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ABELIAN VARIETIES

D. Huybrechts.

in Fourier-Mukai Transforms in Algebraic Geometry

April 2006; p ublished online September 2007 .

Chapter. Subjects: Geometry. 18844 words.

Historically, Mukai's equivalence with the Poincare bundle on the product of an abelian variety and its dual as kernel was the fist Fourier-Mukai transform. The first section in this...

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Algebra of vector bundles

Clifford Henry Taubes.

in Differential Geometry

October 2011; p ublished online December 2013 .

Chapter. Subjects: Geometry. 4008 words.

Any linear operation that can be performed to generate a new vector space from a given set of initial vector spaces can be done fibre-wise with an analogous set of vector bundles to...

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Algebraic and Geometric Surgery

Andrew Ranicki.

September 2002; p ublished online September 2007 .

Book. Subjects: Geometry. 386 pages.

This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been...

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Almost Contact and Contact Geometry

Charles P. Boyer and Krzysztof Galicki.

in Sasakian Geometry

October 2007; p ublished online January 2008 .

Chapter. Subjects: Geometry. 17159 words.

This chapter presents the necessary foundational material on almost contact, contact, and metric contact structures. The key result is an orbifold version of the famous Boothby-Wang...

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Applications of Stochastic Geometry in Image Analysis

Marie‐Colette N.M. van Lieshout.

in New Perspectives in Stochastic Geometry

November 2009; p ublished online February 2010 .

Chapter. Subjects: Geometry. 11152 words.

A discussion is given of various stochastic geometry models (random fields, sequential object processes, polygonal field models) which can be used in intermediate‐ and high‐level image...

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Applications of the Euler characteristic

Simon Donaldson.

in Riemann Surfaces

March 2011; p ublished online December 2013 .

Chapter. Subjects: Geometry. 4464 words.

The genus of a compact oriented smooth surface S can be defined as one-half the dimension of the de Rham cohomology group H 1(S). Another way of defining the genus involves the...

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Basic definitions

Simon Donaldson.

in Riemann Surfaces

March 2011; p ublished online December 2013 .

Chapter. Subjects: Geometry. 6116 words.

This chapter presents definitions of Riemann surfaces and holomorphic maps, and provides examples, including algebraic curves and quotients.

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Branes on Poisson Varieties

Marco Gualtieri.

in The Many Facets of Geometry

July 2010; p ublished online September 2010 .

Chapter. Subjects: Geometry. 13278 words.

This chapter takes a second look at a classical structure in differential and algebraic geometry — that of a holomorphic Poisson structure — which is a complex manifold with a holomorphic...

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Building low-dimensional manifolds

Selman Akbulut.

in 4-Manifolds

August 2016; p ublished online November 2016 .

Chapter. Subjects: Geometry. 4742 words.

Building 4-dimensional handlebodies

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BUNDLES

Andrew Ranicki.

in Algebraic and Geometric Surgery

September 2002; p ublished online September 2007 .

Chapter. Subjects: Geometry. 17231 words.

This chapter brings together the basic properties of fibre bundles, fibrations, and vector bundles required in surgery theory. The tangent and normal bundles of a manifold are introduced....

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