CC BY
541
y^K 515.17
RESIDUES OF LOGARITHMIC DIFFERENTIAL FORMS A.G. Aleksandrov
Institute for Control Sciences, Russian Academy of Sciences,
Profsojuznaja str., 65, Moscow, 117997, Russian Federation, e-mail: ag_aleksandrov@mail.ru
Abstract. In this note we give an elementary introduction to the theory of logarithmic differential forms and their residues. In particular, we consider some properties of logarithmic differential forms related with properties of the torsion holomorphic differentials on singular hypersurfaces, briefly discuss the definitions of residues due to Poincare, Leray and Saito, and then explain an elegant description of the modules of regular meromorphic differential forms in terms of residues of meromorhic differential forms logarithmic along a hypersurface with arbitrary singularities.
Keywords: logarithmic differential forms, residue-forms, residue map, regular meromorphic differential forms, torsion holomorphic differentials.
Introduction
From the historical point of view, the concept of logarithmic differential form had its origin in the classical theory of residues. The term "residue" (together with its formal definition) appeared for the first time in an article by A.Cauchy (1826), although one can find such a notion as implicit in Cauchy’s prior work (1814) about the computation of particular integrals which were related with his research towards hydrodynamics. For the next three-four years, Cauchy developed residue calculus and applied it to the computation of integrals, the expansion of functions as series and infinite products, the analysis of differential equations, and so on ...
Though it was already transparent in the pioneer work of N.Abel, a major step towards the elaboration of the residue concept was made by H. Poincare who introduced in 1887 the notion of differential residue 1-form attached to any rational differential 2-form in C2 with simple poles along a smooth complex curve. Subsequently E. Picard (1901), G. de Rham (1932/36), A.Weil (1947) obtained a series of similar results about residues of meromorphic forms of degree 1 and 2 on complex manifolds; such developments were associated with cohomological ideas, leading to the formulation of cohomological residue formulae. Such cohomological ideas were later pursued by G. de Rham (1954) and J.Leray (1959) who defined and studied residues of d-closed C^ regular differential q-forms on S \ D with poles of the first order along a smooth hypersurface D in some complex manifold S, q > 1.
In 1972 J.-B. Poly [24] proved that Leray residue is well determined for any (not necessarily d-closed) semi-meromorphic differential forms u as soon as u and du have simple poles along a hypersurface.
In fact, for the first time these two conditions were considered by P.Deligne [11]; he introduced the notion of meromorphic differential forms with logarithmic poles along a divisor, normal crossings of smooth irreducible components. In such context this notion was extensively studied in algebraic geometry and in differential equations by many authors (for example, by Ph.Griffiths,
A.G. Alexandrov partially supported by the Russian Foundation of Basic Research (RFBR) and by the National Natural Science Foundation of China (NSFC) in the framework of the bilateral project "Complex Analysis and its applications"(project No. 08-01-92208)
J.Steenbrink, N.Katz). As a result in 1975, Kyoji Saito [25] considered meromorphic differential forms satisfied these conditions in the case of divisors with arbitrary singularities. Somewhat later, his note has been published in a volume [26] of the RIMS-publication series, which is not accessible to many of those interested in the subject. Saito established the basic properties of logarithmic differential forms and studied some applications to computing Gauss-Manin connection associated with the minimal versal deformations of simple hypersurface singularities of types A2 and A3. In 1980 a paper by Saito [27] was published; it contains an essential part of materials of the above mentioned works. Among other things, in this paper a general notion and important properties of residues of logarithmic differential forms are discussed in detail.
At present time the theory of logarithmic differential forms is exploited fruitfully in various fields of modern mathematics. Among them, one can mention the following:
complex algebraic geometry (the cohomology theory of algebraic varieties and Hodge theory [12], [10], [29], etc.),
topology and geometry (the theory of arrangements of real and complex hyperplanes [21], [7], the fundamental group of the complement of a singular hypersurface [19], etc.),
the theory of singularities, the deformation theory and the theory of Gauss-Manin connexion [26], [4], etc.,
the theory of D-modules, the microlocal analysis, the theory of differential equations [11], [22], the theory of flat coordinate systems [28], etc.,
complex analysis (the theory of Abel’s integrals [15], Torelli theorems, the theory of primitive forms and their periods [16], etc.),
the theory of special functions (generalized hypergeometric functions [12], etc.), mathematical and theoretical physics (the theory of Frobenius varieties and the topological field theory [20], etc.)
