GREEN'S FORMS AND MEROMORPHIC FUNCTIONS

Theorem 4. $\gamma[Z]$ is regular in $\mathfrak{M}-|Z|$ and satisfies the differential equations $$(2.11) \delta \gamma[Z](x) = 0,$$ $$(2.12) \delta d\gamma[Z](x) = -w[Z](x).$$ The derived form $d\gamma[Z]$ of $\gamma[Z]$ is regular harmonic in $\mathfrak{M} - |Z|$ if and only if Z is a bounding cycle on $\mathfrak{M}$ . Furthermore $d\gamma[Z]$ has the finite norm: $||d\gamma[Z]|| < + \infty$ and satisfies the integral equations $$(2.13) (d\gamma[Z], d\psi) = \int_{Z} \{ \psi - w[\psi] \},$$ $$(2.14) (d\gamma[Z], \tau) = 0 (\delta\tau = 0),$$ where $\psi$ means an arbitrary $\rho$ -form having continuous first derivatives and $\tau$ an arbitrary $(\rho + 1)$ -form with continuous first derivatives satisfying $\delta \tau = 0$ .

Theorem 5. Let $D = \Sigma_k m_k \Gamma_k$ be a bounding (2n-2)-cycle on $\mathfrak{M}$ consisting of a finite number of closed analytic surfaces $\Gamma_k$ with minimal local equations $f_{k\mathfrak{D}}(z) = 0 \quad (k = 1, 2, \ldots, \kappa)$ . Then the integral $$\Phi_D(z) = \frac{2\pi(-1)^{n(n+1)/2}}{(n-1)!} \Lambda^{n-1}\gamma[D](x) + 2\pi i \int_{-\pi}^z d\gamma[D](x) + \text{const.}$$ is the Picard integral of the third kind with the logarithmic polar cycle D, i.e. $\Phi_D(z)$ is a many valued analytic function on $\mathfrak M$ which is regular in $\mathfrak M - |D|$ and, for every $\mathfrak p \in |D|$ , $\Phi_D(z)$ has, in a sufficiently small neighbourhood $N(\mathfrak p)$ of $\mathfrak p$ , the form $$\Phi_D(z) = \sum m_k \log f_{kn}(z) + regular part,$$ where $\gamma[D](x)$ means the integral of the Green's form $\gamma^{2n-2}(x, \xi)$ over the cycle D: $$\gamma[D](x) = \int_D \gamma^{2n-2}(x, \cdot).$$ The real part $$\Re \Phi_D(z) = 2\pi \left\{ (-1)^{n(n+1)/2}/(n-1)! \right\} \Lambda^{n-1} \gamma[D](x)$$ of $\Phi_D(z)$ is one-valued on $\mathfrak{M}$ . The period $[\Phi_D]_{\xi}$ of the integral $\Phi_D(z)$ over an arbitrary 1-cycle $\xi$ is given by $$[\Phi_D]_{\zeta} = 2\pi i \left\{ I(\zeta, C) + \int_C w^*[\zeta] \right\} \qquad (\partial C = D),$$ where C is a (2n-1)-chain with $\partial C = D$ and w $[\zeta]$ means the harmonic 1-form of the first kind associated with the 1-cycle $\zeta$ in the sense of Hodge's Theorem. 4. Multiplicative meromorphic functions. Now it is obvious how to construct a multiplicative meromorphic function having the given divisor D. Assume that a bounding (2n-2)-cycle $D=\Sigma m_k\Gamma_k$ on $\mathfrak M$ with integral coefficients $m_k$ consisting of a finite number of closed analytic surfaces $\Gamma_k$ is given. Then, constructing the Picard integral $\Phi_D(z)$ of the third kind as above and putting $$F_D(z) = \exp \left\{ \Phi_D(z) \right\},\,$$ we get the multiplicative meromorphic function $F_D(z)$ having D as its divisor. Thus we obtain the following

