By Robert Friedman

A singular function of the e-book is its built-in method of algebraic floor idea and the examine of vector package conception on either curves and surfaces. whereas the 2 topics stay separate in the course of the first few chapters, they develop into even more tightly interconnected because the booklet progresses. hence vector bundles over curves are studied to appreciate governed surfaces, after which reappear within the facts of Bogomolov's inequality for good bundles, that's itself utilized to review canonical embeddings of surfaces through Reider's procedure. equally, governed and elliptic surfaces are mentioned intimately, prior to the geometry of vector bundles over such surfaces is analysed. the various effects on vector bundles look for the 1st time in ebook shape, subsidized by way of many examples, either one of surfaces and vector bundles, and over a hundred workouts forming a vital part of the textual content. aimed toward graduates with an intensive first-year direction in algebraic geometry, in addition to extra complicated scholars and researchers within the parts of algebraic geometry, gauge conception, or 4-manifold topology, the various effects on vector bundles can be of curiosity to physicists learning string conception.

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**Sample text**

The algebraic curves whose Riemann surface has genus 0 may therefore be parametrized birationally using rational functions. For this reason, they are called rational curves. 3 Let T be a compact Riemann surface of genus p 0 with n distinct marked points. Then, the dimension of the complex vector space of meromorphic functions on T with simple poles, all of them contained among the marked points, is equal to the sum of the expression 1 p C n and of the dimension of the complex vector space of holomorphic 1-forms on T which vanish at the marked points.

There, one may discover the essential influence of Gauss on the first directions of research in this domain. For instance (see also [153]), it was Listing, one of Gauss’ students, who invented the term “topology”, which was to replace in the twentieth century that of “analysis situs”, proposed by Leibniz. Chapter 15 Riemann and the Birational Invariance of Genus As explained in the previous chapter, although Riemann developed a very topological vision of surfaces, he did not forget that his fundamental examples came from algebraic functions and their integrals.

Note that this “Riemann–Roch theorem”, which also holds when one interprets “Riemann surface” in the abstract sense of a smooth compact holomorphic curve (see Chap. 23), shows that there exist plenty of non-constant meromorphic functions on them. This implies with a little more work that any smooth compact holomorphic curve embeds in a projective space. On the other hand, starting from complex dimension 2, there exist smooth compact holomorphic manifolds which cannot be embedded in any projective space (one says in this case that they are not projective).