[ ;’ *I ‘6 c $2 p ~-3 $98 ‘b as”8 G. M. Khenkin A.G. Vitushkin (Eds.) K 2 Several Complex Variables II Function Theory in Classical Domains Complex Potential Theory With 19 Figures nger-Ve Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Encyclopaedia of Mathematical Sciences Volume 8 Editor-in-Chief: RX Gamkrelidze Contents I. Multidimensional Residues and Applications L. A. Aizenberg, A. K. Tsikh, A. P. Yuzhakov 1 II. Plurisubharmonic Functions A. Sadullaev 59 III. Function Theory in the Ball A. B. Aleksandrov 107 IV. Complex Analysis in the Future Tube A. G. Sergeev, V. S. Vladimirov 179 Author Index 255 Subject Index 258 I. Multidimensional Residues and Applications L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov Translated from the Russian by J.R. King Contents Chapter 1. Methods for Computing Multidimensional Residues (A.P. Yuzhakov) ............................................ Introduction .................................................. 5 1. Leray Theory. Froissart Decomposition Theorem ............... 1 .l. Leray Coboundary ..................................... 1.2. Form-Residue, Class-Residue, Leray Residue Formula ....... 1.3. Tests for Leray Coboundaries. Froissart Decomposition Theorem .............................................. 6 1.4. Cohomological Lowering of Pole Order .................... 7 1.5. Generalization of the Leray Theory to the Case of Submanifolds ofCodimensionq> 1 ................................... 9 0 2. Application of Alexander-Pontryagin Duality and De Rham Duality ................................................... 10 2.1. Application of Alexander-Pontryagin Duality ............... 10 2.2. Residues of Rational Functions of Two Variables ............ 11 2.3. Application of De Rham Duality .......................... 13 5 3. Homological Methods for Studying Integrals that Depend upon Parameters. Application to Combinatorial Analysis .............. 15 3.1. Analytic Continuation of Integrals Depending on Parameters. Isotopy Theorem ....................................... 16 3.2. Foliation near a Landau Singularity. Picard-Lefschetz Formula 18 3.3. Some Examples of Integrals Depending on Parameters ....... 20 3.4. Application of Residues to Combinatorial Analysis .......... 22 Chapter 2. Multidimensional Logarithmic Residues and Their Applications (L.A. Aizenberg) ................................ 24 5 1. Multidimensional Logarithmic Residues ....................... 24 9 2. Series Expansion of Implicit Functions ......................... 31 I. Multidimensional Residues and Applications 3 2 L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov Chapter 1 0 3. Application of the Multidimensional Logarithmic Residue to Systems Methods for Computing Multidimensional Residues ..................................... 33 of Nonlinear Equations . 37 6 4. Computation of the Zero-Multiplicity of a Holomorphic Mapping 9 5. Application of the Multidimensional Logarithmic Residue to the A.P. Yuzhakov TheoryofNumbers ......................................... 38 Chapter 3. The Grothendieck Residue and its Applications to Algebraic ...................................... 39 Introduction Geometry (A.K. Tsikh) 39 Introduction .................................................. One of the problems in the theory of multidimensional residues is the prob- $1. Integral Definition and Fundamental Properties of the Local Residue ................................................... 40 lem of studying and computing integrals of the form / ............................................ 40 1.1. Definitions 1.2. Representation of the Local Residue by an Integral over the .................................. 41 Jv Boundary of a Domain 1.3. Transformation Formula for the Local Residue ............. 