The classical subject of surfaces containing a hyperelliptic curve (here a double cover of P1) among their hyperplane sections was settled some years ago by the third author and Van de Ven [SV] (see also [Se], [Io]). This paper is devoted to answering the following general question arising very naturally from that problem.

The theory of integral geometry has mainly treated identities between integral invariants of submanifolds in Riemannian homogeneous spaces like as dμg(g) where M and N are submanifolds in a Riemannian homogeneous spaces of a Lie group G and I(M ∩ gN) is an integral invariant of M ∩ gN. For example Poincaré’s formula is one of typical identities in integral geometry, which is as follows. We denote by M(R2) the identity component of the group of isometries of the plane R2 with a suitable invariant measure μM(R2).

In this section we shall present some facts in the background of the problem under our next consideration. Let (X, A, μ) be a finite measure space, and let B be a Banach space with a norm ‖ • ‖. Let M(μ) denote the linear space of all μ-measurable functions from X into R, and let T be a linear operator from B into M (μ).

In this paper we prove the reciprocity law for a Kummer extension of an algebraic number field K. The proof is similar to the proof of the same theorem by Kubota [14, 15]. Such methods were applied by Gauss [6, 7] to the cases K ═ Q, and by Habicht [8] to the case . We now discuss informally the structure of the proof. All definitions and statements made at this stage are only approximations to the truth, and shouldn’t be used as references for the later chapters.

0. In [0-1] it was proved that for any bounded hyperconvex domain D in C2 the Bergman kernel function K(z, w) of D satisfies

In case n ═ 1, this is due to a behavior of sublevel sets of the Green function. The general case then follows by the extendability of L2 holomorphic functions.

(Anti-)self-dual metrics are 4-dimensional Riemannian metrics whose Weyl conformai tensor W half vanishes. The Weyl conformai tensor W of an arbitrary metric on an oriented 4-manifold has in general the self-dual part W+ and the anti-self-dual part W− with respect to the Hodge star operator * and one says that a metric is self-dual or anti-self-dual if W− = 0 or W+ = 0, respectively.

Let X be a non-compact Riemann surface and D ⊂ X an open subset. By a classical result due to Behnke and Stein [2] D is Runge in X (i.e. the restriction map has dense image) iff X\D has no compact connected components. In other words the obstruction to holomorphic approximation is purely topological. This result has been generalized to 1-dimensional Stein spaces by Mihalache in [11].

Note that an extra condition of symmetry of f, namely, the boundedness of f(x) = f(− x) is added to the hypothesis of the Lemma. Lemma 2.1 was used in three places in the paper to prove that the functions F, G*, and f given in pages 140, 141, and 144, respectively, belong to BMO. It can be checked by a standard argument that these functions satisfy the symmetry condition.