Extension of coherent analytic subsheaves

Math. Ann. 188, 128--142 (1970) (~) by Springer-Vertag 1970

Extension of Coherent Analytic Subsheaves*

Yur~-ToNG SIu and G/dNTHER TRAUTMANN

Recently the theory of extending coherent analytic sheaves and subsheaves has been investigated by Frisch and Guenot [1], Serre [10], Thimm [19, 20], and the authors [12-t5, 17, 21, 22, 23]. tt is proved in [1] and [15] that a coherent analytic sheaf which is equal to its (n + 1) th absolute gap-sheaf can always be extended through a subvariety of dimension ~n. The best result for coherent analytic sheaf extension is conjectured to be the following:

(1) If 0 < e t < 1 (1 <i~N), then a coherent analytic sheaf defined on the Hartogs domain

O ' = {(zl , . . . , zN) ~ a:Nllz,I < 1, 1 _< i < N; Iz~l > ej for some n + 1 < j < N}

u{(z l . . . . . zN) ~ ~N[ Izil < 1, 1 N j < N; Izkl <~k, 1 <k__< n}

which is equal to its n th absolute gap-sheaf can always be extended to

Q = {(zl, ..., zN)e ~:NIIz, I < 1, 1 <-i<-N}.

This conjecture has not yet been proved. The closest result obtained so far is the following [15, p. 136, Th. In] :

(2) If 0 < ~ < 1 (n + 1 ~ j ~ N), then a coherent analytic sheaf defined on

Q" = {(z~ . . . . . zN) ~ ~Nllz, t < 1, 1 _< i < N; Izjl > e~ for some n + 1 < j ~ N}

which is equal to its (n + 1) th absolute gap-sheaf can always be extended to Q, where Q is the same as in (1).

In this paper we continue the investigation of extending coherent analytic subsheaves. The following Rothstein type and Osgood type theorems are proved:

(i) (Rothstein type). A coherent analytic subsheaf which is equal to its 0 th relative gap-sheaf can always be extended across a *-strongly 0-concave boundary.

(ii) (Osgood type). Suppose a coherent analytic subsheaf is defined on the complement of a subvariety of dimension n in a domain of ~E N. If the subsheaf is equal to its ( n - 1) tn relative gap-sheaf, then it can be extended through the subvariety if and only if its tensorial restriction to any member of an open family of (N - n + 1)-planes whose union covers the domain can be extended.

The Rothstein type theorem implies the conjecture (1) for the subsheaf case. Since the equality of a subsheaf and its 0 t~ relative gap-sheaf is equivalent

* During the research for this paper the first author was partially supported by NSF Grant GP~-7265 and the second author was a Fulbright Scholar.

Extension of Subsheaves 129

to the dimension of every one of its associated subvarieties being > e + 1, the Rothstein type theorem generalizes the corresponding statement for subvariety extension. The Osgood type theorem is slightly weaker than the corresponding statement for subvariety extension. However, it is the sharpest possible for subsheaf extension. The results in this paper illustrate the general principle that everything which can be done for subvariety extension can also be done for subsheaf extension with suitable modifications.

The Rothstein type theorem enables us to obtain a result for general sheaf extension which is stronger than (2) but weaker than (1).

For the proof of our results we use methods of normalization, meromorphic function extension, subvariety extension, and gap-sheaves.

§ 1. Gap-Sheaves

If (X, (P) is a complex space (not necessarily reduced) and ~ is a coherent analytic sheaf on X, we denote as usual by SQ(ff) the set {x ~ X t codh~x < Q} (e ~ 0). It is proved in [9] that SQ(ff) is a subvariety and dimSQ(ff)< Q.

If A is a subvariety in X, we denote by Jt ~° ~ the (coherent) analytic subsheaf of ~-~ of all section-germs whose supports are contained in A. If Q is a non-negative integer, we introduce the analytic sheaf

~vf0~ = ~ { ~ o ~ [ dimA __< e}

which is the subsheaf of all section-germs of ~ whose supports have dimension --<0. Applying [8, p. 359, Satz I], we obtain

~,~o~ = ~ ( ~ , ~ .

Hence ~ o ~ is coherent. From [9, p. 81, Satz 5] we conclude easily the following (cf. [16, Lemma 4]):

(3) The Q-dimensional component of Supp.,~VQ°~ agrees with the Q-dimensional component of SQ(o.~).

If f# is a coherent analytic sheaf on X which contains ~ as a subsheaf and if 2: f~ ~ f~/~" is the natural sheaf-epimorphism, we define

=

and call ~ the Qth (relative) gap-sheaf of ~ in (¢. If A is a subvariety of X, then we define

[ A ] =

and call ~ [A] the (relative) oap-sheaf of ~- in ff with exceptional subvariety A (cf. [18, 11]). ~ and ~ [ A ] are coherent and dim S u p p ~ / ~ < Q. If ~-' is a coherent analytic subsheaf of f¢ I X - A, we can still define ~ ' [A] as the subsheaf of (9 generated by local sections s E F(U, f¢) satisfying s[ U - A e F ( U - A , ~'). However, in general o~'[A] is not coherent. In fact, ~ ' [ A ]

130 Y.-T. Siu and G. Trautmann:

is coherent if and only if ~ ' can be extended coherently to X as a subsheaf of f~. Hence, a coherent analytic subsheaf defined outside a subvariety can be extended through the subvariety if and only if it can be extended across each point of the subvariety.

