# Download Algebraic Geometry over the Complex Numbers by Donu Arapura PDF

By Donu Arapura

ISBN-10: 1461418097

ISBN-13: 9781461418092

This is a comparatively fast moving graduate point creation to complicated algebraic geometry, from the fundamentals to the frontier of the topic. It covers sheaf idea, cohomology, a few Hodge idea, in addition to a number of the extra algebraic features of algebraic geometry. the writer usually refers the reader if the therapy of a undeniable subject is instantly to be had in different places yet is going into substantial element on subject matters for which his remedy places a twist or a extra obvious standpoint. His situations of exploration and are selected very rigorously and intentionally. The textbook achieves its function of taking new scholars of complicated algebraic geometry via this a deep but huge creation to an unlimited topic, ultimately bringing them to the leading edge of the subject through a non-intimidating style.

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**Example text**

A similar deﬁnition holds for complex manifolds. 15). This means that f ∈ CY∞ (U) if every point of U possesses a neighborhood V ⊂ X such f |V ∩Y = f˜|V ∩Y for some f˜ ∈ C∞ (V ). For a complex submanifold Y ⊂ X, we deﬁne OY to be the sheaf of functions that locally extend to holomorphic functions. 12. If Y ⊂ X is a closed submanifold of a C∞ (respectively complex) manifold, then (Y,CY∞ ) (respectively (Y, OY )) is also a C∞ (respectively complex) manifold. 2 Manifolds 27 Proof. We treat the C∞ case; the holomorphic case is similar.

3 for algebraic varieties over arbitrary ﬁelds. We ﬁrst need a good substitute for compactness. 4. If X is a compact metric space, then for any metric space Y , the projection p : X × Y → Y is closed. Proof. Given a closed set Z ⊂ X ×Y and a sequence yi ∈ p(Z) converging to y ∈ Y , we have to show that y lies in p(Z). By assumption, we have a sequence xi ∈ X such that (xi , yi ) ∈ Z. Since X is compact, we can assume that xi converges to say x ∈ X after passing to a subsequence. Then we see that (x, y) is the limit of (xi , yi ), so it must lie in Z because it is closed.

In analogy with manifolds, we can deﬁne an (abstract) algebraic variety as a k-space that is locally isomorphic to an afﬁne variety and that satisﬁes some version of the Hausdorff condition. It will be convenient to ignore this last condition for the moment. The resulting objects are dubbed prevarieties. 1. A prevariety over k is a k-space (X , OX ) such that X is connected and there exists a ﬁnite open cover {Ui }, called an afﬁne open cover, such that each (Ui , OX |Ui ) is isomorphic, as a k-space, to an afﬁne variety.