Multi-logarithmic differential forms on complete intersections Текст научной статьи по специальности «Математика»

УДК 517.55

Multi-Logarithmic Differential Forms on Complete Intersections

Alexandr G.Aleksandrov*

Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 65, Moscow, 117997,

Russia

Avgust K.Tsikh

Institute of Mathematics, Siberian Federal University, Svobodny 79, Krasnoyarsk, 660041,

Russia

Received 2.02.2008, received in revised form 10.04.2008, accepted 12.04.2008 We construct a complex fi* (log C) of sheaves of multi-logarithmic differential forms on a complex analytic manifold S with respect to a reduced complete intersection C С S, and define the residue map as a natural morphism from this complex onto the Barlet complex uQ of regular meromorphic differential forms on C. It follows then that sections of the Barlet complex can be regarded as a generalization of the residue differential forms defined by Leray. Moreover, we show that the residue map can be described explicitly in terms of certain integration current.

Keywords: complete intersection, multi-logarithmic differential forms, regular meromorphic differential forms, Poincare residue, logarithmic residue, Grothendieck duality, residue current.

Introduction

In 1887, Henri Poincare [21] introduced the notion of residue 1-form associated with rational 2-forms on C2. In the papers of G. de Rham, [23], and J. Leray, [18], this notion was extended to the class of d-closed meromorphic q-form w having poles of the first order along a smooth divisor. They proved that the residue of such a form is holomorphic. Inspired by the ideas of Leray and P. Dolbeault, then M. Herrera, P. Liberman, N. Coleff and others have developed theory of the integral representation and residue currents (see [7], [8], [6]).

The next stage of the development of the theory started in early '70's. It is closely related with the general theory of logarithmic differential forms. In fact, forms of such type were first considered in 1972 in a work by J.-B. Poly [22] who defined and studied the Leray residue for semi-meromorphic differential forms. He had proved that the Leray residue is well defined for any (not necessarily d-closed) semi-meromorphic differential form w, as soon as w and dw have a simple pole along a hypersurface. Somewhat later, K. Saito investigated the class of meromorphic differential forms w satisfying these conditions; in a series of his works,

* e-mail: alexandr@ipu.rssi.ru t e-mail: tsikh@lan.krasu.ru © Siberian Federal University. All rights reserved

these forms were called logarithmic along a hypersurface (see [24], [25]). Saito established the basic properties of logarithmic differential forms and their residues, and considered some important and interesting applications. It should be also remarked that the same notion has occurred also in the book by Ph. Griffiths and J. Harris [12], with no references to previous works; in this book, the authors use this notion only in the case of divisors which are normal crossings of smooth hypersurfaces. From another point of view this case has also been treated in earlier works by P. Deligne, J. Steenbrink, F. Pham, and many others.

Almost at the same time, in 1976, D. Barlet introduced and investigated a complex w*x of subsheaves of the sheaves of meromorphic differential forms on an analytic subvariety X of a complex manifold (see [3], [4]). If dim X = n, then the sheaf ^X is naturally isomorphic to the Grothendieck dualizing module. In a purely algebraic context, a similar notion also appeared in the works by E. Kunz [16], [17], M. Kersken [15], and others on general duality theory. They call ¡¿X the sheaf of regular meromorphic differential forms of degree q. In a joint paper by G. Henkin and M. Passare an analytic interpretation of regular meromorphic forms is given (see [14]). Some of their results are essentially deduced from the basic properties of weakly holomorphic functions, established in a series of earlier works of the second author and his collaborators (see [28], [29], [27]).

In the present paper, we discuss in detail the notion of a multi-logarithmic differential form introduced in [2]. In fact, a meromorphic multi-logarithmic differential form has poles along a divisor D = |Ji=1 Di such that C = p|i=1 Di is a reduced complete intersection. Our main theorem asserts that there is a natural morphism which maps the complex of the multi-logarithmic differential forms onto the complex of the regular meromorphic forms. The latter complex was defined and studied by D. Barlet [3], [4]. We also define the residue map and residue form of a multi-logarithmic differential form, as a generalization of corresponding definitions due to J. Leray, J.-B. Poly and K. Saito. In 1990 the first author investigated similar constructions in the case of logarithmic differential forms along the divisor D. More precisely, it was proved [1] that the residue map defined by K. Saito [26] gives a natural quasi-isomorphism between the complexes of sheaves of the logarithmic differential forms along the divisor D and of the regular meromorphic forms on D. It should also be mentioned that an explicit description of residue map in terms of integration currents is also given here; it is mainly based on results obtained in [28], [29], [27], [6], and [19].

The present paper contains four sections. In the first one we give the basic notation and definitions, and establish some properties of multi-logarithmic differential forms with respect to a reduced complete intersection in a complex analytic manifold. Then we consider the notion of regular meromorphic differential forms and discuss their properties. In Section 3, our main result is proved. The final section contains an explicit description in terms of integration currents of the residue map in the case of multi-logarithmic differential forms.

Acknowledgment. The authors are very grateful to Professor A.Yger for useful and stimulating conversations on the theory of residue currents.