Of course, this list is quite incomplete and can be easily extended by the specialists in related fields of mathematics.
Following our previous work [3] in this note we give an elementary introduction to the theory of logarithmic differential forms and their residues. In Section 1 we recall the basic notations, definitions and properties of logarithmic differential forms along a reduced hypersurface in a complex analytic manifold. In Section 2 we consider some relations of logarithmic differential forms and torsion holomorphic differentials on singular hypersurfaces. In the next sections we briefly discuss the definitions of Poincare, de Rham, Leray and Saito residues, and apply the theory of regular meromorphic differential forms to the case of singular hypersurfaces. Among other things, we obtain a highly elegant description of these modules on an arbitrary singular hypersurface D in terms of residues of logarithmic differential forms.
1 Logarithmic differential forms
Let U be an open subset of Cm, and let D be a hypersurface defined by an equation h(z) = 0, where h(z) = h(z\,...,zm) is a holomorphic function in U, and zi,...,zm is a system of coordinates. Suppose that D is reduced, that is, h(z) has no multiple factors.
Definition 1.1 ([25], [27]) A meromorphic differential q-form u, q > 0, on U is called logarithmic (along a divisor D) if u and its differential du have poles along D at worst of the first order. It means that hu and hdu are holomorphic differential forms on U.
Remark 1.2 In fact, for the first time the above two conditions appeared in a work by Deligne (see [11], Prop. 3.2, (i), p.72) who studied meromorphic differential forms with logarithmic poles along divisors with normal crossings (thus, such a divisor D is the union of its smooth irreducible components).
In practical computations, the second condition is usually replaced by the condition “dh Au is a holomorphic differential form on U”; both conditions are equivalent, in view of the identity d(hu) = dh A u + h ■ du.
Let S be an m-dimensional complex manifold, and let ^S = , d)q=0 1 be the de Rham
complex of germs of holomorphic differential forms on S, whose terms, locally at the point x G S, are defined as follows:
^S,x = Os,x(... ,dz^ A ... A dziq,...), (ii,... ,iq) G [1,m].
Let D be a reduced hypersurface of S, and let h = 0 be an equation of D, locally at the point x G D. A meromorphic q-form u is logarithmic along D at x, if hu and hdu are holomorphic. We denote the OS,x-module of germs of logarithmic q-form at x and the corresponding sheaf of logarithmic differential q-form on S by ^Sx(logD) and Qqs(log D), respectively. Thus, the OS-module ^S(log D) is a submodule of ^S(*D), consisting of all the “differential forms with polar singularities along D.” Obviously, the sheaves ^S(log D) and ^S coincide off the divisor D, for all q > 0. By definition,
fiiyiogB) - n*, as Os,„ tt&OoeD) = in^.
In what follows, when we consider the local situation the point x will be taken to be 0 for simplicity. We shall also assume that U is an open subset of Cm containing the origin.
Example 1.3 Suppose D C U be a hyperplane or, more generally, a smooth hypersurface defined by the equation zi = 0. Then
dz
QsoihgB) = 0s,o(— ,dz2l... ,dz„
zi
is a free OS,0-module of rank m, generated by the forms dz1/z1, dz2,..., dzm. Moreover,
^S,o (log D) - A nS,o (log D), 1 < q < m.
Example 1.4 More generally, let us consider the case when D is the union of k < m coordinates hyperplanes in S = Cm. In other words, D is a strong normal crossing. This case considered in many works published before Saito’s preprint [25]. Then the defining equation of D is written as follows: h = z1 ■ ■ ■ zk = 0, and an easy calculation shows that
fUo(logD) - OsJ — ,dzk+\,..., dzm \ zi zfc
and for all 1 < q < m there are the following isomorphisms
^S;o(l°g D) = AnS,o(log D).
Thus, nS,0(log D) is a free OS,0-module of rank .
