Integration of a Differential Form on an Analytic Complex SubvarietyAuthor(s): Pierre LeLongSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 43, No. 2 (Feb. 15, 1957), pp. 246-248Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/89820 .

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INTEGRATION OF A DIFFERENTIAL FORM ON AN ANALYTIC COMPLEX SUBVARIETY

BY PIERRE LELONG

UNIVERSITY OF PARIS*

Communicated by Marston Morse, December 12, 1956

[. The purpose of this note is to give a precise definition of the operator of

egration

t(')- = f p (1) w

an exterior differential form 'p on an analytic complex subvariety W. The )blem arises because an analytic complex subvariety in a domain D of Cn (or, are briefly, an analytic set in D) is not, in general, a manifold. W e give (a) an stence theorem for t(cp) and (b) a proof that t is a closed current in D, that is, a) = 0 for the forms i/ with compact support, which are homologous to zero in D.

II. A set W is called an analytic set in a domain D of Cn(zI, . . ., Zn) if M e D s a neighborhood (UM) such that (UM) n W is defined by the simultaneous uations

fi(21, ..., Zn) = O, . .. f,(z, . .., Zn) = 0, (2)

ere fk is holomorphic in (UM); M e W is an ordinary point of W if there exists

analytic one-to-one mapping Z = F(z) such that F [(UM) n W] = F[(UM) ] n Cs, aere Cs is a complex linear subspace. We denote by A a complex q-vector given

Zk = Z' + i akui , 1 < < 1 < i< q, (3)

th unitary representation and fundamental form

dr7q = (Aq) =( (-)-)/2 u A di1 ... dut A daq.

s is the maximum of the numbers q such that P can be an isolated point of n AQ, p = n - s is the complex dimension of W in P e W. We have the de-

nposition W = U Wk, where Wk is k complex dimensional in each of its points. W' = W n Aq is not of dimension zero, the boundary of B(P, r) = a- ]

lu I 2 < r2 intersects W'. Starting from this property and applying the

onecker integral to the system (2) after substituting equation (3), where Zk' I aki are considered as complex parameters, we obtain THEOREM 1. If W is an analytic set in D and M(zk?) is an isolated point of Aoq n W, ere Aoq is given by (aik)o and Zkt = zk? in equation (3), there exists a neighborhood

'M) of M, e > O and e' > 0, such that for af - (a,k)0o < e, and zk' - Zk? < e',

, n W n A' is a set of I (<N(Mo)) isolated points, where N(Mo) is the number

intersections of Aoq n W in Mo. [II. Positive Currents.-We denote by *, the adjoint of a form i/; *Q(Aq) is

? fundamental form of the An-" orthogonal to AO.

246

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Definition: A current2 t is called positive of degree q (or belongs to 0q+) if t is ,mogeneous of degree (q, q) in dz,, dZj and if for every complex q-vector Aa, the stribution T [A ]() = f t A *Q(A )f is a positive measure. A form p is called ,sitive of degree q (or belongs to I,q+) if (p has continuous coefficients and 'p e oq. THEOREM 2. A positive current t = (i/2)(--1) q(q-.)/2 ) tg(i)(t)dzai A ... A dzi

(i)(j) a- A ... A dzjq is continuous of order zero; the distributions T(i) ((f) =

't()(j)f dr2n are complex measures: T(,(J) = T(j,() and T(k)(k) ( 0. Remark A: We put I tD = sup It(p)l for (p, (C"), with compact support p) c D, and I| (i)(j)| < 1 for the coefficients of 'p. Then it is possible to find

system A of (Cnq)2 complex q-vectors Aq, with l ak - (aek)o| < e, where e > 0 d Aoq[(aik)o] are given, such that we have Itl D < k(A) sup T[A,P], where k(A)

a constant depending on A. The class of the positive currents possesses the lowing properties: THEOREM 3 (multiplication). If t e O + and 'p c Qbi+, then t A 'p c Op+1+ THEOREM 4 (division). If t A ( = 0, t c Op+, and if c e i1 + satisfies p a- 0 d q+l = 0 in D, then we have t = tl A 'f, and tl e Op_q (tl =- 0if p < q). THEOREM 5. The image of a positive current induced by an analytic and locally a-to-one transformation is a current which is positive.

IV. If W is the analytic set f(zi, ..., Zn) = 0 in D, we have the following two pressions for the current t('p):

(a) t = -2 ddlog If I; (b) t = 0 df A df.

