Vortex equation in holomorphic line bundle over complex manifold

On the Canonical Line Bundle and Negative Holomorphic Sectional Curvature

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differen...

Holomorphic Approximation on Real Submanifolds of a Complex Manifold

Let X be a complex manifold, let 0 be the sheaf of germs of holomorphic functions on X, and let T(K, 0) denote the ring of sections over KQX (holomorphic functions on K, see [5] as a general reference for the terminology used here). If K is a compact set in X, let C(K) be the Banach algebra of continuous complex-valued functions on K with respect to the maximum norm on K. Let A (K) denote the c...

On holomorphic fields of complex line elements with isolated singularities

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The Drinfeld Sokolov Holomorphic Bundle and Classical

Kähler Geometry

De nition 3.1. Let X be a complex manifold. A holomorphic vector bundle of rank k over E is a complex manifold E and a holomorphic map : E ! X such that makes E ! X into a complex vector bundle of rank k, and E admits holomorphic trivializations, i.e. there’s an open cover U of X trivializing E such that for each U 2 U, there’s a biholomorphic map ' : EjU ! U C commuting with projection to U th...