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 k0 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 zV 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 zV, 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.

@article{AIF_1974__24_4_157_0,
     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},
     mrnumber = {51 #928},
     zbl = {0288.32014},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.535/}
}
TY  - JOUR
AU  - Björk, Jean Erik
TI  - On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in ${\bf C}^n$
JO  - Annales de l'Institut Fourier
PY  - 1974
SP  - 157
EP  - 165
VL  - 24
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.535/
DO  - 10.5802/aif.535
LA  - en
ID  - AIF_1974__24_4_157_0
ER  - 
%0 Journal Article
%A Björk, Jean Erik
%T On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in ${\bf C}^n$
%J Annales de l'Institut Fourier
%D 1974
%P 157-165
%V 24
%N 4
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.535/
%R 10.5802/aif.535
%G en
%F AIF_1974__24_4_157_0
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. http://archive.numdam.org/articles/10.5802/aif.535/

[1] P. A. Griffiths, Function theory of finite order on algebraic varieties, Journ. of Diff. Geom., 6 (1972), 285-306. | MR | Zbl

[2] L. Ehrenpreis, A fundamental principle for systems of linear differential equations with constant coefficients and some of its applications, Proc. International Symp. on Linear Spaces, Jerusalem (1960). | Zbl

[3] L. Ehrenpreis, Fourier analysis in several complex variables, Pure and Appl. Math., 17 (1970), Wiley-Intersci. Publ. | MR | Zbl

[4] L. Hörmander, L2-estimates and existence theorems for the <ATT>-operator, Acta Math., 113 (1965), 89-152. | Zbl

[5] R. Narasimhan, Cohomology with bounds on complex spaces, Springer Lecture Notes, 155 (1970), 141-150. | MR | Zbl

[6] V. P. Palamodov, Linear differential operators with constant coefficients, Springer-Verlag, 16 (1970). | MR | Zbl

[7] J.-P. Serre, Géométrie analytique et géométrie algébrique, Ann. Inst. Fourier, 6 (1955), 1-42. | Numdam | MR | Zbl

[8] H. Skoda, dʺ-cohomologie à croissance lente dans Cn, Ann. Sci. de l'École Norm. Sup., 4 (1971), 97-121. | Numdam | MR | Zbl

[9] H. Skoda, Applications des techniques à L2 à la théorie des idéaux d'une algèbre de fonctions holomorphes avec poids, Ann. Sci. de l'École Norm. Sup., 5 (1972), 545-579. | Numdam | MR | Zbl

Cited by Sources: