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On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in 𝐂 n

Jean Erik Björk (1974)

Annales de l'institut Fourier

Let V be an algebraic variety in C n and when k 0 is an integer then Pol ( V , k ) denotes all holomorphic functions f ( z ) on V satisfying | f ( z ) | C f ( 1 + | z | ) k for all z V and some constant C f . We estimate the least integer ϵ ( V , k ) such that every f Pol ( V , k ) admits an extension from V into C n by a polynomial P ( z 1 , ... , z n ) , of degree k + ϵ ( V , k ) at most. In particular lim k > ϵ ( V , k ) is related to cohomology groups with coefficients in coherent analytic sheaves on V . The existence of the finite integer ϵ ( V , k ) is for example an easy consequence of Kodaira’s Vanishing Theorem.

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