We define the Qth absolute gap-sheaf ~ l ° ~ by the following presheaf

U ~li__~m {F(U - A, ~ ) ] dimA < Q}.

It is proved in [13] that ~ 0 ~ is coherent if and only if

dim Supp ~0+ 1 ~ < Q.

If f e F(X, (9) and Z ( f ) = Supp(d~/fO), then (3) implies that for x E Z ( f ) the germ f~ is not a zero-divisor for ~ if and only if

dimxZ(f)c~SQ(~) < Q

for every ~ = 0. Hence we have the following:

(4) If f l . . . . . f k e F(X, (9) and (fi)x is not a unit of (_gx(1 < i ~ k ) , then (fOx . . . . . (fk)~ form an ~x-sequence if and only if

k

for every 0->0 (cf. [24, p. t52, Satz (II. 2.1)] and [15, p. 14i, Lemma 9]). Suppose G is an open subset oftl~" and ~ is a coherent analytic sheaf on G.

Suppose E is an (n - k)-plane in ti2". Let J be the ideal-sheaf on tI2" for E. We denote by ~11 E the coherent analytic sheaf ~ / J ~ on EnG. Suppose ~ is a coherent analytic subsheaf of ff on G. In general, ~ IIE may fail to be a subsheaf of ff II E, i.e. the sheaf-homomorphism

~- ti E--,~II E

induced by ~: ~(~ may not be injective. However (4) implies that

(5) ~ II E is a subsheaf of f~ II E if

dimEnSo(fg/~r ) < Q - k

for every Q __> 0. From (3) and (4) we conclude the following:

(6) Suppose dimEc~S~(fg/~) < Q -- k

for every Q >-- 0. If ~m = ~- for some m > 0, then

(~-tl E) . -k = ~-II E .


§ 2. Subsheaf Extension and Meromorphic Functions

Suppose G C G are connected open subsets of a normal reduced complex space (X, g~) of pure dimension n. Suppose ~ is a coherent analytic subsheaf of OP[ G satisfying ~n_ ~ = ~ . Let ~ = CP/~. ~ is a torsion-free coherent analytic sheaf on G. Let s i ~ F(G, ~ ) be the image of (0 . . . . . 0, 1, 0 . . . . . 0) under the natural sheaf-epimorphism ¢P- ,~ ' , where 1 is in the i th position. The following is clear.

(7) ~ can be extended coherently to G as a subsheaf of t~ ~' if and only if (i) ~- can be extended to a coherent analytic sheaf ~ on G and (ii) s~ can be extended to some element of F(G, ~#) (1 _< i <~ p).

Let r = rank ~-~. Select s~ . . . . . . s~, such that the sheaf-homomorphism

~p : ¢ ~ "

on G defined by these r sections is injective. Let J be the maximum ideal-sheaf on G such that J ~ ~ q~((~0. J is coherent and the zero-set of J has dimension < n - 1. Since J ~ ~ tp((9"), every local section of~, ~ can be lifted back through ¢p to a meromorphic section of the trivial vector bundle associated to ¢r. There- fore we have a sheaf-monomorphism

on G such that ~r ~ ~

~ r

is commutative, where ~ / i s the sheaf of germs of meromorphic functions on X. Let t i = ~p(si). We call t 1 . . . . . tp a set of associated meromorphic vector-functions for ~.

If ~ can be extended to a coherent analytic subsheaf 2~ of ~)Pl G, then we can assume ~ n - t = 2~. We repeat the preceding argument with ~ = g)P/~ instead of ~ (using the same ix . . . . . i,) and obtain associated meromorphic vector-functions t'l . . . . . ?p for ~ . tt is an r-tuple of meromorphic functions on (~ extending ti(1 <__ i <= p).

Conversely, if ti can be extended to an r-tuple t* of meromorphic functions on G (1 _-<iN p), then the subsheaf ~ '* of ~gr[ G generated by tT . . . . . t* is coherent. For, if f is a non-identically-zero holomorphic function on some connected open subset U of G such that f t * is an r-tuple of holomorphic functions on U, then ~-* ~ f ~ * on U and f ~ * is a subsheaf of~ ' [ U generated by f t * . . . . . f t* . If we identify ~ with ~p(~'), then ~ * extends ~ and t* extends s i ( l< i<p) .

Hence by (7) we have the following.

Prolmsition 1. ~ can be extended coherently to G as a subsheaf of 0 p and only if t~ . . . . . tp can be extended to r-tuples of meromorphic functions on G.


§ 3. Rothstein Type Theorem on Subsheaf Extension

Suppose (X, (9) is a complex space. A real-valued function v on X is said to be *-strongly O-convex at x ~ X if there exist a nowhere degenerate holomorphic map 7~ from some open neighborhood U of x in X to some open subset G of ~" and a real-valued C 2 function ~ on G such that v = ~ o 7~ and at

every point in G the Hermit ian matrix \ Oz i O~j ] has at least n - Q + 1 positive

eigenvalues, v is said to be *-strongly O-convex on X if it is *-strongly 0-convex at every point of X.

