no. 4
On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in 𝐂 n
Annales de l'Institut Fourier, Volume 24 (1974) no. 4, pp. 157-165.

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.

Soit V une variété algébrique dans C n  ; pour un nombre entier k nous désignerons par Pol (V,k) toutes les fonctions holomorphes f(z) sur V qui satisfont à |f(z)|≤C f (1+|z|) k pour chaque point z∈V, où C f est une constante. Nous estimons le plus petit nombre entier ε(V,k) tel que toute fonction f∈ Pol (V,k) admet un prolongement de V dans C n par un polynôme P(z 1 ,...,z n ), où le degré de P est majoré par k+ε(V,k). En particulier lim k>∞ ε(V,k) est lié à certains groupes de cohomologie à coefficients dans certains faisceaux analytiques cohérents sur V. L’existence de ε(V,k) est par exemple une conséquence facile du “Vanishing Theorem” de Kodaira.

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     author = {Bj\"ork, Jean Erik},
     title = {On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in ${\bf C}^n$},
     journal = {Annales de l'Institut Fourier},
     pages = {157--165},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {24},
     number = {4},
     year = {1974},
     doi = {10.5802/aif.535},
     zbl = {0288.32014},
     mrnumber = {51 #928},
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     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.535/}
}
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Björk, Jean Erik. On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in ${\bf C}^n$. Annales de l'Institut Fourier, Volume 24 (1974) no. 4, pp. 157-165. doi : 10.5802/aif.535. https://aif.centre-mersenne.org/articles/10.5802/aif.535/

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