GREEN'S FORMS AND MEROMORPHIC FUNCTIONS

THEOREM 1 (HODGE).7 The mapping Z—>w[Z] gives an isomorphism be- tween the p-Betti group {over the real field) of SSR and the space ©p of all real harmonic p-forms of the first kind attached to 9ft. In what follows this theorem will be cited as Hodgës Theorem. The character of the isomorphism Z —> w[Z] will become more clear if we notice that the relation8 (2.2) jtw*[Z] = J(f, Z) holds for an arbitrary (2n — p)-cycle f, where /(f, Z) means the intersection number of f and Z. Now we put w[4>"](t)

THEOREM 2 (DE RHAM [10 pp. 140-143]). For each p, 0 ^ p ^ 2w, there exists on $1 one and only one real double p-form 7P(*, Q = ( - T Y T ; * . . . IXM . . . *(*, 0[d**kfc. . . dxl][d?dp. . . dp] satisfying the following three conditions: (i) for every fixed £ on 5DÎ, yp(x, £) w regular with respect to x1, x2,. . . , x2n except for x — £ and satisfies (2.3) A*7P(*, £) = w'(* 0 ; 6The fact that the space (Sp has a finite dimension can be proved independently of a famous theorem of Hodge. See de Rh$

THEOREM 3 (PRINCIPLE OF ORTHOGONAL PROJECTIONS).10 Let G be an open subset of 5DÎ and \// be a measurable p-form defined in G with 11^| |G < + °° • Then, if \f/ satisfies the integral equation (*, AV)G = 0 for all p-forms r\ C G having continuous third derivatives, \p is regular in G and satisfies A\f/ = 0. Again, if \f/ satisfies the integral equations (*, d\)G= 0, (*, ôrj)G = 0 9This formula is an immediate consequence of the formula (4.5) in de Rham [10]. 10Kodaira [5, pp. 608-609]. The metho

THEOREM 4. y[Z] is regular in 2ft — \Z\ and satisfies the differential equations (2.11) by[Z]{x) = 0, (2.12) ôdy[Z](x) = - w[Z](x). The derived form dy[Z] of y[Z] is regular harmonic in $ft — \z\ if and only if Z is a bounding cycle on W. Furthermore dy[Z] has the finite norm : | |^T[Z]| | < + °° and satisfies the integral equations (2.13) (dy[Z],d4,) = J z { * - * ] j , (2.14) (dy[Z], T) = 0 (or = 0),$

THEOREM 5. Let D = 2kmkTk be a bounding (2n — 2)-cycle on 9JÎ consisting of a finite number of closed analytic surfaces Yk with minimal local equations fkp(z) = 0 (k = 1, 2, . . . , K). Then the integral 2<ir(— i W » H - i ) / 2 *D(*) = \ ' t An-17[D](x)+ 2*i (n - 1) ! *dy[D](x) + const. is the Picard integral of the third kind with the logarithmic polar cycle D> i.e. $D(z)$

THEOREM 6. Let D = XmkTk be a bounding (2n — 2)-cycle on 2)? with in- tegral coefficients mk consisting of a finite number of closed analytic surfaces Tk with minimal local equations fk$(z) = 0 (k = 1, 2, . . . , K). Then https://doi.org/10.4153/CJM-1951-014-1 Published online by Cambridge University Press$

THEOREM 7. A meromorphic function F(z) with the divisor D = dC is one- valued if and only if the congruence /(r, c) + w*[f] s 0 (mod 1) c holds for every 1-cycle f with integral coefficients. This theorem can be considered as a generalization of Abel's Theorem [13, pp. 126-127] in the classical theory of Riemann surfaces. Considered as a functional of 1-cycles f with integral coefficients, XD = XD(£) is a character of the 1-homology group IPÇM) of 5DÎ over the additive group of all integers. The