[Show abstract][Hide abstract] ABSTRACT: We show that a compact complex surface which admits a conformally K\"ahler
metric g of positive orthogonal holomorphic bisectional curvature is
biholomorphic to the complex projective plane. In addition, if g is a Hermitian
metric which is Einstein, then the biholomorphism can be chosen to be an
isometry via which g becomes a multiple of the Fubini-Study metric.
[Show abstract][Hide abstract] ABSTRACT: We show that if a compact complex surface admits a locally conformally flat
metric, then it cannot contain a 2-sphere of non-zero self intersection. In
particular, the surface has to be minimal. Then we give a list of
Houston journal of mathematics 05/2013; · 0.43 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that a compact complex surface together with an Einstein-Hermitian
metric of positive orthogonal bisectional curvature is biholomorphically
isometric to the complex projective plane with its Fubini-Study metric up to
rescaling. This result relaxes the K\"ahler condition in Berger's theorem, and
the positivity condition on sectional curvature in a theorem proved by Koca.
The techniques used in the proof are completely different from theirs.
[Show abstract][Hide abstract] ABSTRACT: In this paper we will prove that the only compact 4-manifold M with an
Einstein metric of positive sectional curvature which is also hermitian with
respect to some complex structure on M, is the complex projective plane CP^2,
with its Fubini-Study metric.
Proceedings of the American Mathematical Society 12/2011; DOI:10.1090/S0002-9939-2014-11929-0 · 0.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study a coupled system of equations on oriented compact
4-manifolds which we call the Bach-Merkulov equations. These equations can be
thought of as the conformally invariant version of the classical
Einstein-Maxwell equations in general relativity. Inspired by the work of C.
LeBrun on Einstein-Maxwell equations on compact Kaehler surfaces, we give a
variational characterization of solutions to Bach-Merkulov equations as
critical points of the Weyl functional. We also show that extremal Kaehler
metrics are solutions to these equations, although, contrary to the
Einstein-Maxwell analogue, they are not necessarily minimizers of the Weyl
functional. We illustrate this phenomenon by studying the Calabi action on
Journal of Geometry and Physics 12/2011; 70. DOI:10.1016/j.geomphys.2013.03.025 · 0.80 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This report is aimed to be a user's guide to Morrison's exquisite article, On K3 surfaces with large Picard number, published in Inventiones Mathematicae in 1984.