1. Multi-Logarithmic Differential Forms

Let U be an open subset of S = Cm, and let D1,..., Dk be a set of hypersurfaces defined by the equations hj(z) = 0, j = 1,..., k, k ^ 1, respectively. Here hj(z) = hj(zi,..., zm),

j = 1,..., k, are holomorphic functions on U, and z = (zi,..., zm) is a system of coordinates. In what follows we shall suppose that hi,...,hk is a regular sequence in Oj, and I = (hi,...,hk)Oj is a radical ideal, that is, I = \[I. In other words, the complex analytic variety C = Di O ... O Dk is a reduced complete intersection. In particular, dim C = m — k and the differential k-form dhi A ... A dhk vanishes identically on no irreducible component of C. Let Qj = (Q j, d) q=0 i be the de Rham complex of germs of holomorphic differential forms on U whose terms, locally at the point 0 G U, are defined as follows:

Qj,0 = Oj,o( . .. ,dzj! A ... A dzjq,. .. ) , (ii, .. ., iq) G [1, m].

Set D = Di U ... U Dk, and Dj = D \ Dj, j = 1,..., k. Let Qj(*Dj) be the OU-module of meromorphic differential forms of degree q ^ 0 consisting of all the differential q-forms with the polar divisor Dj. We write Di = 0 for k = 1, so that Qj(*Di) = Qj.

Proposition 1.1. Let w be a meromorphic differential q-form on U, q ^ k, with poles along the divisor D = Di U ... U Dk. The following conditions are equivalent:

k k

}U(*Di), hjdw G "U

=1 i=i

i) hw G ^ "U(*D), hdw G ^ "U+1(*A), j = 1,..., k ;

kk

ii) hj w G ^ "U (*A), dhj A w G ^ "U+1(*A), j = 1,.. .,k ;

i=i i=i

iii) There exists a holomorphic function g, not equal identically to zero on every irreducible component of C, a holomorphic (q — k)-form £ and a meromorphic q-form n G i=i Qj(*Dj) on U such that

dhi dhk gw = — A... A — A £ + n; hi hk

iv)There exist analytic subsets Aj C Dj, j = 1,...,k, of codimension at least 2 such

k=i(Dj \ Aj)

that the germ of w at any point x G Uk=i(Dj \ Aj) belongs to

dh1 a ... A d^ A "U-k+è "U,x(*^), 1 k i=1 where "U Xk is the module of germs of holomorphic q-forms on U at x.

Proof. The equivalence of i) and ii) follows directly from the Leibnitz rule:

d(hjw) = dhj A w + hjdw.

Let us prove the implication ii) ^ iii). Consider the following representation of the form w :

y^ a/ (z)-dz/

w = l/l=_

h-1 ■ ■ ■ hk '

where I := Iq = (i 1,..., iq), 1 ^ i 1,..., iq ^ m, is a multi-index, dz/ = dzi1 A ... A dziq, and a/(z) is a set of holomorphic functions on U, skew symmetric with respect to the index I.

It is easy to see that the condition

k

dhj A w G Y^ ttU+^+Di)

i=i

is equivalent to the condition

k

dhj A ai (z) -dzi G h£Îlf+i.

I £=1

For an ordered multi-index I := Iq+i = (ii,... ,iq+i), the latter gives us the following system of relations between the coefficients ai and the derivatives dhj/dzi :

q+1 r)h-

E(-!)"-1 q^ ai\u = j hi + ... + bki hk, j = l,...,k, (1)

1=1 '

m

with holomorphic coefficients j, ... ,biiI G Oj .

Let us fix a multi-index Jp = (ji,..., jp) C [1, ...,m], 1 ^ p ^ k, and denote the corresponding minor of the Jacobian matrix Jac(hi,..., hk) = \\dhi/dzj|| as follows:

A j p = Aj1...jp = det

dhi

dzj

T^p

Let I Ç Ou be the ideal generated by the sequence of germs (h1,..., hk). We shall prove the following relations:

Ajpaiq = V sgn (Kjq\Kj Ak açjpJq\K) (mod (I)), p =l,...,k, (2)

Ken ^ , \ '

|K|=p

by induction on p. Suppose that p =1, J1 = ji, and set I = (j,Iq) = (ji, ii,..., iq) in formula (1). One gets

dr ai=J2(-1) ai\i* (m°d(i)^

j e=i %i

and this coincides with relation (2) for p =1.

Assuming relation (2) to be true for p — 1, one can prove it for p as follows. Making use of the cofactor expansion along the p-th row of the determinant Ajp, one obtains:

Ajp ai = V ( — 1)P- ^ Ap-j . ai.

We have

Aji...i...jp ai = V sgn( K K) AK a(ji.Jt...jp,I\K') (mod(I)),

K'eii ^ ' \ '

|K'|=p-1

by the induction hypothesis. By substituting this relation into the previous equation and changing the order of summation, one obtains

/ \ p dh J ai ^ E sgn K'/q\K' AK' dhp (^(2)).

