By Yum-Tong Siu, Günther Trautmann (auth.)

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SO(~) Since for y ~ x we have ~y, Y it follows that N U is either {x} or empty. If m ~ i, assume that dim S k ( ~ ) ~ k for k < m. Let x ~ S m ( ~ ) . Without loss of generality we may assume that x @ S O ( T ) , dim x S m ( ~ ) ~ m, then we can find y ~ x in a neighborhood of x such that dimy S m ( ~ ) ~ m. Since x @ S O ( q ) , that fx is not a zero-divisor for there exists fx ~ ~ x ~x" the sheaf-homomorphism f is injective. ~ ~ F = {y ~ UIf(y) : 0}. 17) - i ~ dim x F n S m ( ~ ) . 2) we conclude that F N Sm(~) ~ U c Sm_l(~/f~) n U and by induction hypothesis we know that dim x S m _ l ( ~ / f ~ ) Hence dim S m ( ~ ) ~ m.

15) C0rollar~. 16) Corollary. ~ I U = 0 for i ~ q. Let X be a complex space and 7 analytic sheaf on X. be a coherent Then the following two conditions are equivalent. (a) dim S k + q + l ( ~ ) (b) For any subvariety A of dimension -<- d in an open subset U of X, ~ Ai F =< k for every k < d. = 0 for i -<- q. Proof. 14). is proved in the following way. k < d. Assume (b) => (a) (a) is not true for some Then we can find a subvariety A in S k + q + l ( ~ ) is an open subset of X and dim A = k + 1 i d.

Then dim Supp s < n - q if one of the following two conditions holds. (a) sx e of n kx © x ° ~x for every x c A0 where ~/vx is the maximal ideal 57 (b) For every x ~ A, sx ~ ~x ~x' where ~ x is the ideal generated by (~q+l - ~q+l(X))x ..... (~n-Zn(X))x in n O x" Proof. Let ~V ~ / ~ V + l ~(v) = ~ k s n _ q _ 1 7 (v) . and let B = Uv:I If x ~ B, then there exists an open neighborhood U of x and sections ' " generating ~v ~lu PV such that the induced homomorDhism (n~/~) pv _ ~ ( v ) is an isomorphism.