By Robert Friedman

This ebook covers the speculation of algebraic surfaces and holomorphic vector bundles in an built-in demeanour. it truly is aimed toward graduate scholars who've had an intensive first-year path in algebraic geometry (at the extent of Hartshorne's Algebraic Geometry), in addition to extra complex graduate scholars and researchers within the components of algebraic geometry, gauge idea, or 4-manifold topology. some of the effects on vector bundles also needs to be of curiosity to physicists learning string thought. a unique function of the ebook is its built-in method of algebraic floor concept and the research of vector package deal concept on either curves and surfaces. whereas the 2 topics stay separate during the first few chapters, and are studied in trade chapters, they turn into even more tightly interconnected because the publication progresses. therefore vector bundles over curves are studied to appreciate governed surfaces, after which reappear within the evidence of Bogomolov's inequality for reliable bundles, that's itself utilized to check canonical embeddings of surfaces through Reider's approach. equally, governed and elliptic surfaces are mentioned intimately, after which the geometry of vector bundles over such surfaces is analyzed. a number of the effects on vector bundles look for the 1st time in e-book shape, compatible for graduate scholars. The e-book additionally has a robust emphasis on examples, either one of surfaces and vector bundles. There are over a hundred routines which shape a vital part of the textual content.

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**Extra info for Algebraic Surfaces and Holomorphic Vector Bundles**

**Example text**

13 Digression: homeomorphism versus diffeomorphism. These constructions relate to two very profound questions in differentiable topology. The ﬁrst question is: given a topological manifold, does it admit a smooth structure? Second: if it admits a smooth structure, is it then unique up to diffeomorphism? The second question was answered negatively by Milnor [35] (1957) who found an example of a smooth 7-manifold (called an exotic sphere) which is homeomorphic to the usual 7-sphere but not diffeomorphic to it.

So here comes the ﬁnal, ofﬁcial deﬁnition of a cobordism: Let 0 and dism from 0 to maps 1 be closed oriented (n − 1)-manifolds. 2 Cobordisms 23 such that 0 maps diffeomorphically (preserving orientation) onto the inboundary of M, and 1 maps diffeomorphically (preserving orientation) onto the out-boundary of M. Note that this is practically the same as the ﬁrst deﬁnition we gave, but it gives more ﬂexibility. We can now have a a cobordism from to , provided we can ﬁnd two diffeomorphisms: one from to the in-boundary, and another from to the out-boundary.

The precise categorical statement is that M0 pushout, or the colimit of M0 ← → M1 (cf. 7 in the Appendix). M1 → In the situation of the arbitrary diagram, the unique map f : M0 X can be thought of as the gluing of the two maps f0 and f1 . So the statement can also be formulated like this: continuous maps glue if they agree on the border between their domains. e. topological spaces equipped with atlases). 2 Gluing topological manifolds. Suppose now M0 and M1 are topological manifolds with a common boundary component .

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