On Holomorphic Sectional Curvature and Fibrations



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In this dissertation, we prove the existence of a metric of definite holomorphic sectional curvature on certain compact fibrations. The basic idea for these curvature computations is to use the already available information on the signs of the holomorphic sectional curvatures along the base and the fibers of the fibration, and construct an appropriate warped metric on the total space. For a few specific fibrations, like Hirzebruch surfaces, isotrivial families of curves, and product manifolds, we shall also comment on the pinching constants of the holomorphic sectional curvatures. All these results are either in the case of strictly positive holomorphic sectional curvature, or in the case of strictly negative holomorphic sectional curvature. At the end of this dissertation, we give a few examples to show that the sign of the holomorphic sectional curvature of a fibration might not be what we would expect in the cases where the base or the fibers have semi-definite holomorphic sectional curvatures.



Holomorphic sectional curvature, Curvature, Fibration, Negative curvature, Positive curvature, Hirzebruch surface, Isotrivial family of curves, Family of curves, Product manifold, Product metric, Covering space, Semi-definite curvature, Warped metric


Portions of this document appear in: Alvarez, Angelynn, Ananya Chaturvedi, and Gordon Heier. "Optimal pinching for the holomorphic sectional curvature of Hitchin's metrics on Hirzebruch surfaces." Contemp. Math 654 (2015): 133-142. Published version contained in DOI: http://dx.doi.org/10.1090/conm/654