The following statement is a direct consequence of the basic definition (see [1], or [2], §1).
Claim 1.5 Let D C U be a reduced hypersurface defined by the equation h = 0. Then for any q > 1 there exists a natural isomorphism of OS,0-modules
^5.0 f| л «£>’) = dh Л П1-; (log D).
Proof. Let us remark at first that there is a natural inclusion
fiS.0 f| ((dh/h) A n»-1) dh A OS-,1 (log D).
If an element u G OS0 belongs to the OS,0-module on the left side, then it can be represented in the form u = (dh/h) A n for some n G OS-,1. Hence, by definition,
(n/h) G nS"01(log D) ^ u G dh A nS"01(log D),
and we obtain the desirable inclusion. On the other hand, h■ nS"01(log D) C nS"01. Multiplication by Adh induces the map
dh
dh A Ql^ilogD) —> —A Qq-0\
Obviously this gives us the inverse map to the first inclusion. This completes the proof of Claim.
Lemma 1.6 ([27], (1.1), iii))) Let u be a meromorphic q-form on U, q > 0, and let D C U be a hypersurface as above. Then u is logarithmic along D if and only if there exist a holomorphic function g defining a hypersurface V C U, a holomorphic (q — 1)-form £ and a holomorphic q-form n on U such that
a) dim C D n V < m — 2,
dh
b) gu= — A £ + )].
Proof. For simplicity let us consider the case m = 2. Suppose that u is a logarithmic q-form. Then we have
a1dz1 + a2dz2 A h1a2 — h2a1 . def,, , , A ,
cj =-------------, dh Auj = -----------dz\ A dz2 = b(z)dz\ A d^2,
hh
where a1,a2 and b(z) are holomorphic, and hi = dh/dz^, i = 1, 2. Further,
h|aidzi + h1a2dz2 h
h1a1dz1 + h2a1dz2 h1a2 — h2a , dh
----------;-----------1------;-----dz2 = -r- A cf-i + b(z)dz2.
h h h
It means that
h
There is analogous representation for h'2u, and hence for any gu, where g G J(h) = (h1,h2), the Jacobian ideal of h. Since D is reduced, there is a function g G J(h) defining a non-zero divisor in OD/(h) as required in the condition a).
Conversely, the relation b) implies that
£ n
huj = dh A —I—, gg
that is, hu and dh A u are holomorphic in codimension > 2, hence, in virtue of the Riemann extension theorem, they are holomorphic everywhere. This completes the proof when m = 2. The general case can be considered analogously.
Corollary 1.7 ([25]) With the preceding notations the following conditions are equivalent:
1) u G OS (log D),
dh dh
2) —— cj G — A f^-1 + for all i = 1,, in.
dz» h
Corollary 1.8 The sheaves OS (log D), q = 0,1,...,m, are OS-modules of finite type; the direct sum ®™0nS(logD) is an OS-exterior algebra closed under the exterior differentiation d.
As a consequence, OS (log D) are coherent sheaves of OS-modules for all q > 0.
2 Torsion differentials
In this section we consider simple relations between logarithmic differential forms and torsion holomorphic differentials on hypersurfaces with singularities. By definition, OD,0 = OS;0/(h)OS;0, and ( )
nD,0 = nS,0/(h ' nS,0 + dh A nS,0 ), q > 1.
Thus, nD,0 is the OD 0-module of germs of holomorphic differential forms on the hypersurface D at the point 0 G D. The module nD,0 is usually called the module of Kahler regular differentials. The standard differentiation d induces the action on nD,0 denoted by the same symbol. Thus, the de Rham complex of sheaves of germs of holomorphic differential forms on D is well defined: ( )
nD = K ,<*) q=0,1....'
For completeness, recall the notion of torsion. Given a commutative ring A with the total ring of fractions F, and an A-module N of finite type, we consider the kernel of the canonical map i^ N ®A F, the torsion submodule of N, and denote it by Tors N; it consists of all the elements of N which are killed by non-zero divisors of A.
