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By M. Aizenman (Chief Editor)

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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. 1. Let ϕ be a map on P3 associated with a translation of W (E7 ).

Ca Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. it Received: 26 March 2003 / Accepted: 14 September 2003 Published online: 13 February 2004 – © Springer-Verlag 2004 Abstract: A torsion-free G2 structure admitting an infinitesimal isometry such that the quotient is a K¨ahler manifold is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations.

The case of i = 7. We have α7 = (E − E1 − E2 − E3 ) − E4 . Let C7re be a real curve on C from C ∩ (E − E1 − E2 − E3 ) (u = −u1 − u2 − u3 , cf. 2) to C ∩ E4 (u = u4 ). Let 7 ∈ S(X − D1 − D2 ) be the set of points such that |x 2 − w| = ε, |y 2 − 4xw + g2 x + g3 | = ε and x is in the projection of C7re to the x coordinate. 1. As in the 2-dimensional case, we can take i from H3 (X − D1 − D2 , Z). Let F1 , F2 be divisors such that αi is written as αi = F1 − F2 as above. For our purpose, it is enough to add singular 3-chains in F1 and F2 to the above i .

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Communications in Mathematical Physics - Volume 246 by M. Aizenman (Chief Editor)

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