Suppose D is an open subset of X and x is a boundary point of D. D is said to be *-strongly O-concave at x if there exists a *-strongly 0-convex function v on an open neighborhood U of x in X such that D n U = {ye U Iv(y)> v(x)}.

We assume that D is *-strongly 0-concave at x. Suppose ~ is a coherent analytic sheaf on D. ~ is said to satisfy condi-

tion (E) if there exist an open neighborhood U o f x in X and a coherent analytic sheaf ~ on D u U such that

(i) ~- is a subsheaf of ~] D and (ii) for every open neighborhood W of x in U every element of F(D n W, ~)

can be extended to an open neighborhood of x. 0 If ~Q_ 1(¢ = ~, then (ii) is always satisfied according to [13, p. 373, Th. 3].

Suppose ~ is generated on D by a finite number of global sections.

Proposition 2. Suppose X is reduced and normal and has pure dimension n. I f ¢ < n - 1 and ~ is torsion-free, then ~ admits a coherent extension to some open neighborhood of x.

Proof. By [2, p. 69, Cor. 5.4] there exists an open neighborhood U of x in X such that D c~ U is connected. By replacing D by D c~ U, we can assume that D is connected. Since ~ is generated on D by a finite number of global sections, we have a sheaf-epimorphism 2 : OP~ ~ on D. Let 6e = Ker2 and let t 1, ..., tp be a set of associated meromorphic vector-functions for 5~. By [2, p. 74, Prop. 6.9], t i can be extended to a meromorphic vector-function ~ on D w V for some connected open neighborhood V of x (1 < i < p). Since ~ is torsion- free, 5a,_ 1 = 5e. By Proposit ion 1, 5 ~ can be extended to a coherent analytic subsheaf 5 ~ of (9 p I D w V. (9P/5 ~ is a coherent analytic sheaf on D w V extending ~ . Q.E.D.

Proimsition 3. Suppose X is reduced and has pure dimension n. I f 0 < n - 1 and ~o_ 1 ~ = O, then ~ satisfies condition (E).

Proof. Consider the normalizat ion r ~ : ) ¢ ~ X and define / ) = n - I ( D ) . /) is *-strongly 0-concave at every point of n - 1 (x). The sheaf ~* ~ o n / ) (which is the inverse image of ~ under n) is coherent o n / ) and is generated by a finite number of global sections.

Let ~-- be the torsion subsheaf of r~* ~: and let fq = (~* ~ ) / J - . f¢ is coherent and torsion-free and is generated by a finite number of global sections. By applying Proposit ion 2 to every point of re- ~ (x), we can extend f# to a coherent

E x t e n s i o n of S u b s h e a v e s 133

analytic sheaf if ' o n / ) w r c - I ( U ) for some open neighborhood U of x. We can assume that f~' is torsion-free. The zero th direct image n , ff ' of (~' under satisfies ~fo_ 1 re, f¢' = 0. Hence ~o_ 1 zt, fg' is coherent on D u U and n , (~ ' can be regarded naturally as a subsheaf of G °_ ~ n , if ' [ D.

We have a natural sheaf-homomorphism 2 : ~ n , n * ~ - . The natural sheaf-epimorphism n* ~ ~ f~ induces a sheaf-homomorphism # o 2 : ~- ~ n , (¢. Since our, ° ~ ~ = 0 and

Supp Kera C S u n ( S u p p ~ - ) ,

where S is the set of all singular points of X, we conclude that a is injective. can be regarded as a subsheaf of G °_ 1 n , ff'l D. Therefore ~ satisfies con-

dition (E). Q.E.D. Suppose ~ ' and ~ " are coherent analytic sheaves on D and

O - ~ f f ' ~ f f - - ~ f f " ~ 0

is an exact sequence of sheaf-homomorphisms on D.

Proposition 4. I f ~ " satisfies condition (E), then there exists an open neighborhood U of x in X such that ~ ' [ D c~ U is generated by a finite number of global sections.

Proof. Let sl, ..., sp ~ F(D, ~ ) generate ~ . Let 2 : ( 9 ~ be the sheaf- epimorphism on D defined by s I . . . . . sp. Since ~-" satisfies condition (E), there exist an open neighborhood W of x in X and a coherent analytic sheaf f9 on D u W such that

(i) ~ " is a subsheaf of fq[ D and (ii) q(si) can be extended to a section g~ ~ F(Dw IV, ~) (1 < i ~ p). Let # : O P ~ f f be the sheaf-homomorphism on Dw W defined by gl, ---, g r

K e r # is coherent on Dw IV. On some open neighborhood U of x in W, Ker# is generated by section t I . . . . . t~ e F(U, Ker#). I t is easily verified that ~ ' is generated on D n U by 2(ti) I D n U, 1 < i < q. Q.E.D.

Proposition 5. I f ~ ' and ~ " satisfy condition (E), then ~ can be extended coherently to some open neighborhood of x.

