result announced
in this paper asserts that any torsion-free holomorphic affine connection
with irreducibly acting holonomy group can be generated by twistor methods.


Keywords: exotic holonomy, G-structure, twistor theory