# Algebraic Geometry over the Complex Numbers (Universitext) by Donu Arapura

By Donu Arapura

This can be a particularly fast moving graduate point advent to advanced algebraic geometry, from the fundamentals to the frontier of the topic. It covers sheaf conception, cohomology, a few Hodge conception, in addition to many of the extra algebraic features of algebraic geometry. the writer usually refers the reader if the therapy of a undeniable subject is quickly on hand somewhere else yet is going into massive element on themes for which his therapy places a twist or a extra obvious point of view. His situations of exploration and are selected very conscientiously and intentionally. The textbook achieves its goal of taking new scholars of advanced algebraic geometry via this a deep but vast advent to an enormous topic, finally bringing them to the vanguard of the subject through a non-intimidating variety.

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Xn ]. Deﬁne a function f : U → k to be regular exactly when f ◦ π is regular. Such a function can be represented as the ratio f ◦ π (x0 , . . , xn ) = p(x0 , . . , xn ) q(x0 , . . , xn ) of two homogeneous polynomials of the same degree such that q has no zeros on π −1U. 13. The Zariski topology is never actually Hausdorff except in the most trivial cases. However, there is a good substitute called the separation axiom. 3. Let X be a topological space. Then the following statements are equivalent: (a) X is Hausdorff.

6 1-Forms, Vector Fields, and Bundles 43 g j ∈ C∞ (U), deﬁnes a vector ﬁeld on U ⊆ X precisely when (p; g1 (p), . . , gN (p)) ∈ TX for a p ∈ U. In other words, vector ﬁelds are sections of TX . It remains to ﬁnd a local trivialization. We can ﬁnd an open cover {Ui } of X by (i) (i) coordinate neighborhoods. Choose local coordinates x1 , . . , xn in each Ui . Then the map ⎛ ⎛ ⎞ ⎞ (p; w) → ⎝ p; ⎝ identiﬁes Ui × R with π n ∂yj ⎠ w⎠ (i) ∂ xk p −1 Ui . Tangent bundles also exist for complex manifolds and nonsingular algebraic varieties.

Thus F ∈ OX (U). An afﬁne algebraic variety is an irreducible subset of some Ank . We give X the topology induced from the Zariski topology of afﬁne space. This is called the Zariski topology of X. Suppose that X ⊂ Ank is an algebraic variety. Given an open set 30 2 Manifolds and Varieties via Sheaves U ⊂ X, a function F : U → k is regular if it is locally extendible to a regular function on an open set of An as deﬁned above, that is, if every point of U has an open neighborhood V ⊂ Ank with a regular function G : V → k for which F = G|V ∩U .