Geometry and Analysis of Manifolds
with Reduced Holonomy
SMI, Cortona, 2007
TOPICS FOR THE AFTERNOON SEMINARS
(Alexei Kovalev)
1.
(suitable for Friday of Week 1 or beginning of Week 2)
Sobolev spaces and conformal geometry:
(a) Show that for a compact Riemannian manifold M, we have Lpk(M)=Lpk,loc(M) as topological vector spaces.
(b) Riemannian metrics g and ĝ on a manifold M are said to
be conformally equivalent if ĝ=efg for some smooth
function f on M. Show that the L2 norm of differential
2-forms on a
four-manifold is conformally invariant (i.e. has the same value for any
choice
of metric in a given conformal equivalence class. More generally, when
is an Lpk norm on an n-dimensional manifold conformally
invariant?
2.
(Week 2) This seminar could begin with introducing Cayley numbers
(octonions)
and then
use these to construct an almost-complex structure on an arbitrary
orientable
immersed 6-dimensional submanifold of R7.
Further related material can be
chosen from S. Salamon, Riemannian geometry and holonomy groups
3.
(Week 2, after the lecture of S.Salamon on the Riemannian geometry in 4
dimensions.) Anti-self-duality for the 4-dimensional oriented
Riemannian manifolds:
Decomposition of the Riemannian curvature in dimension 4.
Specialization to
Kähler complex surfaces. Manifolds with holonomy Sp(1) (=SU(2)) are
anti-self-dual. Kähler manifolds are anti-self-dual iff scalar
flat.
Supporting texts:
A.King and D.Kotschick, The deformation theory of
anti-self-dual conformal structures, Mathematische Annalen 294 (1992)
591--609.
C.P Boyer, Conformal duality and compact complex surfaces,
Mathematische Annalen 274 (1986)
517--526.}
(Both articles can be found at
www.digizeitschriften.de)
4.
(Week 2) The hyper-Kähler and quaternionic geometry. In particular,
show using a
`holomorphic volume form' (but without appealing to the isomorphism of
holonomy groups SU(2)=Sp(1)) that if (X,I,h) is a simply-connected
Ricci-flat Kähler surface (I
is the complex structure, h
is the Kähler
metric) then X has another
complex structure J such that
IJ+JI=0 and h
is Kähler with respect to J.
Supporting texts:
S. Salamon, Riemannian geometry and
holonomy groups, Ch. 7
D. Joyce,
Compact manifolds with special holonomy, Ch. 8
5.
(Week 2 or 3.) Construct a hyper-Kähler K3 surface as a generalized
connected sum
X=S1 #T3 S2 of two
asymptotically cylindrical Calabi-Yau
(hyper-Kähler)
surfaces Si. The two principal tasks are (i) to construct
on~$X$ an
integrable almost-complex structure with trivial canonical bundle and
(ii) to
do the `gluing argument for Ricci-flat Kähler metric, using the
Calabi-type
argument.
This is a follow-up to the material
to be lectured in Week 2; for further
advice consult A.~Kovalev.
Useful text: §2 in N. Hitchin,
The moduli space of special Lagrangian
submanifolds. arXiv dg-ga/9711002
6. Calibrated minimal
submanifolds of Rn.
Explain the concept of calibration. Show that calibrated submanifolds
are minimal (moreover, they are volume-minimizing). Discuss one or two
examples of calibrations -- e.g. deduce a differential equation for the
corresponding type of calibrated submanifolds and give example(s) of
solutions.
The reference text is
R. Harvey and H.B. Lawson, Calibrated
geometries, Acta Math. 148
(1982) 47--157. Sections depend on your choice of example of
calibration. A PDF file of the paper is available
7. Explicit/elementary examples
of the Calabi-Yau metrics on the complexifications of Sn, CPn and HPn. The examples are
obtained by using the large
symmetry group of those manifolds to reduce the nonlinear partial
differential equation governing the Ricci curvature to a simple
second-order ordinary differential equation for a function.
The reference texts are
M.B. Stenzel, Ricci-flat metrics on
the complexification of a compact rank one symmetric space, Manuscripta
Math. 80 (1993) 151--163.
T.-C. Lee, Complete Ricci flat Kähler metric on MnI,
M2nII, M4nIII Pacific J.
Math. 185 (1998) 315--326.
PDF files of these papers are available
8. (Week 3) Topological criterion for holonomy = G2. The torsion-free
G2 structure on a 7-manifold only guarantees that the holonomy of
induced metric is contained in G2. Your task if you choose to
accept it is to explain that the holonomy of a compact G2-manifold will be exactly
G2 if and only if the fundamental group of the manifold is
finite. There is a similar criterion for an asymptotically cylindrical G2-manifold.
Supporting text
for the
compact case: §10.1, 10.2 in D. Joyce, Compact manifolds with
special holonomy, §10.1, 10.2
for the asymptotically
cylindrical case §7 in J. Nordström, Deformations of
asymptotically cylindrical manifolds. arXiv:0705.4444
9. Stable differential forms. Explain how these generalize the concept
of a non-degenerate (symplectic) 2-form and discuss some
result(s) from
N. Hitchin, Stable forms and special
metrics, arXiv:math.DG/0107101
or N. Hitchin, The geometry of
three-forms in six and seven dimensions, arXiv:math.DG/0010054