BEAUVILLE COMPLEX ALGEBRAIC SURFACES PDF

We will begin by introducing the theory of algebraic surfaces; reviewing intersection theory on surfaces, the Riemann-Roch theorem and Picard group. We then move toward understanding the classification, via a number of landmark results. We then give a survey of the principal classes of surfaces which appear in the classification. We will assume some familiarity with the basic notions in algebraic geometry; although the Picard group, amplitude, and intersection multiplicity of curves will all be covered though quite briefly. A basic familiarity with complex manifolds will be assumed. Some tolerance for the language of schemes has advantages, but is not required.

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We will begin by introducing the theory of algebraic surfaces; reviewing intersection theory on surfaces, the Riemann-Roch theorem and Picard group. We then move toward understanding the classification, via a number of landmark results. We then give a survey of the principal classes of surfaces which appear in the classification.

We will assume some familiarity with the basic notions in algebraic geometry; although the Picard group, amplitude, and intersection multiplicity of curves will all be covered though quite briefly. A basic familiarity with complex manifolds will be assumed. Some tolerance for the language of schemes has advantages, but is not required. Lecture Notes Notes will be added here throughout the course.

Lecture 1 Foundations: Schemes vs. Complex manifolds, GAGA. Invariants of complex surfaces, Hodge diamond. Curves and Divisors on surfaces. Lecture 2 Amplitude. First examples. Lecture 3 Anatomy of birational maps. Ruled surfaces. Noether-Enriques theorem. Lecture 5 Stein factorization, Kodaira dimension zero surfaces.

Lecture 7 K3 surfaces and Torelli theorems. Lecture 8 Higher Kodiara dimension. Examples of surfaces of general type. References Main references for the course.

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Beauville surface

This result inaugurated a major effort to classify all possible smooth and topological structures on manifolds of dimension at least 5. As an example of such a qualitative result, a closed, simply connected manifold of dimension 2: 5 is determined up to finitely many diffeomorphism possibilities by its homotopy type and its Pontrjagin classes. There are similar results for self-diffeomorphisms, which, at least in the simply connected case, say that the group of self-diffeomorphisms of a closed manifold M of dimension at least 5 is commensurate with an arithmetic subgroup of the linear algebraic group of all automorphisms of its so-called rational minimal model which preserve the Pontrjagin classes []. Once the high dimensional theory was in good shape, attention shifted to the remaining, and seemingly exceptional, dimensions 3 and 4. The theory behind the results for manifolds of dimension at least 5 does not carryover to manifolds of these low dimensions, essentially because there is no longer enough room to maneuver.

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Other books in this series. Complex surfaces; Appendix C. Volume 1 Ibrahim Assem. The classification of algebraic surfaces is an intricate and fascinating branch of mathematics, developed over more than a century and still an active area of research today. Account Options Sign in. In beauvile book, Professor Beauville gives a lucid and concise account of the subject, expressed simply in the language of modern Check out the top books of the year on our page Best Books of Enriques, but expressed simply in the language of modern topology and sheaf theory, so as to be accessible to any budding geometer.

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