Fall 2014

** Organizers: Ioana Suvaina, ****Rares
Rasdeaconu**

** Tuesdays,
3:10-4:00pm in SC 1308 (unless otherwise noted) **

**
Tuesday, September 16th
**

__Speaker:__** Ioana Suvaina,
Vanderbilt University**

__Title____:__** Yamabe invariant of
symplectic 4-manifolds of general type** **
**

__Abstract__**: **We compute the Yamabe
invariant for a class of symplectic 4-manifolds obtained by

taking the rational blow-down of Kahler surfaces. In
particular, for any point on the half-Noether

line we show that there is a minimal symplectic manifold with
known Yamabe invariant.

**Tuesday, September 23rd**

__Speaker:__** Caner Koca,
Vanderbilt University**

__Title__: **On Conformally Kahler
Surfaces**

__Abstract__: The famous Frankel Conjecture in complex
geometry, which was proved by Siu and Yau in the 1981,

asserts that the only compact complex n-manifold that admits a
Kahler metric of positive (bi)sectional curvature

is the complex projective n-space. In this talk, we prove this
conjecture in complex dimension 2 under weaker

hypotheses: Namely, a compact complex "surface", which admits
a "conformally Kahler" metric g of "positive

orthogonal holomorphic bisectional curvature" is biholomorphic
to the complex projective plane.

Our theorem also has a nice corollary: if, In
addition, g is an Einstein metric, then the biholomorphism can
be chosen

to be an isometry, via which g becomes a multiple of the
Fubini-Study metric. (Joint work with M. Kalafat.)

**Tuesday, September 30th**

__Speaker:__** Caner Koca,
Vanderbilt University**

__Title__: **Compact Complex Surfaces and
Locally Conformally Flat Metrics
**

Differential Geometry. Significant progress has been achieved in recent decades if one looks for Einstein metrics on

complex surfaces. Inspired by this, we are interested in the question of existence of another important family of

canonical metrics, called locally conformally flat metrics (LCF, for short), on compact complex surfaces.

Our first result in this direction reduces the question down to a list of well-known cases: If a compact complex surface

admits a LCF metric, then it cannot contain a smooth rational curve of odd self intersection. In particular, the surface

has to be minimal. We will also give a list of possibilities. Whether or not each possibility in our list is realized by an

example is an interesting open problem. (Joint work with M. Kalafat.)

**Tuesday,
October 7th**

__Speaker:__** Rares Rasdeaconu, Vanderbilt University**

__Title__:
**Balanced metrics on uniruled manifolds**

__Abstract__:
A uniruled manifold is a complex manifold which can be
covered by rational curves. In complex dimension two,

Yau characterized the class of uniruled manifolds in
differential geometric terms, by showing that a complex
surface is

uniruled if and only if it admits a Kahler metric of
positive total scalar curvature. We extend Yau's
characterization in

higher dimensions. (Joint work with I. Chiose and I.
Suvaina)

Old Seminar Web-Pages: Fall 2009, Fall
2010