(a) the bracket denotes the positive current relative to the plurisubharmonic iction log If[. On the other hand, (b) is available only for forms 'p whose )port K(p) contains only ordinary points of W; in (b) the positive current 01 degree zero (measure density) is given by an analytic locally one-to-one mapping = F(z) of W on a complex subspace. We consider now an analytic set Wp of complex dimension p irreducible in D I denote by WoP the connected manifold of the oidinary points of WP, and we t E1 = WP - WoP. The definition of the current t('p) (the integral of 'p on ') will be given in three steps: (1) Definition of the positive current to(p) =

,P p on 'p with support K(9p) c D - El. (2) M ajorization of the norm of to a neighborhood of M e WP, for instance in a sphere B(M, r). (3) Solving of a itinuation problem for to defined in D - El, to obtain t defined in D. -. If M e Wop, to is obviously defined by an analytic mapping of WP n (UM) i complex subspace. a. We use the following result, which is a consequence of Theorem 1 and mark A: -HEOREM 6. If W is an analytic set in D, of complex dimension p in ever'y M e W,

I D1 c c D is a compact domain in D, there exists a constant X(D1) > 0 such that norm I toa M of the current to in the spheres B(M, r) c D, satisfies I toI M < A (Dl)r'P. r. Continuation of a Closed Current.-Given a current t in a domain D - E, mre D c Rm(xl, . .,xm) and E is the subspace xi . = . - X = 0, a necessary

dition that t be continuable to the forms (c with support K(p) c D is the

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nvergence of E t A aio, K(ai) c D - E, where ai is a partition of unity in

- E. We give now conditions that a closed current t have a continuation i by closed current. We denote by ar(xi, . ., x) a kernel (C"), with 0 < a, < 1; = 1 in a neighborhood or of the origin coo (xi = ... = Xs = 0); ar has a compact pport Kr; wr C Kr c c r/, and co/ tends to co when r -- 0. We put 0 = - t, d suppose that t is homogeneous of degree ck. THEOREM 7. A necessary and sufficient condition for the existence of t is (a) i art = 0; (b) the existence of the limit -r(P) = (-l)kt[da A p], for a sequence of o rnels ar with properties listed above. Thus r is a current with support in E and is homologous to zero; the limit is lependent of the sequence r,; we have dO = -, t = t + 0. If r = 0, we choose =0, and we say that 7 is the simple extension of t. If t exists, t is continuous of

ite order q in K n (D - E), where K is a compact domain in D. We put

K = sup t((p), where p satisfies K(p) c K n x,2 < r2] and D("(i,)] < 1,

each derivative of total order <q of the coefficients of 'p. W e choose ar =

Xi/r) and denote (ba/bxj)(x/ir) by j, r(xi), to obtain more precise sufficient nditions: THEOREM 8. A sufficient condition for the existence of i is the existence of a finite -mber L such that we have I t3j, r(x) K < Lr. COROLLARY. If t is continuous of order zero and if r~l tI -- 0, t is obtained by

simple extension of t in K c D. VI. Now we consider E, = WP - Wp and put E = W1 U W2 U ... u Ws, lere Wk is the manifold of the ordinary points of E1 - [W1 U W2 U ... U Wt- I]; k is a manifold of complex dimension vk < p - k. In M e Wk, we consider a ighborhood (UM) and an analytic one-to-one mapping Z = F(z) such that (UM) n Wk] = F[(UM) n CVk] where C(k is a complex subspace. By Theorems ind 6 and the corollary to Theorem 8, we obtain the main result of this note: THEOREM 9. If WP is an analytic set in a domain D of Cn and if Wp is of complex mension p in each of its points, the positive current of degree (p, p) defined by

to = fWP p0

,here WoP is the manifold of the ordinary points of WP, and E1 = WP - WOP) is nvergent in D - El. The simple extension t of to gives a closed positive current

p) on the forms 'p with compact support in D. If W is not homogeneous p complex dimensional in D, the current t(p) = fp

w a finite sum of homogeneous, positive closed currents tI in D whose supports e the components of W in the decomposition W = u Wk, where Wk is k complex mensional in each of its points. * Written while the author was at the Institute for Advanced Study (Princeton, N.J.) under e sponsorship of the Air Force Office of Scientific Research. 1 Cf. R. Remmert and K. Stein, Math. Ann., 126, 263-336, 1953, in particular p. 283. The tion of complex dimension is invariant under one-to-one complex analytic mappings. 2 Cf. G. de Rham, Varidtes diff-rentiables (Paris: Herrmann & Cie, 1955). A current t is con-

lUous of order s if t(pn) -- 0 for every sequence of forms apn, (C" ), with compact supports K(p5n) C

a, and sup ID("),cn, (i) -- 0, for every coefficient pn, (i) and every derivative (a) of total order s.

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