Theorem 6. Let $D = \sum m_k \Gamma_k$ be a bounding (2n-2)-cycle on $\mathfrak{M}$ with integral coefficients $m_k$ consisting of a finite number of closed analytic surfaces $\Gamma_k$ with minimal local equations $f_{kp}(z) = 0$ $(k = 1, 2, ..., \kappa)$ . Then $$F_D(z) = c. \exp 2\pi \left\{ \frac{(-1)^{n(n+1)/2}}{(n-1)!} \Lambda^{n-1} \gamma[D](x) + i \int_{-\pi}^z d\gamma[D](x) \right\} \qquad (c \neq 0)$$ is a meromorphic function having D as its divisor, i.e. $F_D(z)$ is a many valued analytic function on $\mathfrak{M}$ which is regular in $\mathfrak{M} - |D|$ and, for every point $\mathfrak{p} \in |D|$ , $F_D(z)$ is represented in a sufficiently small neighbourhood $N(\mathfrak{p})$ of $\mathfrak{p}$ as $$F_D(z) = U(z) \prod_k \{ f_{kh}(z) \}^m_k \qquad (U(z) \neq 0)$$ where U(z) is a non-vanishing regular analytic function defined in $N(\mathfrak{p})$ . The absolute value $$|F_D(z)| = |c| \exp 2\pi \{(-1)^{n(n+1)/2}/(n-1)!\} \Lambda^{n-1}\gamma[D](x)$$ of $F_D(z)$ is one-valued on $\mathfrak{M}$ ; thus $F_D(z)$ is multiplicative. If one prolongs $F_D(z)$ along an arbitrary closed curve $\zeta$ , then $F_D(z)$ is multiplied by the factor (4.1) $$\chi_D(\zeta) = \exp 2\pi i \left\{ I(\zeta, C) + \int_C w^*[\zeta] \right\} \qquad (\partial C = D),$$ where C is a (2n-1)-chain with $\partial C = D$ and $w[\zeta]$ means the harmonic 1-form of the first kind associated with the 1-cycle $\zeta$ in the sense of Hodge's Theorem. As was already mentioned in §1, a multiplicative meromorphic function is determined by its divisor uniquely up to a multiplicative constant. Hence we get, as a corollary of the above theorem, the following

Theorem 7. A meromorphic function F(z) with the divisor $D = \partial C$ is one-valued if and only if the congruence $$I(\zeta, C) + \int_C w^*[\zeta] \equiv 0 \pmod{1}$$ holds for every 1-cycle & 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 $\zeta$ with integral coefficients, $\chi_D = \chi_D(\zeta)$ is a character of the 1-homology group $H^1(\mathfrak{M})$ of $\mathfrak{M}$ over the additive group of all integers. Then $\chi_D$ will be called the integral character [13, p. 125] associated with the divisor D. As a simplest example, let us consider a "torus" $\mathfrak{T}$ obtained from the n-dimensional complex vector space $\{z; z = (z^1, z^2, \ldots, z^n)\}$ by identifying points which are congruent to each other with respect to the discrete subgroup generated by 2n linearly independent vectors $\pi_k = (\pi_k^a)$ $(k = 1, 2, \ldots, 2n)$ . The Kählerian metric $$ds^2 = \Sigma_{\alpha=1}^n |dz^{\alpha}|^2$$ defined in the vector space is invariant under every translation $$T_v \colon z^a \to z^a + v^a;$$ hence it can be considered as a Kählerian metric attached to T. Putting $$\pi_{k}^{a} = \pi_{k}^{j} + i\pi_{k}^{n+j} \qquad (j = a; a = 1, 2, ... n),$$ $$\pi_{ka} = \frac{1}{2} \bar{\pi}_{k}^{a} = \frac{1}{2} (\pi_{k}^{j} - i \pi_{k}^{n+j}) \qquad (j = a),$$ we introduce real components $\pi_{k^j}$ and covariant components $\pi_{ka}$ of $\pi_k$ . Again, for simplicity's sake, we assume that $|\pi_k|=1$ , so that the volume of $\mathfrak X$ with respect to the metric $ds^2$ is equal to 1. The point $t\pi_k$ describes a closed curve in $\mathfrak X$ when the real parameter t moves from 0 to 1 which will be denoted by $\zeta_k$ . Then $\{\zeta_1, \zeta_2, \ldots, \zeta_{2n}\}$ constitutes a base of the 1-homology group $H^1(\mathfrak X)$ of $\mathfrak X$ . A base of the space $\mathfrak S^1$ of all harmonic 1-forms of the first kind is given simply by $\{dx^1, dx^2, \ldots, dx^{2n}\}$ ; hence we have $$w^1(x, \xi) = \sum_{k=1}^{2n} dx^k d\xi^k.