K'c/q v ' \ y £=1 *

The second sum consists of p summands contained in formula (1) for j = p, I = (i1,...,ip, I \ K '). Hence one can rewrite it as a sum of other q — p +1 summands with opposite signs and an element of the ideal (h1,..., hk)OU. This yields the following relation modulo I :

AJPa/ = £ sgn ( £ W,( —1)p-1 £ ( — 1)#(i;AK')dhPa(ji...jPI/\K'\i), (3)

K'c/q \ , \ / ie/\K' i

|K'|=p-1

where #(i; I \ K/) denotes the number of occurrences of the index i in the set I \ K/. Let us order all the pairs (K/, i) in such a way that the multi-index K/ U {i} coincides with a given K C I. For any such pair, the corresponding coefficient a(j1...jp /\(K'\i)) is equal to a(j,/\k). Thus the contribution of the set ordered as above to relation (3) looks as follows:

aj,\K,(—1 £ sgn (k \ i, Iq \ k, i) (—1)#(i ™'>> £ AK\. =

= sgn (k,/' \ Ak .

This completes the proof of relation (2) for p ^ 1.

It is not difficult to see that relations (2) imply the following equality

Ail...ifc ■ £ a/dz/ = dhi A ... A dhk A ( £ a¿i...¿fc/' dz/') + v,

|i|=q |/'|=q-k

with v G k=i hjk, so that m) holds, with g = Ail...ifc, and with £, n of the required shape. It remains to show that one can choose the function g in m) in such a way that g ^ 0 on each of the irreducible components Cj of C. Let {Aj1...ifc}, (¿i,... .¿k) G [1,..., m], be the set of all k x k-minors, with elements A1,..., AN, where N = (m). By assumption, for every non-singular point of C, there is a minor that does not vanish at this point. Let z(j) g Cj and suppose that z(j) is non-singular, then the function

At(z) = tAi + t2A2 + ... + tN An

regarded as polynomial of t, does not vanish identically for every z = z(j). Hence, there is a value to G C such that At0(z(j)) = 0 for all j; thus one can take g(z) = At0(z).

Here is a purely algebraic proof of this result. Consider the ideal G of generated by all the minors Aj1...jk of maximal order of the Jacobian matrix Jac(hi,..., hk). The condition that dhi A ... A dhk does not vanish identically on every irreducible component of C implies that the image G of G in the ring OC,0 is not equal to Ann OC,0. Thus one can use [5],

K

p-1

Th.2.4. (1) which implies that the OC0-depth of the ideal G is at least 1. Hence there is an element g G Oq satisfying the condition in iii).

The implication iii) ^ iv) is obvious: take Aj = Dj n {g = 0}, j = 1,..., k.

Let us prove the implication iv) ^ ii). It is obvious that hjw can be expressed as a sum of meromorphic forms each of which has singularities along at most k — 1 divisors, for example, on Di, and on the subset Ai C Di of codimension 2. Hence by the Riemann extension theorem, such a meromorphic form has singularities only on Di. As a result, hjw £^2k=i nJ(*Di), j = 1,..., k. Similar considerations are valid for the differential form dhj A w. This completes the proof of the implication iv) ^ ii). □

Remark 1.1. In fact, if k =1 and C = D is a reduced hypersurface then Proposition 1.1 is a generalization of the basic Lemma-Definition due to K. Saito (see [26], (1.1)).

Definition 1.1. A meromorphic differential q-form w, q ^ k, on U is called multi-logarithmic with respect to the reduced complete intersection C = Di n ... n Di if w satisfies one of the conditions i) - iv) of Proposition 1.1.

The localization of this definition leads to the notion of multi-logarithmic differential forms with respect to the complete intersection C at a point x G S. We denote the OS,x-module of germs of multi-logarithmic q-forms at x and the corresponding sheaf of multi-logarithmic differential q-forms on S by nSx(logC) and nq(logC), q ^ k, respectively. Thus the OS-module nq(log C) is a submodule of nq(*D), consisting of all the differential forms with polar singularities along the divisor D.

Corollary 1.1. There are the following natural inclusions

h nq(log c) c nq, q > k.

Proof. Condition i) implies that hw has poles along D, while condition iv) shows that this meromorphic form has poles along the hypersurface defined by the holomorphic function g. Hence, hw is holomorphic outside a subset of S of codimension at least 2 since the functions h and g are in general position. It remains to apply the Riemann extension theorem. □

The structure of nq(log C) seems to be more clear from the following observations. Set Cj = fli=i j,=j Di, where j = 1,... ,k. Then there are the following natural inclusions

i dhj

Is (log Cj) c £ Qqs(*Di) C Q%(log C), -j a lS-i(log Cj ) c Q%(log C).

hj

i=i J

In similar notations one gets

k

iq(logCj) c £iq(*Dî) c iq(logc), a a iq-2(logCj) c iq(logc),

i=i • j

and so on.

The following statement shows that the definition of multi-logarithmic differential forms is quite functorial; in particular, it is compatible with restrictions of certain type.

Lemma 1.1. The sheaves Qqs(log C) and Qqs coincide outside D for all q ^ k. The stalks QS x(log C) and Q% x(log Di) are isomorphic at all points x G Di \ C. Further, set Cij = Di fl Dj. Then QS x(log C) = QS x(log Cij) at all points x G Cij \ C. A similar assertion is valid for triple intersections, and so on.

