Organizers: Ioana Suvaina, Rares
Rasdeaconu
Tuesdays,
3:10-4:00pm in SC 1308 (unless otherwise noted)
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
Abstract: The question of existence of Einstein
metrics on compact smooth 4-manifolds is a classical problem
in
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