Period Mappings with Applications to Symplectic Complex Spaces / by Tim Kirschner.

Format:

Book

Author/Creator:

Kirschner, Tim., Author.

Series:

Lecture Notes in Mathematics, 0075-8434 ; 2140

Language:

English

Subjects (All):

Geometry, Algebraic.

Functions of complex variables.

Categories (Mathematics).

Algebra, Homological.

Algebraic Geometry.

Several Complex Variables and Analytic Spaces.

Category Theory, Homological Algebra.

Local Subjects:

Algebraic Geometry.

Several Complex Variables and Analytic Spaces.

Category Theory, Homological Algebra.

Physical Description:

1 online resource (XVIII, 275 p.)

Edition:

1st ed. 2015.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2015.

Language Note:

English

Summary:

Extending Griffiths’ classical theory of period mappings for compact Kähler manifolds, this book develops and applies a theory of period mappings of “Hodge-de Rham type” for families of open complex manifolds. The text consists of three parts. The first part develops the theory. The second part investigates the degeneration behavior of the relative Frölicher spectral sequence associated to a submersive morphism of complex manifolds. The third part applies the preceding material to the study of irreducible symplectic complex spaces. The latter notion generalizes the idea of an irreducible symplectic manifold, dubbed an irreducible hyperkähler manifold in differential geometry, to possibly singular spaces. The three parts of the work are of independent interest, but intertwine nicely.

Contents:

Intro

Preface

Contents

Notation

Chapter 1 Period Mappings for Families of Complex Manifolds

1.1 Introduction

1.2 The Λp Construction

1.3 Locally Split Exact Triples and Their Extension Classes

1.4 Connecting Homomorphisms

1.5 Frameworks for the Gauß-Manin Connection

1.6 The Gauß-Manin Connection

1.7 Generalities on Period Mappings

1.8 Period Mappings of Hodge-de Rham Type

References

Chapter 2 Degeneration of the Frölicher Spectral Sequence

2.1 Problem Description

2.2 Coherence of Direct Image Sheaves

2.3 The Infinitesimal Lifting of the Degeneration

2.4 Comparison of Formal and Ordinary Direct Image Sheaves

2.5 Compactifiable Submersive Morphisms

Chapter 3 Symplectic Complex Spaces

3.1 Symplectic Structures on Complex Spaces

3.2 The Beauville-Bogomolov Form

3.3 Deformation Theory of Symplectic Complex Spaces

3.4 The Local Torelli Theorem

3.5 The Fujiki Relation

Appendix A Foundations and Conventions

A.1 Set Theory

A.2 Category Theory

A.3 Homological Algebra

A.4 Sheaves

A.5 Ringed Spaces

A.6 Multilinear Algebra

A.7 Complex Spaces

Appendix B Tools

B.1 Base Change Maps

B.2 Hodge Theory of Rational Singularities

Terminology.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-17521-1

OCLC:

921822865