Proof. Since ~ is generated on D by a finite number of global sections, there exists a sheaf-epimorphism 2 : (9 p ~ ~ on D. Since ~ " satisfies condition (E), there exist an open neighborhood U of x in X and a coherent analytic sheaf f~" on Dw U such that ~ " is a subsheaf of ~"ID and every element of F(D, ~") can be extended to some open neighborhood of x. After shrinking U, we conclude that qo 2 can be extended to a sheaf-homomorphism # : ¢ P ~ f f " on D w U. After shrinking U further, we obtain a sheaf-homomorphism v : d~ ~ ~ ¢P on U such that

djq--~ Op-.v-~ f~-

is exact on U.

10 Math. Ann, 188


Since i f ' satisfies condition (E), there exist an open neighborhood V of x in U and a coherent analytic sheaf f~' on D u V such that i f ' is a subsheaf of f~'lD and every element of F(D c~ U, ~') can be extended to some open neighborhood of x. Since Im2v C ~ ' on Dc~ U, after shrinking V, we obtain a sheaf- homomorphism a : (gq ~ f¢' on V such that

(9q ~ , (gP

is commutative on D c~ V. The coherent analytic sheaf (9P/v(Ker a) on V extends ~ I D n V. Q.E.D.

Proposition 6. Suppose X has pure dimension n. I f p < n - t and ~,o_ t f f = 0, then f f satisfies condition (E).

Proof. Let JV be the subsheaf of all nilpotent elements of (9 and let (grca = (9 /J ( Since only the local situation at x is involved, we can assume w.t.o.g. that A/'k= 0 for some positive integer k.

Define y)-(0) = ~- and ~ (0 = the gap-sheaf (JV' i f ( l - 1)),_ 1 in ~ ( l - 1)(1 < 1 < k). k

Let S = (,J Suppff(1)/JVff (*-1). d i m S < n - 1 . Since ~ ( k ) = J V k f f on D - - S , /=1

~f~o_ 1 f f = 0 implies that if(k)= 0. F rom the definition of if(l), we have

{ ~o_ 1 ~ " ) = 0

~°_ 1 ( ~ ( ° / ~ " + 1)) = 0 ( 0 ~ / < k ) .

Since JVff(t)C ~(~+ i), it follows that ff(0/~-(t+ ~) can be regarded as an (grcd- sheaf.

We are going to prove (8)~ and (9)z for 0 _< l < k by induction on l:

(8)1 For some open neighborhood Uz of x, if(*) is generated on D n U t by a finite number of global sections.

(9)t ~(l) /~(t+ 1) satisfies condition (E).

F rom Proposition 3, we conclude that (8)z implies (9)t. (8)0 follows from the assumption concerning f t . Assume that (8)~ and (9)t are true for some 0 < l < k - 1. F rom the exact sequence

(10) 0 --, ~(z + ~) _., ~(0 __. ~ - , ) / ~ , + ~) ~ 0,

we conclude that (8)t+ 1 is a consequence of (8),, (9)~ and Proposition 4. F rom (9),_ ~ it follows that if(k-~) satisfies condition (E). We are going to

prove by descending induction on 1 that if(0 satisfies condition (E) (0 =<l< k). Suppose ~-(~+1) satisfies condition (E) for some 0 < l < k - 1 . Consider the exact sequence (10), F rom (9), and Proposition 5, it follows that if(*) can be extended to a coherent analytic sheaf ~ on Dw U for some open neighborhood

~-(t) 0, w e c a n assume that j f f o _ ~ = 0 . 0 ~ - i s U of x. Since ~o_ 1, = ' ~ a - ~ CO-


herent on D w U and ~(z) is a subsheaf of .~o°_l~-tD. Hence ~(l) satisfies condition (E). The Proposition follows from .~ = ~c0). Q.E.D.

Theorem 1. I f ~ o ,~ = O, then .~ satisfies condition (E).

Proof. Since only the local situation at x is involved, we can assume w.l.o.g, that d imX = n < ~ . For 0 -< l < n define ~t) = ~ 0 ~ and for Q < 1 < n define X t = S u p p ~ t ° / ~ ~t-1). X t is of pure dimension l or empty, because ~ o 1 (~"~ /~"- 1)) = 0 (Q < l __< n).

By [2, p. 68, Th. 5.3], for some open neighborhood U of x in X, Xz can be extended to an empty or purely/-dimensional subvariety -~t in D w U (Q < l < n). By using Hilbert Nullstellensatz and [2, p. 69, Cor. 5.4], we can assume after shrinking U that, for some integer k, (j~l))k(ff~)/f¢~-l))=0 on U, where j(t) is the ideal-sheaf on D n U for Xt( O < l ~ n). ~(o/fq(t-1) can be considered as a coherent analytic sheaf on (Xt, (9/J ~°) (O < 1 ~ n).

We are going to prove (ll)t and (12)t for 0 < l ~ n by descending induction on I:

(11)~ For some open neighborhood U t of x, ~#<t) is generated on DnU~ by a finite number of global sections.

(12)~ ~(l)/(~(t- 1) satisfies condition (E).