It is well-known that torsion differentials Tors OD 0 play a key role in analysis of topology and geometry of singular varieties. In the case of an isolated n-dimensional singularity (D, 0), the torsion modules TorsOD0 are trivial for all q = 1,..., n — 1, while TorsOD0 is a finite dimensional vector space. Furthermore, if D is the quasi-homogeneous germ of a hypersurface or complete intersection with isolated singularities then dim CTorsOD 0 = ^, where ^ is the Milnor number of D; it is a very important topological invariant of the singularity.
The following examples show that generators of the module of logarithmic differential forms are naturally expressed through torsion differentials on D.
Example 2.1 Suppose S = C2 and consider the hypersurface D given by the equation h = xy = 0. It is a plane curve with a node. In other words, it is an A1-singularity, a very particular case of strong normal crossing from Example 1.4. Then
nyiogD) * ojdA *). nyiogD) * 0 J^a)
\ x y / \ xy /
are free OS,0-modules of rank 2 and 1, respectively. In this case there is also the following representation: nS0(logD) = OS,0(dh/h, 9/h), where 9 = ydx — xdy. It is not difficult to verify that 9 G TorsnD,0. Indeed, taking a non-zero divisor (x + y) G OD,0 one obtains the following identities in nD,0 :
(x + y) ■ 9 = xydx — x2dy + y2dx — xydy = — (x — y)dh + 2h(dx — dy) = 0.
Moreover, in this case, Tors OD,0 = OD,0(9) = C(9), ^ =1.
Example 2.2 (cf. [30]) With the preceding notations let D C S be a plane curve with a cusp given by the equation h = x2 — y3 = 0. In other words, it is an A2-singularity. Easy calculations show that
nio(io**;**). ^d)SOm(^)
are again free OS,0-modules of rank 2 and 1, respectively. Notice that the numerator of the second generator of nS0(logD), the differential 1-form 9 = 2ydx — 3xdy, represents an element of the torsion submodule Tors OD,0 C OD,0. Indeed, in our case A = OD,0 = C(t2,t3), N = nD0, F = C(t), and the mapping i is given by the normalization of D, that is, x = t3, y = t2. Thus, i(9) = i(2ydx — 3xdy) = 2t2dt3 — 3t3dt2 = 0, that is, 9 G Ker(i) = TorsOD,0. Equivalently, take a non-zero divisor x G OD,0. One then obtains x-9 = 2xydx — 3x2dy = 5hdx — 3xdh = 0 in nD,0 = nS;0/(h-nS;0 + dh A OS,0). Further calculations show (cf. [30]) that Tors OD,0 = OD,0(9) = C(9, y■ 9), that is, ^ = 2.
Proposition 2.3 ([1]) For q = 1,..., m, there are exact sequences of OS,0-modules
0 —- OS^/h ■ «S-Zoos D) — nS,0/h ■ nS.0 —- «D,0 — 0,
0 —> OS0/dh A OS-)1 (logD) —— OS0/dh A OS"-1 —* OD 0 —* 0,
0 — ^',0 + y A «S1 — ^oOog °) Tors Woo — o,
where the homomorphisms of exterior and usual multiplication are denoted by Adh and by -h, respectively.
Proof. The exactness of the first and second sequences follows directly from the basic Definition 1.1. Let us consider a differential q-form u G OS-)1 represented an element of the quotient OS-iVh-OS'o^logD). Suppose dh A u = h■ n, n G OS 0, and set u = u/h. It is obvious that hu and dh A u are holomorphic, hence u G OS-)1 (log D) by definition. Thus the first sequence is exact from the left. Evidently it is exact from the right too. In the same way, one
can easily prove the exactness of the second sequence. The exactness from the left of the third sequence follows from definition. In view of Lemma 1.6, it is clear that Im(-h) C TorsOD 0 because for a non-zero divisor g one has the following chain of implications:
dh
gu = — A £ + ?7 =>- g(hu) = dh A £ + hi] = 0 =>• hu G Tors O,qD 0.
Now let take an element u G Tors OD,0. By definition, there is a non-zero divisor g G OD ,0 such that gu = 0. We will denote by g and u their representatives in OS,0 and OS 0, respectively. Then one has g-u = dh A £ + h-n, £ G OS'-1, n G OS 0. Since g is a non-zero divisor, the condition b) of Lemma 1.6 is satisfied. This implies that u/h = u G OS 0(logD), that is, u G Im(-h).
