By Donu Arapura

It is a rather fast-paced graduate point advent to complicated algebraic geometry, from the fundamentals to the frontier of the topic. It covers sheaf conception, cohomology, a few Hodge conception, in addition to a number of the extra algebraic facets of algebraic geometry. the writer usually refers the reader if the remedy of a definite subject is instantly on hand in other places yet is going into enormous element on themes for which his remedy places a twist or a extra obvious point of view. His circumstances of exploration and are selected very rigorously and intentionally. The textbook achieves its goal of taking new scholars of advanced algebraic geometry via this a deep but wide advent to an unlimited topic, finally bringing them to the vanguard of the subject through a non-intimidating kind.

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**Extra resources for Algebraic Geometry over the Complex Numbers (Universitext)**

**Example text**

3. A holomorphic function is constant on a nonsingular complex projective variety. Proof. PnC with its classical topology is compact, since the unit sphere in Cn+1 maps onto it. Therefore any submanifold of it is also compact. 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).

A rank-1 vector bundle will also be called a line bundle. The product X × kn is an example of a vector bundle, called the trivial bundle of rank n. A simple nontrivial example to keep in mind is the M¨obius strip, which is a real line bundle over the circle. The datum {(Ui , φi )} is called a local trivialization. Given a C∞ real vector bundle π : V → X, deﬁne the presheaf of sections V (U) = {s : U → π −1U | s is C∞ , π ◦ s = idU }. This is easily seen to be a sheaf. When V = X × Rn is the trivial vector bundle, a section is given by (x, f (x)), where f : X → Rn , so that V (X) is isomorphic to the space of vector-valued C∞ functions on X.

Xn ) be the maximal ideal of the origin. Then there is a one-to-one correspondence as shown in the table below: Algebra homogeneous radical ideals in S other than S+ homogeneous prime ideals in S other than S+ Geometry algebraic subsets of Pn algebraic subvarieties of Pn Given a subvariety X ⊆ Pnk , the elements of OX (U) are functions f : U → k such that f ◦ π is regular. Such a function can be represented locally as the ratio of two homogeneous polynomials of the same degree. 13. This is inherited by subvarieties, and is also called the classical topology.