Since X~ ~ D is *-strongly 0-concave at x whenever x ~ X l, it follows from Proposition 6 that (ll)l implies (12)1. Since f f ( " ) = ~ , (ll),, follows from the assumption concerning ~ . Assume that (ll)l+~ and (12)~+~ are true for some 0 < l < n. F rom the exact sequence

(13) 0 ~ f6(t) --* f¢(~ + 1) ~ f~t + 1)/f¢(~)-~ 0,

we conclude that (11)t is a consequence of (11)~+~, (12)~ + 1, and Proposition 4. Since f~¢°)= ~ o ~ = 0, according to (12)0 +~, ~¢°+ ~) satisfies condition (E).

We are going to prove by induction on l that f#(~) satisfies condition (E) (0 < l =< n). Suppose ff<~) satisfies condition (E) for some 0 < l < n. Consider the exact sequence (13). F rom (12)t+ ~ and Proposition 5, it follows that f~(l+ 1) can be extended to a coherent analytic sheaf ~ on D ~ V for some open neighborhood V of x. Since ~ ° f ¢ ' + ' ) = 0 , we can assume that ~ ° f ~ = 0 . ~ o _ t f~ is coherent on Dw V and re*+ ~) is a subsheaf of ~o_ ~ f~[ D. Hence f¢~+ 1) satisfies condition (E). The Proposit ion follows from f f = re"). Q.E.D.

Corollary. I f ~ o ~ = O, then ~ can be extended coherently to an open neighborhood of x.

Remark. It can be proved that Theorem 1 together with its corollary remain valid if, instead of assuming that o~ is generated on D by a finite number of global sections, we merely assume that ~ is generated by F(D, ~) . The idea of the proof is to pull back the boundary of D to obtain an open subset D' C C D and then extend o~ I D'. If the pullback is small, the domain of the extension will contain x. The difference between o~ and the extension can be taken care of by using gap-sheaves. t0"


Theorem 1 a. Suppose D is an open subset of a complex space (X, (9), x is a boundary point of D, and D is *-strongly Q-concave at x. Suppose f~ is a coherent analytic sheaf on X and ~ is a coherent analytic subsheaf of f~ ] D. I f ~ = ~ , then ~ can be extended coherently to an open neighborhood of x as a subsheaf of~.

Proof. Since only the local situation at x is involved, we can assume w.l.o.g, that we have a sheaf-epimorphism 2 : 0 p ~ ~ on X. Let ~ = (~P/2-1 (~) . ~ ° 6 : = 0. By Theorem 1, there exist an open neighborhood U of x and a coherent analytic sheaf 5 on D w U such that 6: is a subsheaf of 5 1D and every element of F(D, J ' ) can be extended to an open subset of x. After shrinking U, we obtain a sheaf -homomorphism/z : (~P ~ J - on D u U such that

"1 i"

is commutat ive on D, where v is the natural sheaf-epimorphism. 2(Ker#) is a coherent analytic subsheaf of N I D u U extending ~ . Q.E.D.

Theorem 1 b. Suppose V is a subvariety of dimension <= n in a complex space X and ~ is a coherent analytic sheaf on X. Suppose ~ is a coherent analytic subsheaf of N l X - V and o~ = o~. I f ~ can be extended across some point of every n-dimensional branch of V as a subsheaf of ~, then o~ can be extended through V as a subsheaf of ~.

Proof. The conclusion of the Theorem is equivalent to the coherence of ~ ' [ V ] . By using induction on dimV, we can assume w.l.o.g, that V is non- singular and connected.

(a) Suppose dim V < n. For every point x e V we can find an open subset D of X - V such that x is a boundary point of D and D is *-strongly n-concave at x. By Theorem 1 a, o~ t D can be extended to a coherent analytic subsheaf ~ of fal D u U for some open neighborhood U of x. We can assume that ~-, = ft . ~ ' [ V ] agrees with ~ on U. ~ - [ V ] is coherent at x. Hence ~ [ V ] is coherent on X.

(b) Suppose dim V = n. Let W be the set of points of V where ~ [ V ] is coherent. Obviously W is open in V. Since ~ can be extended coherently across some point of V as a subsheaf of f~, W 4: 0. Suppose x ~ V is a boundary point of W. We can find an open subset D of ( X - V)w W such that x is a boundary point of D and D is *-strongly n-concave at x. As in (a), we conclude that ~ - [ V ] is coherent at x. W is closed in V. Hence W = V. ~ [ V ] is coherent on X. Q.E.D.

By employing the exhaustion processes used in [7, § 8] and in the proof of [13, p. 374, Th. 4], we obtain readily from Theorem l a the following two theorems.

Theorem 1 e. Suppose Q' and Q are as in (1). Suppose f~ is a coherent analytic sheaf on Q and ~ is a coherent analytic subsheaf of f#[ Q'. I f ~n = ~ , then ~-~


can be extended uniquely to a coherent analytic subsheaf ~ of (9 on Q satisfying

Theorem l d. Suppose v is a *-strongly O-convex function on a complex space X such that {x e X t ,t < v(x) < p~} is relatively compact in X for any two real numbers 2 < p. Suppose fg is a coherent analytic sheaf on X and ~ is a coherent analytic subsheaf of f~IXx for some real number 2, where Xx = {x e X [ v(x) > 2}. I f o~ o = o~, then J: can be extended uniquely to a coherent analytic subsheaf ~" of f# on X satisfying .~,n = ~'.

