On Holomorphic Sectional Curvature and Fibrations

Abstract: The interaction between curvature and geometry of a manifold is a well known phenomenon in the area of differential geometry. For the purpose of this talk, we shall focus on a particular type of curvature: holomorphic sectional curvature. Suppose the geometry of a compact complex manifold is that of a fibration. Then it is possible to comment on the sign of the curvature of the manifold under certain conditions. Given any fibration, we can always talk about its two "directions'': base and fibers. The "conditions'' mentioned above, in order to comment on the curvature of the manifold, are on the signs of the curvatures of the base and fibers of the manifold. If the base and fibers of the manifold carry metrics of definite curvature signs, then we shall see that the same sign carry over to the curvature of the total space. However, in the case of semi-definite curvature on the base or fibers, we might not get the expected result.