41 where o is a closed differential form of degree p on a complex analytic manifold 1.4. Local Duality Theorem .................................. 42 X with a singularity on an analytic set S c X, and where y is a compact p- 0 2. Using the Trace to Express the Local Residue .. 43 dimensional cycle in X\S. A special case of this problem is computing the 2.1. Definition of the Trace and its Fundamental P...r.o..p..e..r.t.i.e..s.. ....... 43 integral (1) when o is a holomorphic (meromorphic) form of degree p = n = 2.2. Algebraic Interpretation ................................. 44 dim, X; in local coordinates the form can be written as w = f(z) dz = 9 3. The Total Sum of Local Residues ............................. 45 f(z 1, **., z,) dz, A ... A dz,, where f is a holomorphic (meromorphic) function. 3.1. The Total Sum of Residues on a Compact Manifold. The Euler- According to the Stokes formula, the integral (1) depends only on the homology Jacobi Formula ........................................ 45 class’ [y] E H,(X\S) and the De Rham cohomology class [o] E HP(X\S). Thus 3.2. Applications to Plane Projective Geometry ................. 47 in integral (1) the cycle y can be replaced by a cycle y1 homologous to it (yr - y) 3.3. The Converse of the Theorem on Total Sum of Residues ...... 47 in X\>S and the form o can be replaced by a cohomologous form w1 (w I - o) 3.4. Abel’s Theorem and its Converse .......................... 48 which may perhaps be simpler; for example, it could have poles of first order on 3.5. Residue Theorem for Vector Bundles ...................... 50 S (see 0 1, Subsection 4). If (yj} is a basis for the p-dimensional homology of the 3.6. The Total Sum of Residues Relative to a Polynomial Mapping manifold X\S, then by Stokes formula for any compact cycle y E Z,(X\S) the inC” ................................................. 51 integral (1) is equal to 0 4. Application of the Grothendieck Residue to the Algebra of P P ......................... 52 J Polynomials and to the Local Ring 0, o = 2 kj ‘0, (2) 4.1. Macauley’s Theorem .................................... 52 V i J .......................... 52 J vj 4.2. Noether-Lasker Theorem in CP” where the kj are the coefficients of thetycle y-as a combination of the basis ................ 53 4.3. Verification of the Local Noether Condition elements {yj}, y - cj kjyj. Formula (2rshows that the problem of computing ......................... 54 4.4. A Consequence of Global Duality integral (1) can be reduced to 1) studying the homology group H,(X\S) (finding its dimension and a basis); 55 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2) determining the coefficients of the cycle y with respect to a basis; 3) computing the integrals over the cycles in the basis. Solving problems 1) and 2) is a difficult topological problem in the multi- dimensional case and requires the machinery of algebraic topology. In some ’ In this chapter we will denote by HP the group of compact singular homology; this group was denoted by Hi in the contribution of Dolbcault (Dolbeault, 1985) in Volume 7 of the Encyclopaedia of Mathematical Sciences. 4 L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov I. Multidimensional Residues and Applications 5 cases, to solve this it helps to apply the dualities of Alexander-Pontryagin If a family of S,, . . . . S,,, of submanifolds of codimension 1 is in general and De Rham ($2). Simple and multiple Leray coboundaries (Subsection 1.1) position, the multiple Leray coboundary is defined: give a construction of standard cycles in X\S. The general structure of the 6”: HP-,(S, n”.n S,) + H,(X\(S, u*-*u S,)), homology group H,,(X\S) is described in “good cases” by the decomposition theorem of Froissard (Subsection 1.3). Integrals on coboundary cycles can be which is anticommutative with respect to