Remark 2.4 It is well-known [14] that Tors OD 0 = 0, 0 < q < c, where c = codim (Sing D, D) and Sing D is the singular locus of D. On the other side, any reduced hypersurface (or complete intersection) D is normal if and only if c > 2 by virtue of Serre’s criterion (“R1 and S2 conditions imply normality”). Hence, when D is normal then the exact sequence of Proposition 2.3 implies the following isomorphisms
dh
^l,o (l°g-^) = ^l,o + A ^l,o > I < q < c.
It is not difficult to see that the support of Tors OD is contained in the singular locus Sing D of the hypersurface D. Moreover, there is a system of generators of OD-module Tors OD containing at least m — 1 elements.
Corollary 2.5 There are the following long exact sequences of OS,0-modules
0 —>■ ^|,o + X A ^s,o > ^d,o ^D,o/Tors —► 0,
0 —► dh A Hf^log D) —► Qqso © ^ A &qs~Q —► Q|i0(log D) —^ Tors ^lqD 0 —► 0.
Proof. This is an immediate consequence of Proposition 2.3 and Claim 1.5.
Remark 2.6 The last sequence is very useful in computing the torsion modules in the case when OS 0(log D) is a free OS,0-module; it gives us an OS,0-free resolution of the torsion module. Following P.Cartier [9] a hypersurface D C S is called Saito divisor or, more often, Saito free divisor if for some q > 1 and, consequently, for all q, the OS-module OS (log D) is locally free. For example, the discriminants of the minimal versal deformations of isolated hypersurface or complete intersection singularities are Saito free divisors.
3 Poincare residue
The following construction [15] is a direct generalization of the original Poinacare definition of the residue 1-form associated with any rational 2-form in C2.
Let u 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:
f (z)dz1 A ... A dzm
u =------
h(z)
where f and h are germs of holomorphic functions, and h is a local equation of D. By definition, the Poincare residue resD (u) 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 G D \ Sing D at least one of the derivatives of h does not vanish:
dh
dzi
. = °.
Then the Poincare residue of u in a small neighbourhood of x is defined as follows:
%-i f (z)dz1 A ... A dzi A ... A dzm
resD (w) = (-1)r
dh(z)/dz^
D
It is not difficult to verify that this restriction depends neither on the index i nor on the local coordinates and on defining equations of D. Moreover, the Poincare residue is holomorphic on the complement S \ D. When D is smooth, one can take h(z) = zm, and then, as usually,
resD ^ ''' A = f(z)dzi A ... A dzm_i,
y zm I
that is, resD(u) is holomorphic on D. As a result one has the following sequence of sheaves
res
0 —► —► ft?(D)----------► Qm-1 —► 0 ,
where nm?(D) denotes the sheaf of meromorphic forms on S having a simple pole along the divisor D. In particular, one concludes that the germ of every holomorphic (m — 1)-form on the nonsingular divisor D is a Poincare residue. It is obvious that this is true globally when the first cohomology group vanishes: H 1(S, n?) = 0.
4 Leray residue-form
As remarked in Introduction De Rham and Leray considered d-closed C^ regular differential forms on S \ D having simple poles on D, where D is a submanifold of codimension 1 in a smooth manifold S. In particular, they proved that locally for such a form there is the following represenation:
dh
(*) U= ~h A^ + 'h
where £ and n are germs of regular differential forms on S. In fact, £|D is globally and uniquely determined; it is closed on D. If u is holomorphic on S \ D then the form £|D is holomorphic on D. The form £|d is called the Leray residue-form on D; it is denoted by res[uj. It is not difficult to see that the definition of the Leray residue-form generalizes the Ponacare residue described above.
Similarly to the construction from the end of the previous section, making use of local representation (*), for any q = 1,...,m one gets (see [23]) the exact sequence
res
° —► (D) —► ^D-1 —► °,
which is equivalent, since the divisor D is a smooth hypersurface, to the sequence
res
0 —► Q| —► Q|(log D) ----------► —, 0 .
