Heier, Gordon2017-07-252017-07-25December 22016-12December 2Portions 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/654http://hdl.handle.net/10657/1937In 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.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Holomorphic sectional curvatureCurvatureFibrationNegative curvaturePositive curvatureHirzebruch surfaceIsotrivial family of curvesFamily of curvesProduct manifoldProduct metricCovering spaceSemi-definite curvatureWarped metric2017-07-25Thesisborn digital