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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. 19346 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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About the André-Oort Conjecture

Umberto Zannier.

in Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)

March 2012; p ublished online October 2017 .

Chapter. Subjects: Geometry. 25946 words.

This chapter discusses some aspects of the so-called André–Oort conjecture. It observes how this resembles the pattern of the statements discussed so far, recalling also a brief history....

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The action functional and the harmonic oscillator

Jean-Michel Bismut.

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

August 2011; p ublished online October 2017 .

Chapter. Subjects: Geometry. 5707 words.

This chapter solves explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean...

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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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Algebraic Constructions of Minimal Compactifications

Kai-Wen Lan.

in Arithmetic Compactifications of PEL-Type Shimura Varieties

March 2013; p ublished online October 2017 .

Chapter. Subjects: Geometry. 23692 words.

This chapter first studies the automorphic forms that are defined as global sections of certain invertible sheaves on the toroidal compactifications. The local structures of toroidal...

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Algebraic Constructions of Toroidal Compactifications

Kai-Wen Lan.

in Arithmetic Compactifications of PEL-Type Shimura Varieties

March 2013; p ublished online October 2017 .

Chapter. Subjects: Geometry. 46005 words.

This chapter explains the algebraic construction of toroidal compactifications. For this purpose the chapter utilizes the theory of toroidal embeddings for torsors under groups of...

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Algebraic Surfaces and the Miyaoka-Yau Inequality

Paula Tretkoff.

in Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

February 2016; p ublished online October 2017 .

Chapter. Subjects: Geometry. 9429 words.

This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is...

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Almost complex structures

Dusa McDuff and Dietmar Salamon.

in Introduction to Symplectic Topology

March 2017; p ublished online June 2017 .

Chapter. Subjects: Mathematics; Geometry. 20797 words.

The chapter begins with a general discussion of almost complex structures on symplectic manifolds and then addresses the problem of integrability. Subsequent sections discuss a variety of...

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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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