At last, for the sake of convenience of notation we set

k

QS(logC) = ^QS(*A), q = 0,...,k - 1. (4)

i=i

It should also be remarked that QS(log C), q ^ 0, are coherent sheaves of OS-modules. Moreover, they are OS-modules of finite type, and the direct sum ©^qQS(log C) is an Os-exterior algebra closed under the standard action induced by the exterior de Rham differentiation d on QS.

In what follows, we let for simplicity x = 0 when considering the local problems in the neighbourhood of x. We shall also assume that U is an open subset of S = Cm containing the origin.

Definition 1.2. In the notation of iii), the restriction to the complete intersection C = Di fl ... fl Dk of the form £/g is called the residue form of w :

e

res. w = -g

Proposition 1.2. The residue morphism res. is well-defined.

Proof. Let w G QS (log C), q ^ 0. If w has two different local presentations

dhi dhk gew = — A ... A — A & + I =1, 2, hi hk

then

dhi A ... A dhk A (gi^2 - g2^i) = hi ■ ■■ hk (gi^2 - g2Vi) G (hi,..., hk)QS.

Hence

dhi A ... A dhk A (gi e2 - g2%i) = 0 (mod (hi,..., hk)).

Now one can apply the part i) of the Theorem from [24] for R = OC}o, M = QS 0 ® OC,o. It follows then that

Ge(gii2 - g2ii) G dhi A QS-0k + ... + dhk A QS-0k, e G Z, e > 1,

in QS o ®Oc,o, where the ideal G of Os,o is generated by all minors of maximal order Aj1...jk of the Jacobian matrix Jac(hi,... ,hk). Since C is reduced, the image G of G in the ring OC,o is not equal to AnnOC,o. Thus one can use [5], Th.2.4. (1), which implies that the Ocoo-depth of the ideal G is at least 1. Hence there is a maximal minor A = Ai1 ..iik which is not a zero-divisor in Oc,o, and

Ae(gie2 - g2ei) G dhi A QS-oi + ... + dhk A QS-/ in QS o ® OC,o. Therefore the class of Ae(gie2 - g2ei) in QC o is equal to zero. Thus the two elements — ei and — e2 define the same class in MC o ®oc 0 QC o. n

gi g2 ' '

Lemma 1.2. The map res. commutes with the exterior de Rham differentiation d and defines a homomorphism of Os-modules

res. : (log C) —► Mc ®oc ^C"', q > k,

so that ^S(*D,) C Ker(res.).

Remark 1.2. In particular, for q = k we have a short complex k

£ nS(*A) fiS(log C) —+ Mc = MC,

i=i

where n: C —► C is the normalization of C. It was observed by K. Saito that if k =1 and C = D is a hypersurface which is a normal crossing outside a subvariety of codimension at least 2, then the image Im(res.) coincides with ); it consists of the so-called weakly

holomorphic functions on D, that is, meromorphic functions whose preimage becomes holo-morphic on the normalization (see [26], (2.8)). Some interesting properties of the class of weakly holomorphic functions have been discussed in [28].

Lemma 1.3. The image res. ^S(log C), q ^ k, of the residue map are Oc-modules. Proof. The definition of multi-logarithmic forms implies that

k

hj • (fiS,o(logC)/£ns(*A)} =0, j = 1,...,k.

^ i=1 '

Hence, the ideal (h1,..., hk ) C OS annihilates Im (res. ). □

Remark 1.3. The requirement m) of Proposition 1.1 is equivalent to the following condition:

Aj1...jfc • u € dh1 A ... A dhf A + £ nu(*A), (ji, . . ., j) G [1, . . ., m],

= 1

for any minor Aj1...jk of maximal order of the Jacobian matrix Jac(h). That is, these minors can be considered as universal denominators for complete intersections in the usual sense.

2. Regular Meromorphic Differential Forms

Let X be an analytic reduced subspace of a complex manifold S, let dim X = n, and dim S = n + k. The OX0-module

=ExtOs,0(Ox,o, ^)

is said to be the Grothendieck dualizing module of the germ (X, 0).

Definition 2.1. The sheaf wX, q ^ 0, of regular meromorphic differential q-forms is locally given as follows:

wX,0 = HomOx,0 , wX,0);

equivalently, the sheaf wqx o consists of all the meromorphic forms w of degree q on X such that w A n € w\ o for any n € fiX o •

We shall study the case when X is taken to be the reduced germ of a complete intersection C = D1 n • • • fl Dk, where D1,..., Dk is a set of hypersurfaces defined by the equations hj(z) = 0, j = 1, • • •, k, k ^ 1, respectively (in the notation of Section 1). In this case,

wn = Extk (n n"+fc) ~ n ( dzi A ••• A dZn+k \ W°,0 = EXt°s,o(Oc,0, QS,0 ) = Oc,0 ^ dhi a ... A dhk )

is a free module of rank one, and

wq,0 = HomOc,0(OC;0q, Oc,0), 0 < q < n.