the order of S,, . . ., S,,, (for the reduced to integrals of lower degree by the simple and multiple Leray residue cohomological multiple coboundary, see (Dolbeault (1985), Sect. 03). formulas (Subsection 1.2). The computation of an important class of residues, the Grothendieck residues, and a special case of them, the logarithmic residue, 1.2. Form-Residue, Class-Residue, Leray Residue Formula. As was pointed is considered in Section 2 and Section 3 of this article; $3 is devoted to the out in (Dolbeault (1985), Sect. 03), if 4 is a closed regular differential form in application of residues to the study of integrals depending on parameters and to X\S with a pole of first order on S, then in some neighborhood of any point combinatorial analysis. a E S the form 4 can be represented as 0 1. Leray Theory. Froissart Decomposition Theorem where s = s,(z) is the defining function of the manifold S in U, and $, 8 are forms Here we will pause to study in more detail the computational side of the which are regular on U,,. Here the form tils is globally defined, is closed, and is Leray theory of residues expounded in (Dolbeault (1985)). To start with, we uniquely determined by the form 4. This restriction Ic/I s is called the form-residue consider the case of codimension 1. of the form 4 and is denoted by res[f. We remark that if 4 is holomorphic on 1.1. Leray Coboundary. We give a constructive description of the X\S, then the form residue res[d] is holomorphic on S. coboundary homomorphism 6 which was introduced in (Dolbeault (1985), Sect. Example 1. Let X = C”, with S = {z E Cc”: s(z) = 0} and 4 = f(z) dz, A 0.3). In the one-dimensional case the simplest cycle (contour) of integration is a ... A dz,/s(z). Since C$= (- ly’-’ ds A dzJs. s;,, where dz,, = dz, A *** A circle of sufficiently small radius around an isolated singular point. Leray (1959) dz,+ A dzj+l A ... A dz,, then res[4] = (- ly’-‘f(z) dzb.,/s:,ls at the points constructed the analog of this for complex analytic manifolds, the coboundary where sij # 0. homomorphism 6. The construction of 6-l was first considered by Poincare (1887). Let X be a complex analytic manifold of complex dimension IZ. Let S be a Remark. The map complex-analytic submanifold of X of codimension 1. We consider a tubular f(z) W(z) + ( - 1 y’ -‘f(z) dzu,ls;, Is neighborhood V of the submanifold S, which is a locally-trivial fiber bundle with base S and fiber V,, a E S, homeomorphic to the disk. In order to construct is called the Poincare residue map and is denoted by P.R. If we denote by !&, such a fiber bundle we choose a Riemannian metric on X and take as V, the Q;(S), C&!-l, the sheaves of germs, respectively, of holomorphic n-forms on X, union of geodesic segments of length p(a), beginning at a and orthogonal to S, meromorphic n-forms having only simple poles on S, and holomorphic (n - l)- where p(a) is sufficiently small. We assume that the function p(a) is smooth; this forms on S, then there is an exact sequenc9 of sheaves implies the smoothness of aV. To each (p - 1)-dimensional element of a chain (a --.._ o-+Q;+Q;( $+sz;-Lo simplex, a rectangle) r.rpml in S we associate a p-dimensional chain in X\S. The 7 chain is 60,~~ = UOE,Op-,, 6a where 6a = al$; it is homeomorphic to aV, x ape1 which defines an exact sequence on cohomology with the natural orientation. Thus a homomorphism of homology groups is HO(X, f2;(s))p2. HO(S, q-l) 5 H’(X, La;). defined, Therefore, the Poincare residue map is surjective on global sections if 6 : f&-,(S) + H,(X\S), H’(X, sZ;t) = 0. In particular, this is true for projective space X = CP”, n > 1. since a6 = - 68. Then the Leray homology exact sequence is defined: Thus for n > 1 every holomorphic form of degree n - 1 on the submanifold S is the Poincare residue of a meromorphic