COMPLEX GEOMETRY Course notes

Construction: Let X be an affine variety over K, and M a module over K[x]. Then
X = zeroes of {f 1 , . . . , f l } over C N .
We have
K[X] := C[z 1 , . . . , z N ]/ 〈f 1 , . . . , f l 〉
Then U ↦→ M ⊗ K[X] O X (U) is an O X -module and this correspondence preserves tensor product, exactness,
etc. Call this O X -module ˜M. In part, ˜M is quasi-coherent (and coherent if M is of finite type) and any
quasi-coherent O X -module is of this form.
Example 4.1.5 (Important). Ideal sheaf I Y
Y ↩→ X (i.e., Y is an algebraic subvariety).
↩→ O X corresponding to the subsheaf of O X vanishing on
We normally assume that the ground field K is algebraically closed. Then the Nullstellensatz tells us that
V (I) := sup(O X /I) ⊆ X is nonempty if and only if I ≠ O X . Hence the ringed space (V (I), O X /I) is
identified with the subscheme of X corresponding to I. A ringed space locally isomorphic to subschemes
of affine spaces is called al algebraic scheme.
Definition 4.1.9 (Associated fibre spaces). Let A be a quasi-coherent sheaf of a O X -algebra of finite type
(i.e., locally generated by finitely many sections as O X -algebras). We define a scheme S = Specm X A and
a morphism π : S −→ (X, O X ) as follows: Let X be affine. Then set
Specm X A := SpecmA(X) := {maximal ideals in A(X)}.
Recall that if R = A(X) and M is a maximal ideal, then R/M = K if K is algebraically closed.
Let f : S −→ K be a regular function. Then {f = 0} c = a basis of open sets, form the Zariski topology.
Let π be the dual to the K-algebra homomorphism K[X] −→ A(X). If D(f) = {x ∈ X / f(x) ≠ 0}
for f ∈ O X (X) then by the quasi-coherence of A, we have A(D(f)) = A(X) ⊗ K[X] K[D(f)]. So that
SpecmA(D(f)) = π −1 (D(f)).
Special case: Given a coherent sheaf of O X -modules F, let A = Sym OX
F. Then the associated fibre space
is called the vector fibre space, denoted by π : V(F) −→ X. Here, Sym means the symmetric product
⊕ k≥0 (Sym k F). Note that
V(F) x = (F x /M x F x ) ∨
Example 4.1.6 (for algebraic geometry).
(I) Definition of normal and tangent bundles (cones):
– The model for tangent vector space at a point is given by Specm(K[ɛ]/ɛ 2 ).
– The Zariski tangent space: Let x ∈ X be a point of al algebraic variety. Then T x X :=
(M x /M 2 x) ∨ .
– The normal bundle of Y ↩→ X (algebraic subvariety): Let I be the ideal sheaf of Y , then
I/I 2 = I ⊗ OX O Y . The normal vector bundle (or normal bundle) of Y in X is I/I . It is denoted
by N Y |X ( −→ π
Y ).
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