Here are some properties of regular meromorphic differential forms:

1) wqC = 0 for q < 0 and for q > dim C;

2) the de Rham differential d is extended to the modules wC, 0 < q < n, so that (wj, d) is a complex;

3) in the case when C is normal, the sheaves wC have the following well-known interpretation:

wq = j i*nc

wC = j*j 'LC,

where j : C\ Sing C —> C is the canonical embedding. More precisely, in such a case, for any q > 0 there is an exact sequence of local cohomology groups with supports in Z = Sing C :

0 HZ(fiC ) wC Eh(Qqc) 0.

Let k =1, so that C = D is a hypersurface. Then both the module w°D 0 and the Od-module res. fiS0(log D) contain all the germs of locally bounded meromorphic functions on D (cf. [4], §2, ex. i) ). Moreover, these two sets coincide if the local fundamental group of the complement S \ D is abelian (see [24]).

Theorem 2.1 ([4]). There is the following exact sequence of Oc-modules

0 wqc ExtOS (Oc , fi|+k) — (ExtOS (Oc , fi|+k+1)) k, q > 0,

where wC C j*j*fiC, the map C is multiplication by the fundamental class of C in S, and E (e) = (e A dh1, .. ., e A dhk).

Thus C(v) corresponds to a Cech cocycle w/h such that w = v A dh. It is easy to see that

wC = ExtOS (OC, Qns+k)•

In particular, wqC is isomorphic to the Grothendieck dualizing module of the n-dimensional complete intersection C C S. In conclusion, we establish a remarkable property of regular meromorphic differential forms in terms of the trace map associated with the Noether normalization. In fact, this property in a slightly different context has been proved by D. Barlet

(see [4]).

Theorem 2.2. The Oc,o-module w^ o, q > 0, consists of all the meromorphic forms w of degree q on C satisfying the following condition: for any n € ^Co', the trace

Trace C/Cn(w A n)(z) = ^^ (w A n)(x)

xEn-1 (z)

is the germ of a holomorphic form on C.

Proof. Taking the Noether normalization of the germ C, one can suppose that the germ (S, 0) is isomorphic to the direct product (Cn, 0) x (Ck, 0) with coordinates (z,w) so that the natural projection n: (C, 0) ^ (Cn, 0) is a finite covering. Let w € wC o, so that for any n € ^C o' one has

a

w A n =

dhi A • • • A dh'

7C,o,

where a € . By definition, one can view the differential form w A n as an element of a Cech cochain in the open set {A(z, w) = 0} :

y(z,w) , w A n = —-dz, A

where y is a holomorphic function and A is the determinant of the Jacobian matrix of the sequence of functions hi(z, w), .. ., h'(z, w) with respect to w, that is,

A = det

Thus

dwj

Trace C/Cn(w A n)(z) = ^^ y(z, w£(z)) dz.

A

t

It is well-known (see [29], Ch.1) that the trace of the meromorphic function y/A, whose denominator is equal to the Jacobian of the map n, is holomorphic. The proof of this fact is based on the integral representation of the trace as

/ 1 \k f y(z, w)dw

V2nv/-I/ ./r hi(z, w) ••• h'(z, w)' where the integrand is holomorphic on the parameters z and

r = {w € Ck : |h1(0,w)| = ... = |hk(0,w)| = e}

is a compact cycle.

To prove the converse, for simplicity, we consider the case n = 1, so that C is a complete intersection curve. The general case can be studied in the same way. Let v € ^So, and

( v N . . ,

suppose that Trace y — is holomorphic, for any holomorphic function y € Oco. Then

V 9/ ,

one has to prove

y

dhi A ... A dh'

C

C

that is,

v A dh = y9 + hifi + ... + hfc /, (5)

where /i,..., / € Os,o are holomorphic functions. By assumption,

Trace ( y — ) (z) = yV(z, w£(z)) V 9/ V 9

is holomorphic for any y. Let us consider the Grothendieck residue

yv A dhi A ... A dh'

Rh,g [yv A dhi A ... A dhfc] = J

re„, 9hi ... hfc

where re,a = {(z,w) € (S,0) : |hi| = e,..., |h'| = e, 191 = 5}. The local variant of the Hilbert Nullstellensatz shows that for some N > 1, one can write

zN = qihi + ... + qfc hfc + Q9.

Therefore after applying the residue transformation formula one obtains

[yv A dhi A ... A dhfc] = [Qyv A dhi A ... A dhfc],

where r = zN. The contour of integration in the latter integral

7M) = {(z,w) € (S, 0): |hi| = e,..., |hfc | = e, |z| = 5}

is fibered under the projection n over the circle |z| = 5. Hence, the integration with respect to w along the classical formula of the logarithmic residue yields

[yv A dhi A ... A dh'] =

f Trace (yv?) (z) = ' f TVace (y9N („,

./|z| = 5 V rJ 2^-1 J |z| = 5 V 9/

since the restrictions of q/r and 1/9 to C are equal. By our assumption the last integral is equal to zero, hence the local duality theorem of Grothendieck implies (5). This completes the proof in the case n = 1. □

3. The Poincare Residue Map

First recall the definition of the classical Poincare residue, following [11]. Let w be a meromorphic differential m-form on an m-dimensional complex analytic manifold S with a polar divisor D C S. Thus, locally we have a representation:

/(z)dzi A ... A dzm

w = hz .