n-form on CP”. . . . -+ H,+,(X) 2 H,-,(S) 5 H,(X\S) 5 H,(X) 4”’ (3) By the theorem of (Dolbeault (1985), Sect. 0.3), for every closed regular differ- where i is the homomorphism induced by the inclusion X\S c X and CC entiable form 4 on X\S, there is a form 4 cohomologous to it which has a pole in induced by the intersection of chains in X, transversal to S, with the of first order on S. In this case the cohomology class of the form res[J] depends submanifold S. only on the cohomology class of the form 4. I. Multidimensional Residues and Applications 7 6 L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov Proposition 1.3 (Grifliths (1969)). Zf S is an algebraic manifold in complex The cohomology class of the form res[&j is called the class residue of the form projectioe space CP”, then 6 : H,-,(S) + H,(@P”\S) is always surjectiue and is 4 and is denoted by Res[f. Since the operator res is linear, Res : HP(X\S) + injective for even n. HP-‘(S) is a homomorphism. We observe that the form residue and the class residue also exist in the case Proposition 1.4. For a cycle y E Z,(X\(S, u * * * u S,)) to be a compound Leray when S is an analytic subset. However in this case the form residue has singu- coboundary ([r] E 6”Hp-,(S, n ..* n S,,,)), it is necessary (and sufficient in the larities on the set S* of singular points of S. If the singular set S* of the set S is case when X is a Stein manifold) that resolved to a divisor with normal crossings, then the form residue only has simple poles on the resolution of S* (Gordan, 1974). The abstract residue formula (0.3.2) of (Dolbeault (1985)) is written thus: Under some assumptions about X and S,, . . . , S,,,, the structure of the homology group H,(X\(S, u * * * u S,,,)) is described by the following theorem. Theorem 1.1 (Leray residue formula). For an arbitrary closed form q4 of de- gree p on X\S and a cycle ct E Z,-,(S) there is a formula Theorem 1.5 (Froissart decomposition (Fotiadi et al. (1965)). Let So, S,, . . . , S,,,and.E,,..., C, be two families of submanifoldso f codimension1 in the complex projective space CP” such that the families are in general position. S, = @Pi-;’ = (5) ClF’“\@” is the hyperplane at infinity. Let Y = z, n ... n 2,. X = @” n Y = 1 Y\S,. Then the cohomology group If the form 4 E Zp(X\(S, u *. . u S,,,)) has a pole of first order on S,, . . . , S,,,, then by applying formula (4) one can define iterated form residues res”[#] E zp-“(S, n . . * n S,,,) and a homomorphism HP (fihfsjnx)) Res”’ : HP(X\(S, u . +. u S,)) + HP-“‘(S, n . . . n S,,,), =H,(X)O f JH,-,(SjnX)@ C 6Hp-,(XnSjnSq) j=l lsjsqsm as the composition of homomorphisms HP(X\(S,u~~~uS,,,))~HP-‘(S1\(Szu~~~uS~))+~*~ where 1h i is the number of elementsi n the set h. + HP-“+‘(S, n...nS,-,\S,,,) R-W-f HP-“‘@, n . . . n S,). Example 2. Let X = C”, and let Sj = {Z E C” : Lj(z) = Ci=l Uj~z, + bj = 01, Iterating Theorem 1.1 we obtain the compound Leray residue. j=l 7 . ..> m be analytic hyperplanes in general position (if Sjl n . . . n Sjk # 0, then L,,, . . ., Lj~ are linearly independent). Since H,(P) = 0, Theorem 1.2 (Leray (1959)). For an arbitrary form 4 EZ~(X\(S~ u ... u S,,,)) Hn- rCSj, n . . . n SjJ = 0 for r < n, and H,(Sj, n * ** n Sj,) 2: Z for Sj, n.. . n and any cycle o E Z,-,(S, n *.. n S,,,), a compoundL eray formula holds: Sjn # a, then there is a basis of the group H,(@“\(S, u.. . u S,,,)) consisting of cycles of the form 8’(Sj, n ... n Sj,) = {z : IL,(#)1 = E, v = jr, . . . , j.}. The residue dm~c j = (2rci)” Res”[#]. (6) of the form 4 = h dz/L’,’ . . . L>, where h E A(C”), rebtive to a basis cycle s s CT P(Sj, n ... n Sj,) is the Grothendieck residue (see Chapter 3) at the point Sj, n The Leray formulas (5) and (6) allow one to lower the degree of the multiple . . . n Sjn. After a linear change of varia es, it Kcomputed as the derivative of a integral (1) when the cycle of integration belongs to a coboundary class: y E multiple Cauchy integral. Y 6Hp-,(S), y E 6”Hp-,(S, n..