Algebra and geometry in several complex variables by Palamodov V.

By Palamodov V.

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Fix a point b ∈ B\∆ and consider the fundamental group π1 (B\∆) of loops through b. 2). , αm (z ) along this loop. , αm . , αm . , m] . 4) called the monodromy; Sm is the group of substitutions. e. , m] invariant with respect of the action of π1 (B\ ) . , αm for z = b and arbitrary root β of the polynomial P (z1 , z ) for any z ∈ B. The continuation of α1 defines a curve in the set 7 Z\P −1 (∆) where p : D × B → B is the projection. This set is connected, since we can reach any point (β, z ) by a curve from (α1 , b) .

Of order < m. The equation φ (a) = 0 is equivalent to p˜ a1 + h2 a2 + ... + ht at = 0 in a nbd of w where a ˜1 = a1 + g2 a2 + ... + gt at . Apply Weierstrass Lemma again and get aj = cj p + bj where bj is a pp. of order < m. Rewrite the syzygy relation in the form . pb1 = h = −h2 b2 − .. − ht bt , b1 = a ˜1 + c2 h2 + ... 2) The right side is a pp. of order ≤ 2m − 2. The function b1 is a pp. of order ≤ m − 2. Indeed, we can blow up the circle ∂D in the formula b1 (z1 , z ) = 1 2πı ∂D h (λ, z ) dλ p (λ, z ) (λ − z1 ) Therefore it is easy to see that b1 = O |z1 |m−2 for large |z1 | .

Definition. , zn } is the algebra of convergent power series, and I is an ideal in this algebra. e. there is only one maximal ideal m = m (A) . It is equal to the image of the maximal ideal m (O n ) under the surjection p : O n → A. The canonical mapping A → A/m (A) ∼ = C is called the residue morphism. The number n is not an invariant of the algebra A; for example the field C admits the surjection O n → C for arbitrary n. The minimal number n is called embedding dimension emdim A of the analytic algebra A.

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