On the Positive Holomorphic Sectional Curvature of Projectivized Vector Bundles over Compact Complex Manifolds

Date

2016-05

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Abstract

In complex geometry, there are few known examples of, and few known results about, manifolds with metrics of positive curvature. For instance, the geometry of fiber bundles and total spaces of fibrations over positively-curved complex manifolds is mysterious and not well-understood. In this dissertation, we study the existence of (pinched) metrics of positive curvature on a particular type of fiber bundle---namely metrics of positive holomorphic sectional curvature on projectivized vector bundles over compact complex manifolds. We first prove a general theorem for projectivized vector bundles, then we discuss a curvature pinching result for projectivized rank 2 vector bundles over complex projective space of dimension 1.

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Keywords

Holomorphic sectional curvature, Complex Manifolds

Citation

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.