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Caner Koca

Publications and source records attributed to Caner Koca.

9 recordsLinked to original sources
arXiv2019

Hermitian metrics of constant Chern scalar curvature on ruled surfaces

It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature Kähler metric \cite{ACGT,G,M17}. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page--Bérard-Bergery's ansatz \cite{P78,B82}. We also construct the interesting case of Hermitian metrics of zero Chern scalar curvature on some ruled surfaces. Furthermore, we discuss the problem of the existence in a conformal class of critical metrics of the total Chern scalar curvature, studied by Gauduchon in \cite{G80,G84}.

math.DG↗
arXiv2019

Einstein Hermitian Metrics of Positive Sectional Curvature

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.

math.DG↗
arXiv2019

On the curvature of Einstein-Hermitian surfaces

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact 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 Kähler condition in Berger's theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.

math.DG↗
arXiv2018

Compact Complex Surfaces of Locally Conformally Flat Type

We show that if a compact complex surface admits a locally conformally flat metric, then it cannot contain a smooth rational curve of odd self-intersection. In particular, the surface has to be minimal. Then we give a list of possibilities of such surfaces.

math.DG↗
arXiv2016

Strongly Hermitian Einstein-Maxwell Solutions on Ruled Surfaces

This paper produces explicit strongly Hermitian Einstein-Maxwell solutions on the smooth compact $4$-manifolds that are $S^2$-bundles over compact Riemann surfaces of any genus. This generalizes the existence results by C. LeBrun in arXiv:1411.3992 and arXiv:1504.06669. Moreover, by calculating the (normalized) Einstein-Hilbert functional of our examples we generalize Theorem E of arXiv:1504.06669, which speaks to the abundance of Hermitian Einstein-Maxwell solutions on such manifolds. As a bonus, we exhibit certain pairs of strongly Hermitian Einstein-Maxwell solutions, first found in arXiv:1504.06669, on the first Hirzebruch surface in a form which clearly shows that they are conformal to a common Kähler metric. In particular, this yields a non-trivial example of non-uniqueness of positive constant scalar curvature metrics in a given conformal class.

math.DG↗
arXiv2015

Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature

We show that a compact complex surface which admits a conformally Kähler 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.

math.DG↗
arXiv2011

Extremal Kaehler Metrics and Bach-Merkulov Equations

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 Hirzebruch surfaces.

math.DG↗
arXiv2007

Irreducible Heegner Divisors in the Period Space of Enriques Surfaces

Reflection walls of certain primitive vectors in the anti-invariant sublattice of the K3 lattice define Heegner divisors in the period space of Enriques surfaces. We show that depending on the norm of these primitive vectors, these Heegner divisors are either irreducible or have two irreducible components. The two components are obtained as the walls orthogonal to primitive vectors of the same norm but of different type as ordinary or characteristic.

math.AG↗