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Moreover, U is a Hamiltonian isometry of the K¨ahler structure (27) and the corresponding momentum map is −t. The integrality assumption for ω2 implies that there exists a principal S 1 bundle over 1 N, classified by [− 2π ω2 ] ∈ H 2 (N, Z). We denote by Y the corresponding 7-dimensional total space and take η to be a connection 1-form satisfying (12). 54 V. Apostolov, S. Salamon Corollary 4. With the above assumptions, the 3-form ϕ = t ω1 ∧ η + dt ∧ ξ ∧ η + t 4 ω3 ∧ dt + t ω2 ∧ ξ defines a torsion-free G2 structure on Y .

1. For case (i-2) and (i-3), we shall write the results only. Replace x/z, y/z, w/z by x, y, w. 1. We denote the set of piece-wise smooth singular 3-chains in X − D1 − D2 by S(X − D1 − D2 ). Using the holomorphic 3-form on X \ (D1 ∪ D2 ), ω= c dx ∧ dy ∧ dw (x 2 − w)(y 2 − 4xw + g2 x + g3 ) (29) (the constant c ∈ C× is determined later), we define the map χX : S(X −D1 −D2 ) → C by the pairing ω, ( ∈ S(X − D1 − D2 )). Let C denote the elliptic curve D1 ∩ D2 . We define a map Q(α) → S(X − D1 − D2 )/H1 (C, Z).

Conservation Law In this section we prove the following theorem. 1. 1. 1. 1 is given by the linear system |− 21 KX | in generic terms. 2), each fiber is preserved by translations associated with the Weyl group W (E7 ), because (i) each surface is an elliptic surface (ii) each map is birational (hence continuous except at the indefinite points) and preserves the fibration (iii) the modulus of elliptic curve is preserved (iv) at least the intersection curve of the pencil is preserved. 1 the corollary below follows.