Hodge theory of stratified varieties Lecture notes for 18.157, Spring 1997

Richard Melrose

Author address: Department of Mathematics, MIT

E-mail address: rbm@math.mit.edu

Contents

Introduction 5

Chapter 1. Hodge theory of compact C^{∞} manifolds 7

Chapter 2. Compact manifolds with boundary 15 1. Boundaries 15 2. Metrics 16 3. Relative and absolute decompositions 18 4. Boundary values 21 5. Abstract Hodge Decomposition 23

Chapter 3. Cones, horns and cusps 29 1. Conic metrics 29 2. Metric Horns 37 3. Cusplike metrics 43

Chapter 4. Resolved spaces 47 1. Manifolds with corners 47 2. Blow-up 48 3. Total boundary blow-up 49 4. Iterated fibrations 52 5. Associated metrics 53

Bibliography 55

4 CONTENTS

Introduction

In these lectures I plan to describe the Hodge theory for a certain class of spaces which includes stratified varieties; these general spaces are sometimes called pseudomanifolds^{1}. Most of the new material here represents joint work with Rafe Mazzeo.

1The definition I take is a little wider than the usual one, so as to include manifolds with boundary and even corners.

6 INTRODUCTION