Geometry Seminar

                                                                                                                      Vanderbilt University
                                                                                                                             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

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, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014