By Kenji Ueno (auth.), Alberto Conte (eds.)
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Extra resources for Algebraic Threefolds: Proceedings of the 2nd1981 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Held at Varenna, Italy, June 15–23, 1981
Is bimeromorphically 31 equivalent to an analytic is an elliptic c) If ~ is a surface with A(M) whose connected Remark curve of genus k(M) I) If = O, Example g~ 2. fibre Mx Example are K 3 are complex equivalent of ~ tori but ~ to an analytic is bimeromorphically K 3 threefold threefold Such threefold M fibres fibre bundle threefold belonging is not over A(M). which belongs t(M). to the class of the Albanese mapping is obtained by the threefold invnlution in of each ~ ~t : Mt---~ C, e U' and resolving its singularities.
When no c o n f u s i o n +C2) - 2 ~ v . D V . (here we use the fact that we have V by a p p l y i n g , 0 -< i -< n . (-| In the f o l l o w i n g instead on is the second e) K o d a i r a are o b t a i n e d Let = hn-i(~7-D)_ b) R i e m a n n - R o c h we have OF FANO T H R E E F O L D S the irregularity q(V) = hl(OV ) = 0 . K 41 COROLLARY 4. Pic(V) ~ H2(V,Zg) . PROOF. Look to the exponential sequence 0 --§ ~ --+ exp r) --+ O* --+ h I (0 V) = h 2 (0 V) = 0 . and use COROLLARY Kv. C2(V) 5. = -24 i PROOF.
Hitchin, N. , and I. M. , Self-duality in four-dimensional Riemannisn geometry. Proc. Royal Soc. London A 362 (1978), 425-461. , Sur les varietes analytiques comples. Ann. Sci. Ecole Norm. Sup. 73 (1956), 157-202. , Application de l'espace des cycles a la classification bimeromorphm des espaces analytiques Kahleriens compacts, preprint 1980. , On the structure of compact complex manifolds in C, to appear. 6 Fujita, T. On K~hler fibre spaces over curves, J. Math. Soc. Japan, 3_~0 11978), 779-794.