By definition, the Poincare residue res_D(w) is a meromorphic (m — 1)-form on D whose singularities are contained in the singular locus Sing D C D. To define this form explicitly, let us note that at each point x € D \ Sing D one of the derivatives does not vanish:

dh . . ,

dz (x) = °.

Hence, locally one can set

f (x)dzi A . .. A dzi A .. . A dzm

resD (w) = ( —1)m

S (x)

D

It is not difficult to verify that this expression does not depend on the index i, on the local coordinates, and on the defining equation of D. Moreover, the Poincare residue resD (w) is holomorphic on the complement S \ D. When D is smooth, one can take h(z) = zm, then

, (f (x)dzl A ... A dzm\ ,, a . t

resD - = f (x)dz1 A ... A dzm-1

\ zm I

coincides with the usual residue.

For brevity, the Poincare residue resD(w) will be denoted by res(w). When D C S is a nonsingular hypersurface, the Poincare residue induces an exact sequence

res

0 —► O^ —► O^D)-► Om-1 —► 0 ,

where Om(D) denotes the set of meromorphic forms on S having a pole of the first order along the divisor D. In particular, it follows that the germ of every holomorphic (m — 1)-form on D is a Poincare residue. It is obvious that this is true globally when the first cohomology group vanishes: H 1(S, OSS) = 0.

More generally, following J.Leray (see [13]) one can apply the above construction to meromorphic differential q-forms on S with a smooth polar divisor D C S for any 1 ^ q ^ m, and write locally for any d-closed differential q-form w on S \ D, having poles of the first order along D, the following presentation:

dh

w = -r A £ + n, h

where £ and n are holomorphic. The restriction of £ to D is a holomorphic form independent of the choice of the local equation h = 0 of D; we denote it by res(w) as before. As a result, one gets (see [20]) an exact sequence

res

0 —► OS —► OS(D) -► OD-1 —> 0, 1 < q < m,

which is equivalent to the sequence

res

0 —► OS —► OS (log D) -► OD-1 —► 0 , (6)

since the divisor D is a smooth hypersurface.

Remark 3.1. As it has been already remarked by de Rham and Leray (cf. [13]), for a meromorphic q-form w with poles of the first order along D the above definition gives us the holomorphic (q — 1)-form res(w) in cases when the singular locus Sing D of the divisor D consists of isolated double quadratic points only and either q < m, or q = m, and the coefficient of m-form h ■ w vanishes on Sing D. This phenomenon can be readily explained by considerations which we use in the next and previous sections.

An extension of the exact sequence (6) to the case when the divisor D has arbitrary singularities was described in [1]

res.

0 —► ^S —► ^S(logD) -+ wD-i —► 0, q > 1. (7)

The following theorem is devoted to further generalization of this construction to the case of complete intersection.

Theorem 3.1. Let C C S be a reduced complete intersection. Then under notations of Section 1 there is a short exact sequence

k

res

£ ^S(*A) —► (log C) -+ w£Tk —► 0, q > k,

=1

and a natural isomorphism of Oc -modules

res. ^S(log C) = , q ^ k.

Proof. The proof goes similarly to [1], §4. Thus, it is sufficient to verify our statements locally. Remark 1.3 implies

Дл...л • res. ^(logC)|u С ^rk|cnu,

for all maximal minors Д^...^, (ji, ...,jk) € [1,...,m] of the Jacobian matrix Jac(h). Since о — Oc,o(dzi Л ... Л dzn+k/dhi Л ... Л dhk), the definition of the modules of regular meromorphic differential forms implies immediately that

res. ^S(log C) С .

Let now K.(h) be the usual Koszul complex associated with the sequence h = (hi,..., hk) :

dk-l di

0 ^ Os,o(eo Л ... Л ek-i) -► ... -► Os,o(eo) + ... + Os,o(ek-i) ^

do d-i

-► Os,o-► Oc,o ^ о ,

where

Kk(h) = OS,o(eo Л ... Л ek-i), ..., Ki(h) = Os,o(eo) + ... + Os,o(ek-i),

and Ko(h) = Os,o, do(ej) = hj+i, i = 0,..., k — 1, d-i(1) = 1. From the dual exact sequence one obtains, by definition,

ExtOso(Oc,o, ^S+oi) =HomOs,0(K(h), ^/^(Homo^(Kk-i(h), ^+1)).

This means that any element of ExtOso (Oc,o, ^S+j1 ) can be represented as a Cech (k — 1)-cochain (more explicitly, a (k — 1)-cocycle)

€ Homos (Kk(h), ^ — Cfkk)i(^S"+oi),

hi---hk € os,ov"s,o ) — ^(k) ^'s,o

where v G . Choose now an element v G ^S+O such that

h-i ■ ■ - hi

A dhj G ExtOS 0 (Oc,o , npo2), j = 1,...,k,

corresponds to the trivial element. This means that for any j = 1,... ,k the form v A dhj/h\ ■ ■ ■ hk, is defined by an element of dk-1(HomoS 0 (Kk-1(h), ^s+q)) . This yields that

v A dhj G ^ hiQ

q+2 S ,

j = 1,...,k,

or, equivalently,

1 A dhj G^^ QS+1 (*A), j = 1,...,k, where u =

hi■■ ■ hk '