Theorem le . Suppose Q' and Q are as in (1) and ~ is a coherent analytic sheaf on Q'. I f 2,~°+ 2 ~ = 0 and ~l ° ~ = ~ , then ~ can be extended uniquely (up to isomorphism) to a coherent analytic sheaf ~" on Q satisfying o ~ ~,, ~ =.3 .

Proof. Since ~¢go+ z o~ = 0, o ~ , + ~ ~ is coherent. By (2), ~o+ 1 .NI Q" can b~ extended uniquely (up to isomorphism) to a coherent analytic sheaf o~ on Q satisfying ~ o ~ - = o~, where Q" is as in (2). By [15, p. 135, Prop. 20], o~- can be uniquely embedded as a subsheaf of ~" I Q'. By Theorem 1 c, o~ can be extended uniquely to a coherent analytic subsheaf ~r of ~ on Q satisfying ~,~, = ~ . It is clear that 0 ~ = ~ . ~ , ~" and o&- is uniquely determined up to isomorphism. Q.E.D.

§ 4. Osgood Type Theorem on Subsheaf Extension

The set P,,,k of all k-planes in ~" can be regarded canonically as an open subset of the Grassmann manifold G.+ l,k+ 1" We give P.,R the topology induced from G.+Lk+I, A subset 8 of P.,k is called an open family of k.planes if g is open in P,,k. We denote w {EIE ~ 8} by I~[.

The following lemma follows from [5, p. 84, Hauptsatz] and [3, p. 654, Satz 4].

Lemma 1. Suppose G is an open subset of lE", V is a subvariety of dimension <= n - 1 in G, and f is a meromorphic function on G - V.

(a) Suppose 8 is an open family of 1-planes in IE" such that G C I¢I- If for every E ~ ~ the restriction o f f to Ec~(G - V) can be extended to a meromorphic function on Ec~G, then f can be extended to a meromorphic function on G.

(b) I f as a meromorphic function f can be extended across some point of every (n - 1)-dimensional branch of V, then f can be extended to a meromorphic function on G.

The following lemma follows from [6, p. 340, Satz c 3].

Lemma 2. Suppose G is an open subset of qY, V is a subvariety of dimension < n - 1, and Z is a subvariety of pure dimension n - i in G - V. Suppose 8 is an open family of 1-planes in ten such that G C Igl. If for every E ~ g, Zc~E can be extended to a subvariety in E n G , then the closure of Z is a subvariety in G.

The following lemma follows from results in [4].

138 Y.-T. Siu and G. Trau tmann :

Lemma 3. Suppose DCC/) are open subsets of ~L", V is a subvariety of dimension <_ k in D, and Z is a subvariety in D - K Suppose 8 is an open family of (n - k)-planes in C" such that D C [8[. There there exists an open subfamily 8' of 8 such that (i) 18'[ intersects every k-dimensional branch of V mD and (ii) for every branch Y of Z r~ D and every E e 8', dim E r~ Y =< dim Y - k.

We denote the structure sheaf of •" by ,~.

Proposition 7. Suppose D is a domain in ~", V is a subvariety of dimension n - 1 in D, and ~ is a coherent analytic subsheaf of ,(PP[ D - V. Suppose 8 is an open family of 1-planes in C" such that D C t81. Suppose for every E e 8, Im(~t [[ E--*,~ p l[ E) can be extended coherently to E n D as a subsheaf of ,(gP LL E. Then ~ can be extended coherently to D as a subsheaf of ,(9 p.

Proof. Choose arbitrarily a relatively compact subdomain G of D. Let G = (~ - V and let 9 ~ = ~ [ G. We need only show that ~ can be extended to (~ as a subsbeaf of .0 p. We use the notations of § 2.

By Lemma 3, we can choose an open subfamily 8 ' of 8 such that (i) [8't intersects every (n-1)-d imensional branch of Vr~G and (ii) d imE n (SQ(~-) u S~(Coker ~p)) __< Q - n + 1 for every E e 8 ' and every 0 --> 0.

Fix arbitrarily E e 8'. By using (5), we conclude the following: (i) :~ tl E is a subsheaf of,(gPllE on Er~G;

(ii) the sheaf-homomorphism ,(9 r It E - ~ tIE induced by ¢p is injective; and (iii) rank(~" tl E) = r. Hence ti[Er~G, 1 <__i<=p, form a set of associated meromorphic vector-

functions for ~ [[ E. Since ~ H E can be extended coherently to E n G as a subsheaf of ,OP[[E, by Proposition 1, t i lEmG can be extended to an r-tuple of meromorphic functions on E n G(1 __< i _< p).

Since E is an arbitrary element of 8 ' and [8'[ intersects every ( n - 1 ) - dimensional branch of V n G , by Lemma 1 (a), tl . . . . . tp as r-tuples of meromorphic functions can be extended across some point of every (n - 1)-dimensional branch of V n (~. By Lemma 1 (b), t~, ..., tp can be extended to r-tuples of meromorphic functions on G. By Proposition 1, 9 ~ can be extended coherently to G as a subsheaf of,/9 p. Q.E.D.

