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It is a really fast-paced graduate point advent to complicated algebraic geometry, from the fundamentals to the frontier of the topic. It covers sheaf concept, cohomology, a few Hodge idea, in addition to many of the extra algebraic facets of algebraic geometry. the writer usually refers the reader if the therapy of a undeniable subject is instantly on hand in other places yet is going into substantial element on subject matters for which his remedy places a twist or a extra obvious perspective. His situations of exploration and are selected very conscientiously and intentionally. The textbook achieves its objective of taking new scholars of complicated algebraic geometry via this a deep but wide creation to an unlimited topic, ultimately bringing them to the vanguard of the subject through a non-intimidating variety.

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**Additional info for Algebraic Geometry over the Complex Numbers (Universitext)**

**Example text**

Since X is compact, | f | attains a maximum somewhere, say at x0 ∈ X. The set S = f −1 ( f (x0 )) is closed by continuity. It is also open by the maximum principle. So S = X. 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.

Let π : Cn+1 − {0} → Pn be the natural projection that sends a vector to its span. In the sequel, we usually denote π (x0 , . . , xn ) by [x0 , . . , xn ]. Then Pn is given the quotient topology, which is deﬁned so that U ⊂ Pn is open if and only if π −1U is open. Deﬁne a function f : U → C to be holomorphic exactly when f ◦ π is holomorphic. 16), and the pair (Pn , OPn ) is a complex manifold. In fact, if we set Ui = {[x0 , . . , xn ] | xi = 0}, then the map [x0 , . . , xn ] → (x0 /xi , .

Therefore φ ( f ) depends only on the germ fx . Thus φ induces a map Px → M as required. All the examples of k-spaces encountered so far (C∞ -manifolds, complex manifolds, and algebraic varieties) satisfy the following additional property. 3. We will say that a concrete k-space (X, R) is locally ringed if 1/ f ∈ R(U) when f ∈ R(U) is nowhere zero. Recall that a ring R is local if it has a unique maximal ideal, say m. The quotient R/m is called the residue ﬁeld. We will often convey all this by referring to the triple (R, m, R/m) as a local ring.