In view of Proposition 1.1 this means that u G ^S0(log C). Now we shall use the notation and exact sequence of Theorem 2.1. Set P = C-1(v/hi ■■■ hk). Then C(v) corresponds to the Cech cocycle v/hi ■ ■ ■ hk such that v = v A dhi A ■ ■ ■ A dhk (take v = P, w = v in the description of u^C in terms of multiplication by the fundamental class of C in S). This implies

hi ■■ ■ hk

dhi dhk P A—1 A ■ ■ ■ A—1, hi hk

and res.

hi hk

Thus, for any element P G uiqc k there is a preimage under the multi-logarithmic residue map represented by v/hi ■ ■ ■ hk. □

v

v

v

v

v.

4. On Residues of Multi-Logarithmic Forms and (9-closed Currents

We first remark, that in view of Theorem 3.1, the sections of u}*c can be considered as "regular meromorphic forms." More precisely, in complex analysis "regular" or "holomorphic" differential forms are usually described as (9-closed forms (cf. [14]).

The key idea and further considerations in this section are closely related to the methods developed in [28], where the theory of weakly holomorphic functions is studied. Moreover, they lead to another proof of the statement of Theorem 3.1 concerning the representation of sections of Barlet sheaves u)*c as residues of multi-logarithmic differential forms. The main result of this section is the following theorem.

Theorem 4.1. There is a natural one to one correspondence between the sets of residues of multi-logarithmic forms and meromorphic d-closed currents on the complete intersection C = D1 n ... n Dk.

The proof will be given below. We start with the definition of the integration current associated with a meromorphic differential form, and recall the basic properties of residue currents.

Let — = £/g be a meromorphic differential form on a reduced analytic subset C of a domain U C Cm; here £ is a holomorphic differential form and g is a holomorphic function. Such a form defines the following current

y

.9. C

y

lim f - A y, y G D*(U),

(8)

where y is a test ^^-differential form with compact support on U, and the real dimension of £ is equal to the sum (deg y + deg —) of degrees of y and —. This is an integration current on C in the principal value sense.

A residue current Rh associated with a holomorphic map h = (hi,..., h^) : U ^ Ck is a functional on the space D*(U) of C^-differential forms with compact supports on U. It is defined as follows (see [6] or [29]):

Rh(y) = lim /

y

y gD

,2m — k

(U ),

/Te(h) hi • • • h' ' where the integral is taken over the tube

Te(h) = {z e S : |hj(z)| = £j, j = 1,...,k}

with a fixed orientation. Here the approach of the radii e = (ei,..., £&) to zero is required to be along a so-called admissible path: a path e = e(r) is called admissible, if e^ ^ 0 as t ^ 0 and for each j = 1,..., k — 1, the coordinate ej (t) approaches zero faster than any power of the next coordinate ej+i(T).

The principal value of the residue current Rh with respect to a function g: U ^ C is defined as the limit

y

Pg Rh(y) = lini Î

Its (h)n{|s|>5| gh1 • • • hk

along the admissible path (e(r), J(t)) : [0,1] ^ Rfc+1.

When the set h-1(0) = {h1 = ... = hk = 0} is a complete intersection of codimension k then

Rh = d

1

hi

1

"1" " 1 " " 1 "

d A ... A d

.9. .hi. ,hk.

A ... A d

Respectively, when g ^ 0 on every irreducible component of the analytic set h-i(0) then

Pg Rh

When w = a/hi • • • h' is a meromorphic differential form of degree i then there is the current

Rh[w](y):= Rh(a A y), y eD2™-'-1 (U). Analogously for w = a/g•hi • • • h' we set

PgRh[w](y) = PgRh(a A y). Let us assume that the zero set defined by the system of equations

{g = hi = ... = hfc =0}

is a complete intersection. Then the residue currents Rh and their principal values Pg with respect to g have the following properties:

1) RhM = PgRhM;

2) Pg Rh

dhi

A ... A

-

g. _g_ C _g_

A [C];

where [C] is the integration current (a holomorphic chain) on C. In particular, when £ and g are identically equal to the constant 1, then

Rh

dh- dhk IT A ... A -h-h- h-

= dht A ... dhk A d

1

h-

A... Ad

1

hk

= [C ];

3) hj Pg Rh =0, j = 1,...,k;

4) If U is a domain of holomorphy (or a Stein manifold) then for v G Ou one has

vRh = 0

G (h-,..., h-)Ou.

In particular, this property is valid in the local rings Ou,x, x G U.

Properties 1) and 3) are proved in [6], Theorem 4.4.1 and Theorem 1.7.6, respectively, property 2) is obtained in [27], Theorem 2.1; property 4) is obtained in [19], Corollary 6.1.2. Proof of Theorem 4.1. Let v G Q,S+0k (logC), so that

a dhi dhk

v = T-r~, and gv = —— A ... A —— A £ + n,

hi •• • hk hi hk

where a, £ are holomorphic and n satisfies the condition iii) of Proposition 1.1. In view of property 1) one has

RhM = Pg Rh[g^] = Pg Rh

dht dhk £ —1 A ... A —- Ah- h- g_

+ Pg Rh

The second summand corresponds to the current which is trivial by property 3). Property 2) implies

" £"

RhM

g

Hence, the current

_g_ C

£

defined by the residue res. v = -

g

of the multi-logarithmic form

w coincides with the residue current RhM which is <9-closed (see [6]).