Proposition 8. Suppose f9 is a coherent analytic sheaf on an open subset D of C N, V is a subvariety of dimension n in D, and ~ is a coherent analytic subsheaf of fg[D - K Suppose 8 is an open family of (N - n)-planes in C N such that D C 18[. Suppose for every E e 8, Im (~- tt E ~ fq [l E) can be extended coherently to Ec~D as a subsheaf of ~ 11 E.

(a) I f ~ = ~ , then ~ can be extended coherently to D as a subsheaf of f#. (b) I f ~ _ 1 = ~'~, then Supp~,~J~ - can be extended to a subvariety of D.

Proof. (a) By Theorem 1 b, ~ + 1 can always be extended coherently to D as a subsheaf of fq. Hence, by replacing ~ by ~ + 1, we can assume w.l.o.g, that f~ = ~ + 1- We can also assume that ~ 4: fg.

Let X = Suppfq/~'. X is a subvariety of pure dimension n + 1 in D - V. By [4, p. 299, Satz 13], the closure X - of X is a subvariety in D. Let X - = U xi

i


be the decomposit ion into branches. Let J i be the ideal-sheaf on D for Xi. By Hilbert Nullstellensatz, there exists an integer k~ such that jk , f~ C ~ at some point of X~ which is a regular point of X. Let J = Hi Jk'. Since ~ , = ~ , J r # C ~ . Let (9 = (N(9/J)IX- . ~ / J f ~ can be regarded as a coherent analytic sheaf on (X, (91 X) and f # / J f~ can be regarded as a coherent analytic sheaf on ( x - , (9).

Take arbitrarily x ~ X m V. After a coordinates transformation, we can assume the following:

(i) x = 0 . (ii) {zl . . . . . z,+ f = 0 } intersects X - in a subvariety of dimension 0.

(iii) {zi . . . . . z, = 0} is an element of S. We can choose an open neighborhood U of x in X - and a connected open

neighborhood G of 0 in ~ ,+1 such that the p r o j e c t i o n / / : C N ~ E "+1 defined by H(z 1 . . . . . zN)=(zl . . . . . z , + 0 induces a nowhere degenerate proper holomorphic map rc from U onto G.

Let V ' = rc(Vc~ U) and V " = n-a(V') . Let ~ ' be the zero t~ direct image of ( , ~ / J f ¢ ) I U - V " under nl U - V " and let fg' be the zero t~ direct image of (fC/,,cf~)lU under n. f¢' is a coherent analytic sheaf on G and ~ ' is a coherent analytic subsheaf of fq'i G - V'. Since o~,, = ~ , o~,~ = ~:'.

By shrinking G, we can assume that we have a sheaf-epimorphism q: ,+ l (gv~f# ' on G. Let 6~=q-~(~- ' ) . 5 ~ is a torsion-free coherent analytic subsheaf o f ,+ 1(9Vl G - V'.

Since {(z~ . . . . . z~,,)e~Nlzl . . . . . z , , = 0 }

is an element of S, there exists an open family S ' of 1-planes in tE "+1 such that

{(z a . . . . . z .+ l ) e~S '+ l Izl . . . . . z~=O}

belongs to 8 ' and H - ~ (E') e S for every E' e S'. By shrinking G, we can assume that G C IS'l,

Since Im(~" II E ~ f ~ I1 E) can be extended coherently to E n D as a subsheaf of f9 II E for every E e S , Im(SPll E '~,+~(9VlIE ') can be extended coherently to E ' n G as a subsheaf o f , + atOP I[ E' for every E ' e g ' . By Proposit ion 7, 6 e can be extended to a coherent analytic subsheaf 57 o f , + 1 (gP I G.

Let 6 e * = n*~/(6~). 5 a* is a coherent analytic subsheaf of f¢/J(~ on U. 9°* agrees with ~ / J f ~ on U - V " . Since ( ~ - / J f ¢ ) , = ~ / J f q and d i m V " < n , 5~*[ V"] is a coherent analytic subsheaf of f~ / J ff on U extending (. .~/J fC) tU - V . Let 2 : (# --, ( # / J f¢ be the natural sheaf-epimorphism. 2-1(6e* [V"]) is a coherent analytic subsheaf of ff on ( G - X - ) u U extending ~ - I ( G - X - ) w U - V. Since (G - X - ) u U is an open neighborhood of x, ~ can be extended coherently across x as a subsheaf of ft. Since x is an arbi trary point of X - c~ V,

can be extended coherently to D as a subsheaf of ft. (b) Let Z = S u p p ~ , / ~ . W.l.o.g. we assume that Z# : 0. Let Z - be the

closure of Z. Take arbitrarily a relatively compact open subset D' of D. We need only show that Z - c~D' is a subvariety of D'.

140 Y.-T. Siu and G. Trautrnann:

Let ~ = f f /~ . Since ~ _ 1 = ~ , by (3), Z is the n-dimensional component of S,(~).

By Lemma 3, we can choose an open subfamily 8 ' of 8 such that (i) 18'1 intersects every n-dimensional branch of V riD' and

(ii) dimEnS~(~l )nD' < Q - n for every Q > 0 and every E e 8'. Take arbitrarily E e 8'. By (4),

S0(~ II E )nD ' = E n S , ( ~ ) n D ' .

