International Congress of Mathematicians

252 P. Biranbe used to obtain many more examples.Here is another typical application: let C be a projective curve of genus > 0.It was observed by Silva [34] that C x CF" can be realized as a hyperplane sectionin various smooth varieties. Note that by Theorem J, C x CF" cannot have anyLagrangian spheres. It immediately follows that the only smooth varieties X thatsupport C x CF" as their hyperplane section must have small dual. For n < 5results of Ein [17, 18] make it even possible to list all such X's.We conclude with a remark on the methods. The symplectic approach outlinedabove gives coarser results. Indeed Sommese [35] provides examples of varieties thatcannot be ample divisors whereas the methods above only rule out the possibilityof being very ample. On the other hand the symplectic approach has an advantagein its robustness with respect to small deformations (see [9], cf. [36]).5.2. Degenerations of algebraic varietiesThe methods of the previous section can also be used to study degenerationsof algebraic varieties. Let Y be a smooth projective variety. We say that Y admitsa Kahler degeneration with isolated singularities if there exists a Kahler manifoldX and a proper holomorphic map n : X —t D to the unit disc D C C with thefollowing properties:1. Every 0 ^ t £ D is a regular value of n (hence, all the fibres X t = 7r -1 (i),t ^ 0, are smooth Kahler manifolds).2. 0 is a critical value of n and all the critical points of n are isolated.3. Y is isomorphic (as a complex manifold) to one of the smooth fibres of n, sayXt 0 ,to^0.As in the previous section this situation is related to symplectic geometrythroughthe Lagrangian vanishing cycle construction. As pointed out by Seidel [39]one can locally morsify each of the critical points in X 0 = 7r _1 (0) and then byapplyingMoser's argument obtain for each critical point of IT at least one Lagrangiansphere in the nearby fibre X f _. Since all the smooth fibres are symplectomorphic weobtain Lagrangian spheres also in Y.Applying results from Section 4 to this situation we obtain examples of projectivevarieties that do not admit any degeneration with isolated singularities. Forexample, let Y be any of the following:• CF", n > 2. Or more generally CF" x M, where M is a smooth variety withn 2 (M) = 0 and dime M ^ n + 1 (mod n + 1).• Any variety whose universal cover is C", (n > 2), or a domain in C".Then by the results in Section 4, Y has no Lagrangian spheres, hence does notadmit any degeneration as above. More examples can be found in [9].This point of view seems non-trivial especially when H n (Y;Z) = 0, wheren = dime Y. In these cases the vanishing cycles are zero in homology and it seemsthat there are no obvious topological obstructions for degenerating Y as above.^From the list above, the first non-trivial example should be CF" with n = odd > 3.It would be interesting to figure out to which extent the above statement could be