* n S,,,). Since, for a form 4 having a first-order pole, the form-residue res[f (resm[d]) is found constructively, the problem 1.4. Cohomological Lowering of Pole Order. In Subsection 1.2 only the exis- arises of how to lower the order of poles of semimeromorphicfo rms 4 E Zp(X\S) tence of the class residue of a form 4 E Zp(X\S) was discussed, b&no algorithm (4 l Zp(X\(S, n ..* n S,,,))); in other words, how can one find a form d1 was demonstrated for computing it, that is, for finding a form d1 - 4 having a cohomologous to 4 which has a first-order pole. pole of first order. The same is true for the compound class residue of a form f$ E ZP(X\(S, u ... u S,,,)). In some cases the problem of the cohomological re- 1.3. Tests for Leray Coboundaries. Froissart Decomposition Theorem. From duction of a semimeromorphic form (see (Dolbeault (1985)) Subsection 3.5) to a the Leray exact sequence (3) it follows that a cycle y E Z,(X\S) is a coboundary form having a pole of order 1 can be solved constructively. For example, (y - 60 for some r~ E Z,-,(S)) if and only if y - 0 in X. If H,(X) = 0, then Proposition 1.6 (Pham (1967)). Let S = {z : s(z) = 0}, where s is a holomorphic H,(X\S) = 6Hp-,(S), i.e., every cycle in X\S is a coboundary. If H,+,(X) = 0, function in a neighborhood V(S) of the manifold S, with grad sIs # 0, then any then 6 is a monomorphism. 8 L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov I. Multidimensional Residues and Applications 9 We remark that in @P” it is not always possible to lower the order of the pole form 4 E Z*(X\S) having a pole of order k on S can be represented in the form to one. This follows from the fact that in H”(@[FP”\S) there can exist classes which contain no rational forms with first order poles. For example, let (7) s = {[E @P2: [; + r: + r: = O}. where 8 and $ are regular forms in V(S) and e1 = dt+k/(k- 1) + 8. Since S is a curve of genus 1, then Theorem 1.7 (Leray (1959)). Let the submanifoldsS j in a neighborhood V of the set S, n... n S,,, be defined by the equations sj(z) = 0, where the sj are dim H2(@lP2\S) = dim H,(CP’\S) = 2. functions holomorphic in V,j = 1, . . . , m. Zf the form 4 E Z*(X\(S, u .=au S,)) is Since any rational form w with a pole of first order on S can be written as represented in V as const . sZ/(ci + c: + [i), the rational forms with first order poles cannot generate 4 = ds, A -‘. A ds, A w/s;‘+~...s;-;;“+I, the entire group H2(@P2\S). In the general case, if S is an algebraic submanifold of CP”, with q(z) = then r,+...+r, 4tz1, . . . . z,) = 0 being its equation in afline coordinates, then any differential 1 a form of degree n in @P”\S with a pole of order one on S has the form w = Res”[f 3 r,!...r (yl r: Ill’ 1 . ..as. S,n...nS,’ p(z) dz/q(z) in the coordinates z = (z,, . . . , z,), where deg p I deg q - n - 1. The latter condition means that w does not have a pole on the hyperplane at where the drl+“‘+rm/i%;l.. . &km are found recursively from the equation infinity. The class residues of such forms are represented by PoincarC residues: do = ds, A w1 + 1.. + ds, A w,, wjdgf aw/asj, (8) P.R.