Below we show that a generalization of this sequence to the case when the divisor D has arbitrary singularities requires more delicate considerations.
5 Saito residue map
In fact, Leray considered d-closed forms on S\D in order to construct a natural homomorphism of cohomology spaces Hp(S \ D) ^ Hp-1(D), and then the co-boundary homomorphisms of homology groups Hp-1(D) ^ Hp(S \ D), the main ingredient of his famous residue-formula.
Furthermore, in 1972 J.-B. Poly [24] proved that the representation (*) are valid for any semi-meromorphic differential form u as soon as u and du have simple poles along a smooth hypersurface D C S. By definition, a differential form u is called semi-meromorphic when locally all its coefficients can be represented as quotient of smooth and holomorphic functions. Hence, the Leray residue is also well determined for such forms without assumption on their d-closedness.
Following Saito [27] we describe a natural generalization of the Leray residue for meromorphic differential forms satisfying the above two conditions for a divisor D with arbitrary singularities, that is, for logarithmic differential forms in the sense of Definition 1.1.
Let D C S be a hypersurface, and let the sheaf Md be the OD-module of germs of meromorphic functions on D.
Definition 5.1 (see [27], (2.2)) The (logarithmic) residue morphism is a homomorphism of OS-modules
res. : QSS(log D) —► Md ®od ^D-1,
defined locally as follows: taking the representation of the basic Lemma 1.6, for any u £
Q| o (log D) we set
’ 1 t
res. to = -•£.
g
Thus, the residue res. u is the germ of the meromorphic (q- 1)-form in the module Md,0®oD 0 Qq—1
QD ,0 .
Claim 5.2 ([27], (2.5)) Let D C S be a hypersurface. Then for any q > 1 there exists the following exact sequence of OS-modules
0 Q| Q|(log D) -^ Md ®od QD—1.
Proof. Making use of the representation of logarithmic forms as in the definition of the symbol res. above, one obtains
res. u = 0 ^ gu £ Q| 0 ^ u £ Q| 0.
This completes the proof.
Remark 5.3 In particular, for q =1 one has
0 —► QS —► QS(log D) —^ Md — Md,
where D is the normalization of D. Moreover (see [27], Lemma (2.8)), if n: D —> D is the morphism of normalization, then the image Im (res.) contains п*(Од) consisting of the so-called weakly holomorphic function on D, that is, of meromorphic functions, whose preimage becomes holomorphic on the normalization.
Remark 5.4 By this way we can consider the image of the logarithmic residue res. (log D) as an OD-module. Indeed, the definition of logarithmic forms implies that h ■ (QS о (log D)/QS o) = 0. Hence, the multiplication by h- annihilates Im (res.).
6 Regular meromorphic forms and Saito residue map
We are going to describe the image of the Saito residue map in terms of regular meromorphic forms for logarithmic differential forms with poles along a divisor D С S with arbitrary singularities together with a generalization of the exact sequences from Section 3 and Section 4.
Now we consider the sheaves of OD-modules uD, q > 0, called regular meromorphic differential q-forms on the hypersurface D. So let X, dim X = n > 1, be the germ of an analytic subspace of an m-dimensional complex manifold S, and let uX = Ext"-n(OX, Q") be the Grothendieck dualizing module of X.
Definition 6.1 ([18], [8]) The sheaves uX, q = 0,1,..., n, of regular meromorphic differential q-forms on X are defined as follows: uX consists of all meromorphic differential forms of order q on X such that u Л n £ uX for any n £ QX~9 or, equivalently, uX = HomOx (QX~9, WX).
Let us apply this Definition in the particular case when X = D is a hypersurface, that is, n = m — 1.
Claim 6.2 Let D С U be a reduced hypersurface. Then res. Q|+1(log D) С uD for all q = 0, 1 , . . . , m — 1 .
Proof. Set dz = dz1 Л... Л dzm. Then with preceding notations one has a natural isomorphism uD — OD(dz/dh). That is, uD = HomoD(QD~9, OD(dz/dh)) for all q = 0,1,...,n. Then Corollary 1.7 implies that ^--res. Q|(logD)|[r С QqDl\Dr]U f°r * = 1, • • • ,m, or, equivalently, dh Л res. Q|(logD)|^ С QD|DnU. This completes the proof.
