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This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a course given at the Institut de Mathematiques de Jussieu in 2004 and 2005. Aimed at postgraduate students with a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. Including notions from other areas, e.g. singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs are given and exercises aid the reader throughout.

Algebraic geometry --- Fourier transformations --- Transformations de Fourier --- Géométrie algébrique --- Fourier transformations. --- Géométrie algébrique --- Geometry, Algebraic --- Transformations, Fourier --- Transforms, Fourier --- Geometry --- Fourier analysis --- Transformations (Mathematics) --- Geometry, Algebraic.

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Geometry --- Functional analysis --- landmeetkunde --- functies (wiskunde)

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Cubic hypersurfaces are described by almost the simplest possible polynomial equations, yet their behaviour is rich enough to demonstrate many of the central challenges in algebraic geometry. With exercises and detailed references to the wider literature, this thorough text introduces cubic hypersurfaces and all the techniques needed to study them. The book starts by laying the foundations for the study of cubic hypersurfaces and of many other algebraic varieties, covering cohomology and Hodge theory of hypersurfaces, moduli spaces of those and Fano varieties of linear subspaces contained in hypersurfaces. The next three chapters examine the general machinery applied to cubic hypersurfaces of dimension two, three, and four. Finally, the author looks at cubic hypersurfaces from a categorical point of view and describes motivic features. Based on the author's lecture courses, this is an ideal text for graduate students as well as an invaluable reference for researchers in algebraic geometry.

Surfaces, Cubic. --- Hypersurfaces. --- Equations, Cubic. --- Geometry, Algebraic --- Superfícies cúbiques --- Hipersuperfícies --- Geometria algebraica --- Geometry, Algebraic.

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Now back in print, this highly regarded book has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces, which include moduli spaces in positive characteristic, moduli spaces of principal bundles and of complexes, Hilbert schemes of points on surfaces, derived categories of coherent sheaves, and moduli spaces of sheaves on Calabi-Yau threefolds. The authors review changes in the field since the publication of the original edition in 1997 and point the reader towards further literature. References have been brought up to date and errors removed. Developed from the authors' lectures, this book is ideal as a text for graduate students as well as a valuable resource for any mathematician with a background in algebraic geometry who wants to learn more about Grothendieck's approach.

Moduli theory --- Sheaf theory --- Surfaces, Algebraic --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Algebraic surfaces --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Sheaf theory. --- Moduli theory. --- Surfaces, Algebraic.

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Calabi-Yau manifolds --- Mathematical physics