Hodge Theory and Complex Algebraic Geometry II

Abstract

Introduction page I Preliminaries 1 Holomorphic Functions of Many Variables 1.1 Holomorphic functions of one variable 1.1.1 Definition and basic properties 1.1.2 Background on Stokes' formula 1.1.3 Cauchy's formula 1.2 Holomorphic functions of several variables 1.2.1 Cauchy's formula and analyticity 1.2.2 Applications of Cauchy's formula 1.3 The equation ∂g ∂z = f Exercises 2 Complex Manifolds 2.1 Manifolds and vector bundles 2.1.1 Definitions 2.1.2 The tangent bundle 2.1.3 Complex manifolds 2.2 Integrability of almost complex structures 2.2.1 Tangent bundle of a complex manifold 2.2.2 The Frobenius theorem 2.2.3 The Newlander-Nirenberg theorem 2.3 The operators ∂ and ∂ 2.3.1 Definition 2.3.2 Local exactness 2.3.3 Dolbeault complex of a holomorphic bundle 2.4 Examples of complex manifolds Exercises v vi Contents 3 Kähler Metrics 3.1 Definition and basic properties 3.1.1 Hermitian geometry 3.1.2 Hermitian and Kähler metrics 3.1.3 Basic properties 3.2 Characterisations of Kähler metrics 3.2.1 Background on connections 3.2.2 Kähler metrics and connections 3.3 Examples of Kähler manifolds 3.3.1 Chern form of line bundles 3.

On tangential deformations of homogeneous polynomials

“…Then S is an irreducible hypersurface, see for instance [10], Section 2.1. Write a general f ∈ P(S n,d ) as…”

Laplace operators on Sasaki-Einstein manifolds

“…Since the proof is based on the equivalent considerations in the Kähler case, our discussion will follow [1,2] very closely.…”

“…2 Furthermore, there is a two-form J = 1 2 dη with i ξ J = 0. J defines an endomorphism on T S which satisfies…”

The decomposition theorem, perverse sheaves and the topology of algebraic maps

“…Recall that we use cohomology with rational coefficients. A standard textbook reference for what follows is [92]; see also [175,44].…”