Conversely, under the same assumptions, let £/g be a meromorphic form on C, and the corresponding current (8)

V

J

_g_ C

V

lim f - A V, V G V(U)

JCn{|g|>e| g

is d-closed. Then in virtue of property 2)

0 = d =<

.g. C _g.

A [C] .

k

v

C

Since <9[C] = 0, applying the Leibnitz rule and property 2), one obtains

0 = d

Property 4) yields that is,

A [C] = d

A dhi A ... A dhf A d

1

hi

A ... A d

1

hfc

dh1 A ... A dhf A £ G (g, h1,. .., hf)^U+0f,

dhi A ... A dh* A £ = ag — hi ni — ... — h* n*,

(9)

where a, ni,..., ^fc € are holomorphic forms in the domain U C S. Here property 4)

is used in computing the coefficients of the form £ A dh. Let us consider the meromorphic form w = a/hi • • • h^. We have

g •w = g •

dh A £ + hini + ... + hfcnf

hi • • • hf

hi hf

dhi dhf

A ... A —* A £ + n, hi hf

where

n=

E

ni

hi hf

G E<0*(*A).

-| 1 ' "k - -i

This completes the proof of Theorem 4.1. □

One can deduce the main statement of Theorem 3.1 from the following theorem.

Theorem 4.2. A meromorphic differential q-form on the complete intersection C = Di O ... O Dk defines a meromorphic d-closed current on C if and only if it is a section of the Barlet sheaf w^ of regular meromorphic forms.

Proof. Let w = £/g be a meromorphic q-form C. It is not difficult to see that

d(w) = 0 ^^ d(w A v) = 0 for any v G .

Making use of property 4) and of similar considerations of the converse part of the proof of Theorem 4.1 we conclude that the property of <9-closedness of w is equivalent to the existence of the relation dhA(£ A v) = ag — hini —... — hknk, where a, ni,..., nk G n™ are holomorphic forms in the domain U C S. This is equivalent to the condition

£ A v g

dhi A • • • A dh*

G wn

£

Hence, by definition, w = — G w^ as required. □

Remark 4.1. This proof is based on a similar idea from [14], Theorem 2.

Remark 4.2. Thus, any section of the Barlet sheaf w^ is the residue of a multi-logarithmic form. Such meromorphic forms on the complete intersection C are characterized by the following two conditions:

a) they are holomorphic on the set of non-singular points of C (cf. [13]),

a

a

c

c

b) they have no residues with respect to the singular locus of C (this is the so-called d-closedness property).

A differential form ^ which satisfies condition a) is not a priori meromorphic. Nevertheless it should define the following integration current (see [27]):

b], p) = lim / ^ A p, (10)

where dh = dhi A ■■■ A dhk. This integration current differs from (8) by the choice of the "principal value" of the integral. Recall that under our assumptions the singular locus Sing C of C is defined by the condition dhi A ■■■ A dhklc = 0. Then a natural question arises: Whether the class of differential q-forms ^ which are holomorphic on the non-singular part of C and define the d-closed currents by means of formula (10), and the class of sections of Barlet sheaf wqc coincide?

In connection with this problem it should be noted that there are non-exact d-closed currents corresponding to non-holomorphic differential forms (see [9]).

If one supposes that ^ is meromorphic then Proposition 1 of [27] asserts that it is true. Thus one gets the following elegant description of the Grothendieck dualizing module on a reduced complete intersection in terms of d-closed currents. We remark that a priori all sections of the dualizing module are meromorphic on C and holomorphic on the non-singular part of C.

Corollary 4.1. Any element ^ of the Grothendieck dualizing module of the reduced complete intersection C = D\ O .. . O D¡~ defines a meromorphic d-closed current on C by formula (10) and vice versa.

In conclusion we pose a problem which is a stronger version of the above question. Let be the set of differential q-forms ^ on U such that they are holomorphic on the non-singular part of C and satisfy the following condition (cf. formula (10)) :

( p ) = lim / ^ A p = 0.

(11)

Conjecture 4.1. The classes of differential q-forms defined by the set and by the sections of the Barlet sheaf Wq coincide.

Remark 4.3. It is well-known that any pure dimensional analytic set can be regarded locally as a component of some reduced complete intersection (see [10], p.52). In fact, this leads to a generalization of the notion of the multi-logarithmic differential form with respect to an arbitrary pure dimensional analytic set.

The work was supported in part by the Russian Foundation of Basic Research RFBR and by the Japan Society for Promotion of Science JSPS in the framework of the joint project "Geometry and Analysis on Complex Algebraic Varieties" (project No. 06-01-91063) and by the grant of the Siberian Federal University in the framework of HM project No. 45.2007. The first author was partially supported by the International Association INTAS (project No. 05-96-7805) The second author was partially supported by the Russian Foundation of Basic Research RFBR (project No. 08-01-00844)

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