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Convex Analysisの二,三の進展について

Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

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The geometry of moduli spaces of sheaves

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. 
We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. 
Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.

Generalized complex geometry

心臓病診療プラクティス 16. 心不全を癒す

Generalized complex geometry encompasses complex and symplectic ge- ometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deforma- tion theory, relation to Poisson geometry, and local structure theory. We also dene and study generalized complex branes, which interpolate be- tween at bundles on Lagrangian submanifolds and holomorphic bundles on complex submanifolds.

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Preface 1. Triangulated categories 2. Derived categories: a quick tour 3. Derived categories of coherent sheaves 4. Derived category and canonical bundle I 5. Fourier-Mukai transforms 6. Derived category and canonical bundle II 7. Equivalence criteria for Fourier-Mukai transforms 8. Spherical and exceptional objects 9. Abelian varieties 10. K3 surfaces 11. Flips and flops 12. Derived categories of surfaces 13. Where to go from here References Index

Fourier-Mukai transforms in algebraic geometry

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

Local Theory.- Complex Manifolds.- Kahler Manifolds.- Vector Bundles.- Applications of Cohomology.- Deformations of Complex Structures.

Daniel Huybrechts

Papers

The geometry of moduli spaces of sheaves

Fourier-Mukai transforms in algebraic geometry

Lectures on K3 Surfaces

Complex Geometry: An Introduction

Calabi-Yau Manifolds and Related Geometries