Since Im(~. ~ It E--,f¢lI E) can be extended coherently to E n D as a subsheaf of f¢ II E, ~11E can be extended to a coherent analytic sheaf ~ on EnD. The subvariety S0(~) n D' therefore extends E c~ S,(~) n D'.

Since E is an arbitrary element of 8 ' and 18'1 intersects every n-dimensional branch of VnD' , by Lemma 2, Z - riD' is a subvariety in D'. Q.E.D.

Theorem 2. Suppose c~ is a coherent analytic sheaf on an open subset D of IF. N, V is a subvariety of dimension n in D, and ~ is a coherent analytic subsheaf of ~91D - V satisfying o~,_ 1 = .~. Suppose 8 is an open family of ( N - n + 1)- planes in ff~N such that D C [8I. Then ~ can be extended coherently to D as a subsheaf of (~ if and only if, for every E e 8, I m ( ~ II E ~ f~ II E) can be extended coherently to E n D as a subsheaf of ~9 tl E.

Proof. We need only prove the "if" part, because the "only if ' part is trivial. Let 8 ' = {E' e PN,N_.tE' C E for some E e 8}. 8 ' is an open family of (N - n)-

planes. For every E' e S', Im(~ t I E'~f~II E') can be extended coherently to E' n D as a subsheaf of f¢il E'.

Let Z = Supp °~-,/~ ". By applying Proposition 8 (b) to 8', we conclude that the closure Z - of Z is a subvariety of D.

~ , = J r on D - Z - . By applying Proposition 8 (a) to D - Z - and 8', we conclude that ~- [ D - Z - - V can be extended to a coherent analytic subsheaf ~-' of f f l D - Z - . We can assume that ~, ' = ~ ' .

Let V * = V n Z - . d i m V * < n - 1. Let o~-, be the subsheaf of ~ I D - V* which agrees with ~- on D - V and agrees with o~-, on D - Z - . ~ * is a coherent extension of ~-. ~ * = ~-* on V - V*.

Take arbitrarily a relatively compact open subset D* of D. To finish the proof, it suffices to show that ~ * I D * - V* can be extended coherently to D* as a subsheaf of f~.

By Lemma 3, we can choose an open subfamily 8* of 8 such that (i) 18"1 intersects every ( n - 1)-dimensional branch of V*nD*,

(ii) d i m E c ~ ( V - V*)nD* -< 1 for every E ~ 8 ' , and (iii) dimEc~SQ(fa/~:*)nD* < e - n + 1 for every Q > 0 and every E eS* . Take arbitrarily E~ 8*. ~ It E can be extended to a coherent analytic

subsheaf 6e of ~# tl E on Ec~D. We can assume that 6 e [ E n V] = 6e. Let W = E n ( V - V * ) n D * . Since #-,* = ~ - * at V - V * , by (6), (#-*IIE)I

= (~:* 11 E) at W. Since dim W < 1, (#:* II E) [ W ] = #-* II E. Since ~ar, II E and Se agree on E n ( D - V), ~ * tl E agrees with 6e on En (D * - V*). The coherent analytic subsheaf 6e of f~ ti E on E n D * therefore extends #:* I1E.


Since E is an arb i t ra ry e lement of 8 " and I~*t intersects every ( n - 1 ) - d imens iona l b ranch of V*c~D*, by Propos i t ion 8 (a), ~-* can be extended coherent ly across some poin t of every ( n - 1 ) - d i m e n s i o n a l b ranch of V * n D * as a subsheaf of f#. Since o~*_ 1 = ~ * , by Theorem 1 b, ~ * can be extended coherent ly to D* as a subsheaf of ~ . Q.E.D.

Remark. The theorem for subvar ie ty extension cor responding to Theorem 2 requires only that g is an open family of (N - n)-planes. F o r subsheaf extension such an assumpt ion is no t enough. The following is a counter-example.

D = II22 . V = {Z 1 = 0}. cff= 2(92.

~ is the subsheaf of f q lD-V generated by (1, e x p ~ ) , (zl-z~,O), and

(O, zl-z~), g=P2,1. For every E e g , Im(o~ltE--*fal lE) differs from fgllE only at a finite n u m b e r of points and hence can be extended coherent ly to EnD as a subsheaf of f# II E. o~ canno t be extended coherent ly to D as a subsheaf of f9, otherwise F//(z 1 - z~)f¢ can be extended coherent ly to D as a subsheaf of f~/(z 1 -z~)f#, which implies that the ho lomorph ic funct ion on

1 {z~ = z~} - {0} induced by exp - - can be extended to a meromorph ic funct ion

z 1 on {z, = z~}.

References

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J. reine u. angew. Math. 234, 123--15t (t969). 20. - - Fortsetzung von koh~renten analytischen Modulgarben.Math. Ann. 184, 329--353 (1970). 21. Trautmann, G.: Ein Kontinuit~tssatz ffir Fortsetzung koh~irenter analytischer Garben.

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Yum-Tong Siu Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556, USA

Gfinther Trautmann Mathematisches Seminar der Universit~it D-6000 Frankfurt a. M.

(Received October 20, 1969)

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Extension of coherent analytic subsheaves