[w] = [(p dz&(aq/azj)ls]. According to the remark in 1.2, this coincides with the cohomology of holomorphic forms of degree n - 1 on S. In general, on which is a consequenceo f the condition do A ds, A *a* A ds, = 0 (d4 = 0). a compact manifold S this part of the cohomology comprises a proper subspace We observe that in order to find the forms I/ and 0 in (7) and Wj in (8) it is of H”-‘(S). necessary to employ a partition of unity. The apparatus of partial derivatives for Let A{(S) be the set of rational p-forms with poles of order k on a exterior differntial forms was developed by Norguet (1959). submanifold S c CP”, J& = A;(S)/dA;I:(S). Then the image of the mapping P.R. : Z1 + H”-‘(S) lies in what is called the primitive subgroup H:,;:(S) (see Theorem 1.8 (Leinartas, Yuzhakov (cf. Aizenberg-Yuzhakov (1979)). Let Qt, Grifliths (1969)). Moreover there exists a natural map R, : sk --+ F(“-l*k-l)(S), . . . , Q, be irreducible polynomials in @” and let the Sj = {z E C” : Qj(z) = 0}, j = 1, mapping zk to the Hodge filtration . . . . m be manifolds in general position. Then the form p-Lo’(s) c p-‘.“(s) c . . . c p-l.n-1)(s) = fp(S) w = P dz, A a. * A dz,/Q’,’ . . . Q’,- of the group H”-’ (S), where R, = P.R. and ty image &(.%$) is the primitive is cohomologousi n @“\(S, v * * * v S,,,) to a form of type subgroup FEi-,kk’k-l)(S). W* = C PJ dz, A .ff A dz,JQj, . . . Qj,, J (I 1.5. Generalization of the Leray Theory t&the Case of Submanifolds of where J = (j,, . . . . j,}, k I n, and the PJ are polynomials. Codimension q > 1 (Norguet (1971)). I.&t X be a complex analytic manifold, dim, X = n, and let S be a complex submanifold of codimension q (in some Remark. The form w* is found constructively using elimination theory (the neighborhood U,, of any point a E S the set S n U,, = {z E U, : sl(z) = f.. = Hilbert Nullstellensatz). s&z) = 0}, where the sj are holomorphic functions in U, and the vectors grad sj, Theorem 1.9 (Grihiths (1969)). Let Q be an irreducible homogeneousp olyno- j= 1, . . . . q are linearly independent). There is an exact homology sequence mial and let S = {[ E CP” : Q(c) = 0} be a manifold. Then any closed rational analogous to (3), where the homomorphism 6 is induced from a fiber bundle n-form w in CP” with poles on S, with base S and fiber S, homeomorphic to the (2q - l)-dimensional sphere. A differential form 4, regular in X\S, is called a simplef orm (cf. [Dolbeault (1985), w = PtWKYQ”(O~ (9) 4.11) if there exists a form II/, regular on X, such that for any point a E S, in some where52([) = cT=I (-ly’cj dc, A ***[j]**. A dc,, can be replacedb y another form neighborhood U, of the point, the form 4 can be represented as of type (9) which is cohomologous to it but which has m I n - E(n/q), where q = deg Q (the symbol [j] signifies that the term dcj is omitted). 10 L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov I. Multidimensional Residues and Applications 11 where 0, is a form regular in U,, and the basis cycles are found from this system of linear equations, K, = -’ “tI (- l)y-‘3, ds, A . ..[v]... A d3, A ds, A *a. A ds,. Jc~=$lk,vJyv~~ j=l,--.,q, where kjy = o(c,,,, rj). We observe that det [lkjyll # 0 is the condition of homo- Theorem 1.10 (Norguet). Every form 4 E ZP(X\S) is cohomologous to a logical independence of the cycles r,, . . . , rq. simple form ~+4Ei Zp(X\S). Thus for any cycle o E Zp-2q+1(S), the residue formula Let p = n and let o = f(z) dz be a holomorphic form in D. From the preced- holds: ing this then follows: 4 = WY Theorem 2.1 (On residues). bet the function f be holomorphic in the domain s do (q - l)! s o 4’S* D c C”, let T = C”\D, and let T = T u {a} be a subpolyhedron in the spherical compuctificution of C”, 43 u (co). Zf {oj}~=l is a basis for the (n - 1)-dimensional homology of the singular set T, and if (rj}3=1 is the basis dual to this in the 3 2. Application of Alexander-Pontryagin Duality and n-dimensional homology of D, then for any cycle y E Z,(D) this equation holds: De Rham Duality f(z) dz = (2xi)” 5 kjRj, s Y j=l 2.1. Application of Alexander-Pontryagin Duality.’ To find the dimension where kj = o(oj, y) and Rj = (2xi)-” jrj f(z) dz. and a basis for the group H,(X\S), it is sometimes useful to apply the topologi- cal Alexander-Pontryagin duality theory which establishes an isomorphism of The application of Alexander-Pontryagin duality is especially effective in the homology groups: case when n = 2 and T is an analytic set. In this case the study of the two- dimensional homology group of a domain in real four-dimensional space re- duces to the study of the one-dimensional homology group of a surface (a complex curve). As an example we consider the residue of a rational function of where S” is a manifold homeomorphic to the n-dimensional sphere and T is a two variables. compact subset of S”, where p + r = n - 1. There exist dual bases (yj}, {~j> such that o(oj, yk) = hjk, where o(aj, yk) is the linking coefficient of the cycles Oj and yk. 2.2. Residues of Rational FuncWtions 4of TJ wo ,Var iables. We apply the ap- Let the form o be regular in the domain D = C”\T (T is the singular set of proach described above to integrals of the form the form 0). We compactify the space @” to form the sphere S2” = 6” = C: u {co} by attaching a single point {co} at infinity. Since T is closed in C”, T = (10) T u {co} is compact, and D = ‘I?\ f Then by Alexander-Pontryagin duality, Q(w, z) dw A dz' H,(D) N H,,-,-i(f). Thus to find the dimension and a basis {yj} of the group where P and Q are polynomials (or we may ass&me that P is an entire function), H,,(D), it is necessary and sufficient to find the dimension q and a basis Cj of the the cycle y E Z2(C2\T) and T = (Q(w, z) = 0). Let Q = Q’,’ . . . Q>, where Ql,l, (2n - p - l)-dimensional homology group of the singular set f Then the coeffi- . . ) Q’,- are irreducible polynomials. Then’“T = uyxl q, where Tj = (Qj(W, z) = cients kj of an expansion of an arbitrary cycle y E Z,(D) with respect to the basis O}. Using the Euler-Poincare formula, it-is not difficult to compute the dimen- {yj}4=i, the basis dual to (~j)4=1, are the linking coefficients of the cycle y with sion of the group H, (?) and consequently that of its dual group H2(C2\ T). . the cycles of the dual basis: Theorem 2.2 (Yuzhakov, see Aizenberg-Yuzhakov (1979)). The dimension of the homology groups H,(T) N H2(C2\ T) is defined by the formula To find the integrals with repect to the basis cycles, jyj o, it is sufficient to take 4 = j$l 2Pj + i (4i - 1) + 1 - m, i=l q homologically independent p-cycles in D, r,, . . . , r,. Then the integrals over where pj is the genus of the surface (the genus of the Riemunn surface defined by the equation Qj(W, z) = 0) and qi is the number of irreducible elements of the subset T intersecting at the point Ai; s is the number of such points of self-intersection, 2The first to apply Alexander-Pontryagin duality to complex analysis was Martinelli (1953), who and 1 is the number of elements at infinity of the set T (the number of connected used it to deduce a generalization of the Cauchy integral formula to C” for multiple integrals of cmponents of the set T n {I WI’ + Iz12 > R2} for R sufficiently large). degreen+!,O<I<n-1. 12 L.A. Aizenbexg, A.K. Tsikh, A.P. Yuzhakov I. Multidimensional Residues and Applications 13 We construct the dual bases of H,(f) and Hz(C2\ T). For this, on each HI(?). For any cycle y E Z,(@*\T) the integral irreducible component Tj we take 2pj canonical cycles c$, s = 1, . . . , 2pj, (a basis of the one-dimensional homology of the corresponding Riemann surface) so P(w, z) dw A dz/(wz - 1) = (2ni)‘k. R, that the cycles G;, 2k -1 and r~j:2 k intersect each other only at one point and do not ss V intersect the other c,\. Moreover we assume that the curves CJ~~d o not pass where k = o(y, ol), through any self-intersection point Ai. We take Y,!