String theory and balanced metrics
One of the main motivations for considering balanced metrics, in addition to the
considerations already mentioned, has to do with the theory of what are known as
heterotic strings with super-symmetry. The physics here is
Building Geometric Structures
Summary of Profesor Yaus Lecture 2
Thursday, April 5,2007
Notes and supplementary remarks ( in [ ]s ) by Robert E. Greene
Last time:
General idea: build geometric structures using methods of (usually nonlinear) partial
differ
Lecture no. 3, Professor S.T. Yau , April 10, 2007
Notes and supplementary comments (in [ ]s) by Robert E. Greene
Last time: Real 4-manifolds M4 with almost (many) complex structures but with no
integrable almost complex structure, no complex structure. I
Lecture no. 4, Professor S.T. Yau , April 12, 2007
Last time we explained how, for a compact complex surface, b1 even implies that the
surface is Khler. The argument we used made use of the classification of surfaces of
Kodaira. But in fact it is possible
Appendix on Details of Inoue Surfaces
In the Kodaira classification of compact complex surfaces already discussed, a special position is
occupied by what are called the Class VII0 surfaces. They are (minimal) surfaces with first Betti
number b1= 1.
These
An Interlude on Curvature and Hermitian Yang Mills
As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have
already noted, these are nearly the same thing).
Suppose we wanted somehow to find a metric on S2, conformal to a giv
After the interlude:
Details of Minimization of Total Curvature and Hermitian Yang Mills
In the interlude on Hermitian Yang Mills ideas, it was remarked that an Hermitian Yang
Mills connection was obtained as the result of minimizing a curvature (squared)
Hermitian Yang Mills Metrics on Vector Bundles and Stability
Previously, we discussed the application, to Innoue surfaces in particular, of the
idea that a stable holomorphic vector bundle admits a Hermitian Yang Mills metric, or
what is often called a He
Lecture 5 -Part I
Professor S.T. Yau
April 17, 2007
One of the crucial differences between Khler and non- Khler geometry has to do with
the difference between a closed form of type (p,p) being exact in the usual sense of being
in the image of d versus the
Lecture 6 Part I
Professor Yau
Allowable singularities and Compactifications
To treat the class VII surfaces with b1=1 and b2 =0 but with curves on the surface, one needs to
think about extending the idea of sections of bundles to sections with specified
Summary of Profesor Yaus Lecture #7
Tuesday May 2,2007
Notes and supplementary remarks ( in [ ]s ) by Robert E. Greene
[Preliminary remarks on Levi forms and so on: Let M be a complex manifold with
Hermtian metric g i j in (z1,zn ) holomorphic coordinates
Twistor Spaces and Balanced Metrics on Complex Manifolds
Complex manifolds of complex dimension 1 (Riemann surfaces) are of course
always Khler, that is admit Khler metrics, on account of the obvious dimension
situation: d=0 simply because it is a 3-form!
Building Geometric Structures:
Summary of Prof. Yaus lecture, Monday, April 2 [with additional references and
remarks] (for people who missed the lecture)
A geometric structure on a manifold is a cover by coordinate systems [a sub-atlas] in
which the tran