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**Extra info for Algorithmic Geometry [auth. unkn.]**

**Example text**

P zj Z k − zkZ j = 0 where z 1 , . . , z n are coordinates in Bn and Z 1 , . . , Z p are homogeneous coordinates in Pp−1 . 3 Complex vector bundles and divisors We recall brieﬂy the deﬁnitions and the operations on vector bundles. 1. 13) where γab : Ua ∩ Ub → GL(d, C) is a C ∞ mapping. The mapping π is called projection, the mappings γab are called transition matrices and the mappings ϕa are called trivializations. Hence F is covered by the open sets π −1 (Ua ) Cd × Ua . We denote Fm = π −1 (m) the ﬁber 26 Classical Hodge theory of a point m ∈ M .

Then H ∩ Ua is described by the holomorphic equation n sa ≡ uk k=0 because Zk Za Zk =0 Za are the holomorphic coordinates in Ua . One has in Ua ∩ Ub sa = Zb sb Za b Z so that L(H) is a line bundle with transition functions ϕab = Z a . It is called the OPn (1)-bundle. It does not depend on the choice of H. Indeed, obviously Complex manifolds, vector bundles, diﬀerential forms 31 ÈÈ H ∼L H for any two hyperplanes because H − H is the divisor of the global k rational function uuk Z k. Z k The holomorphic sections of O(1) are the homogeneous polynomials of degree 1 in the Z k .

We can restrict L to Y , obtaining L|Y , and then extend L|Y by zero to X. We denote by LY such extension, so that LY |Y = L|Y and LY |X\Y = 0.

### Algorithmic Geometry [auth. unkn.]

by Mark

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