$$ The harmonic 1-form $w[\zeta_k]$ associated with $\zeta_k$ is therefore represented as $$w[\zeta_k](x) = \pi_{ka}dz^a + \bar{\pi}_{ka}d\bar{z}^a.$$ Now, suppose an irreducible closed analytic surface $\Gamma$ in $\mathfrak T$ to be given and consider the (2n-2)-cycle $$D = m\Gamma_v - m\Gamma$$ , $\Gamma_v = T_v\Gamma$ , m being a positive integer. Obviously D is homologous to zero; moreover, constructing the "cylinder" $$C(v, \Gamma) = \{tv + z; 0 \leqslant t \leqslant 1, z \in \Gamma\}$$ over the surface $\Gamma$ , we have $$D = m\partial C(v, \Gamma).$$ From (4.1) follows therefore $$\chi_D(\zeta) = \exp 2\pi i \left\{ m \int_{C(v, \Gamma)} w^*[\zeta] \right\}.$$ By a simple calculation, we obtain (4.2) $$\int_{C(v, \Gamma)} w^*[\zeta_k] = i \sum_{\alpha, \beta=1}^n A_{\alpha\beta}(\bar{\pi}_{k\alpha}\bar{v}^{\beta} - \pi_{k\beta}v^{\alpha}),$$ where $$A_{a\beta} = \left(\frac{i}{2}\right)^{n-1} \int_{\Gamma} (-1)^{n+a+\beta} [dz^1 \dots dz^{a-1} dz^{a+1} \dots dz^n d\overline{z}^1 \dots d\overline{z}^{\beta-1} d\overline{z}^{\beta+1} \dots d\overline{z}^n].$$ $C(\pi_j, \Gamma)$ is a (2n-1)-cycle which is homologous to the direct product $\zeta_j \times \Gamma$ . Hence, putting $$Q_{jk} = i \sum_{\alpha, \beta=1}^{n} A_{\alpha\beta} (\pi_{k\beta} \pi_j^{\alpha} - \bar{\pi}_{k\alpha} \bar{\pi}_j^{\beta}),$$ we get by (4.2) $$Q_{jk} = \int_{C(\pi_j, \Gamma)} w^*[\zeta_k] = I(\zeta_j \times \Gamma, \zeta_k) = I(\Gamma, \zeta_j \times \zeta_k);$$ thus Qjk are integers. Finally, putting $$v^{a} = \sum_{i=1}^{2n} \delta^{i} \pi_{i}^{a},$$ 5 j being real numbers, we get from (4.2) and (4.3) $$\int_{C(v, \Gamma)} w^*[\zeta_k] = \Sigma_j Q_{jk} \delta^j$$ and thus obtain the formula (4.4) $$\chi_D(\zeta_k) = \exp 2\pi i \left\{ \Sigma_j Q_{jk} m \delta^j \right\},$$ where &•\ = 7(r, frX J"\)- From this formula we can deduce several conclusions concerning meromorphic functions on %. First, since Qjk are integers, we can choose, for given T, the vector v and the integer m so that Yv^ Y and 2jQjkmô^j= 0 (mod 1). Hence if X contains an irreducible closed analytic surface Y then there exists on X a one-valued meromorphic function with the divisor of the type D, — mYv — mT. As is well known, we can choose the periods -wk so that there exists on X no one-valued meromorphic function other than constants. Such X contains therefore no closed analytic surface. On the other hand, in case Y is not a "cylindrical surface,"¹⁷ we have $$|Q_{jk}| = |A_{\alpha\beta}|^2 \neq 0,$$ and therefore we can choose v and m so that Yv y\ Y and that XD coincides with an arbitrarily preassigned character x of IP(X). Thus, if X contains an irreducible closed analytic surface Y which is not cylindrical, then there exists on X a multiplicative meromorphic function with the divisor of the type D = mYv — mT whose "multiplier" XD coincides with an arbitrarily preassigned x of IP(X).*