Geometry and Analysis of Manifolds
with Reduced Holonomy
SMI, Cortona, 2007
TOPICS FOR THE AFTERNOON SEMINARS
(Simon Salamon)
1. Kähler manifolds. Equivalence of the various
definitions. Properties of Kähler manifolds (possibly including
[ρ]=2πc1)
2. Reducible holonomy. Applying the theory to compact Ricci-flat
Kähler manifolds [J 3.2+6.2]
3. The continuity method. Overview of the Calabi conjecture;
stating the basic estimates and explaining how they are used [J
5.2] (Maybe in collaboration with the existence seminar)
4. Nearly-Kähler metrics. Definition, characterization in
terms of exterior forms, and relatonship to G2 holonomy.
See the Tour, §6
5. Spin(7) metrics. Elementary properties of metrics with
holonomy Spin(7) on an 8-manifold [J §10.5]
6. Isothermal coordinates. Existence of these on a real
surface, as a special case of the Newlander-Nirenberg theorem.
Possible source: Spivak's vol IV, from the Palazzone's library
7. A metric with holonomy SU(m+1). Constructed on the total
space of the canonical bundle of a Kähler-Einstein manifold;
see the red book, pages 107-109
8. More on K3 surfaces. Realizations, topology, Hodge numbers,
complex moduli. Source: [J §7.3] and other books
9. The moduli of holonomy G2 metrics. Deformations of
3-forms; relevance of b3 [J 10.4]
10. The topology of compact G2 manifolds. Possible
reducible holonomy subgroups of G2, refined Betti
numbers. Main source: [J 10.2]. The seminar should also include some
Hodge theory from [J 3.5.2] OR background on the first Pontrjagin
class of a smooth manifold
11. A modification of the simple example
T7/Γ. Study of fixed points and singular sets of
the example in [J §12.3], following a similar presentation in
lectures (the details are fiddly but elementary). Discussion of the
resolutions in the same section OR the further examples from [J
§12.4]
12. The structure of the singularities of
T7/Γ. The notation and cases of [J §11.3],
described carefully with the help of at least one example from Chapter
12
Another topic may follow