~= 6oj’,, where wj’si s obtained from oj,, r = s - (- l)“, by a small perturbation, and 6 is the Leray coboundary. Further, suppose that through the point Ai pass qi elements S,., r = 1, . . . , qi, of R = (2.rri)-* P dw A dz/(wz - 1) ss the set T. On each of these, except for Sqi we construct a simple closed curve oi VI surrounding the point A,. We set yi = &I$.. As the dual cycle c$. we take a i a*vyo, 0) simple closed curve which begins as a curve on S, with initial point Ai and which = mzo (27ci)-* P dw A dz/(wz)“‘+l = nTt= o~(m !)2 aw*azm . ends as a curve which returns to Ai on Sqi. We obtain x:=1 (qi - 1) pairs of cycles. Analogously we construct I - m = cyC1 (Ij - 1) pairs of cycles rj”y = Swj”y 2.3. Application of De Rham Duality. In some cases, for the study of residues and $,,, v = 1, . . . , lj - 1, j = 1, . . . , m, in a neighborhood of the point {co}. it is useful to apply De Rham’s Theorem, which establishes a duality between the homology groups and the cohomology of exterior differentiable forms. The Theorem 2.3. The cycles {Y;~> and0{ $}, r = 1, 2, 3, firm dual baseso f the theorem can be stated thus: Let X be a differentiable manifold. For any homology groups H2(C2\T) and H,(T). homomorphism A : H,(X) --f @, there exists a unique element h = [co] E HP(X) We may assume that aQj/aw # 0 at the points of the curves 0;. Then the of the De Rham cohomology group such that n(g) = j, h = 1, o for any g = integral over a basis cycle is equal to [y] E H,(X). From De Rham’s Theorem we get the following proposition, which is useful in applications. P(w, 4 pdw Adz= Res[P dw A dz/Q], (11) ss vjs Q(w, 4 s 0;. Proposition 2.5. If for cycles rj E Z,(X), j = 1, . . . , q, and forms Oj E Zp(X), j=l 7 . . . . q the determinant det llajkll # 0, where ajk = lvj 0,; and if any form where w E Z”(X) can be representedi n the form o N ‘& cjoj, where the cj are complex Res[P dw A dz/Q] = Res[F dQ, A dz/Q] 3 [l/(r, - l)!] numbers,t hen dim HP(X) = dim H,(X) = q and the {rj}j=l and {Oj}4=1 are bases x WlF/i?Qf-’ dzJQV=,, of the p-dimensionalh omology and cohomology of the manifold X. F = PQ:lQ. (aQ,lW, If ajk = ~jk, then these bases are called dual in the sense of De Rham. Theorem 2.6. Let {yj} and {oj} be bases of the p-dimensional homology and if lo;J c TV. Since a ‘v-‘F/aQ:-’ is a rational function, we have cohomology of the manifold X, dual in the senseo f De Rham. Then for any cycle Theorem 2.4 (Poincart (1887)). The integral of a rational function of two y E Z,(X) and any cocycle o E Zp(X), the integrdl variables over an arbitrary cycle y E Z,(C’\T) can be expressed in terms of the periods of abelian integrals on the Riemann surfaces defined by the equations a= ‘.ijRj,w. Qj(w,z)=O,j=l,..., m. j s V 5 It is clear that for r = 2, 3 the computation of the integral (11) reduces to the where the kj = J, Wj are the coefficients of the expansion y N CSCI kjrj of the computation of the residue of an algebraic function relative to its pole. cycle y with respect to the basis {yj} and where the Rj = svj o are the coefficients of the expansion o N Cj4,1 RjWj of the De Rha m classo f the form o with respect Example 3. If p and q are relatively prime integers, the integral to the basis {oj}. s j, P(w, z) dw A dz/(zP - w”) = 0 for any cycle y E Z,(@*\T), where T = {z” - wq = 0}, since f is homeomorphic to the two-dimensional sphere and In the case of complex analytic manifolds (for example, domains in C”) the next theorems are also useful. (See, e.g., the paper of Onishchik in Volume 10, I H,(C*\T) N H,(f). of this series.) Example 4. Let Q(w, z) = wz - 1. Then f is homeomorphic to the Riemann sphere with two points identified (z = co and z = 0, the pole of the function Theorem 2.7 (Serre). Zf X is a Stein manifold (for example a domain of w = l/z). Thus H,(f) 1: Z. The cycles y1 = {Iwl = IzI = 2) and c1 = {w = l/t, holomorphy in C”), then any cohomology class h E HP(X) contains a holomorphic z = t, 0 I t I co} form dual bases of the homology groups H2(C2\T) and form o E h.