Below we use an equivalent description of the regular meromorphic differential forms uD, q >
0, on the hypersurface D obtained by D.Barlet in a more general context (see [8], Lemma 4). In fact, there is the following exact sequence of OD, 0-modules:
where uD 0 C j*j*QD,0 and C is induced by the multiplication by the fundamental class of D in S. Thus, C(v) corresponds to the Cech cocycle w/h such that w = v A dh.
Theorem 6.3 ([2], §4) Let D C S be a reduced hypersurface. Then for any q > 1 there is the following exact sequence
0 ^ Q|+1 —► Q|+1 (log D) -^ uD ^ 0.
In particular, uD and res. Q|+1(log D) are isomorphic OD-modules.
Proof. It is sufficient to verify the statement locally. In view of Claim 6.2 it remains to prove that any element of uD can be represented as the residue of a logarithmic q-form.
Let K.(h) be the ordinary Koszul complex associated with h, that is,
0 ^ OS,0e0 —^ OS,0 -> OD,0 ^ 0
where K1(h) = OS,0e0, K0(h) = OS,0 and d0(e0) = h, d—1(1) = 1. Then we have the following piece of the dual exact sequence
-------► Hom0s,o(K0(h), QS+01) -^ Hom0s,o№(h), QS+1)
---> ExtOS,0 (OD,0, QS+o1) ^ °.
Hence, any element of ExtOS0(OD,0, Qfo1) can be represented as a Cech 0-cochain (more explicitly, a 0-cocycle) in the following way
v/h £ Homoso(*1(h), QI+01) =
where v £ Q|H01. Choose now an element v £ Q|H01 such that
^ A dhe Extos o(0D,0,
corresponds to the trivial element. That is, v A dh/h is defined by an element of d0(HomOs0(K0(h), Q|H02^. This means that v A dh = h ■ n for some form n £ Q|H02. The first exact sequence of Proposition 2.3 implies that v £ h ■ Q|H01(log D). Set v = C—1(v/h). By definition, C( v) corresponds to a Cech cocycle v/h such that v = v A dh (take v = v, w = v in the above description of uD with the help of multiplication by the fundamental class). This yields C( v) = v/h = v A dh/h, and res. (v/h) = v. Thus, for any element v £ uD there is a preimage under the logarithmic residue map represented by /h. This completes the proof.
Remark 6.4 In fact, the representation (*) implies directly that res. Q^(logD) = uD — OD(dz/dh), in view of the formal decomposition dz/h = (dh/h) A (dz/dh). Further, it is not difficult to verify that in the case of plane node of Example 2.1 there is natural isomorphisms res. QS(log D) — n*(O_D) — uD (cf. Remark 5.3). A similar result is also valid in a more general situation (see [27], Theorem (2.9)).
Remark 6.5 It should also be underlined that there is a far reaching generalization of main results of this section to the case of complete intersections. In papers [5] and [6] it was developed the theory of multi-logarithmic differential forms and their residues with applications to the general theory of multidimensional residue and residue currents on complex spaces.
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ВЫЧЕТЫ ЛОГАРИФМИЧЕСКИХ ДИФФЕРЕНЦИАЛЬНЫХ ФОРМ
А.Г. Александров Институт проблем управления РАН, ул. Профсоюзная, 65, Москва, 117997, Россия, e-mail: ag_aleksandrov@mail.ru
Аннотация. В этой заметке излагается элементарное введение в теорию логарифмических дифференциальных форм и их вычетов. В частности, рассматриваются некоторые свойства логарифмических форм, связанные с кручением голоморфных дифференциалов на особых гиперповерхностях, кратко обсуждаются понятия вычета, данные Пуанкаре, Лерэ и Саито, а затем приводится красивое описание регулярных мероморфных дифференциалов в терминах вычетов мероморфных дифференциальных форм, логарифмических вдоль гиперповерхности с произвольными особенностями.
Ключевые слова: логарифмические дифференциальные формы, форма-вычет, регулярные мероморфные дифференциальные формы, кручение голоморфных дифференциалов.
