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Download Algebraic geometry 01 Algebraic curves, algebraic manifolds by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov PDF

By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

"... To sum up, this publication is helping to benefit algebraic geometry very quickly, its concrete kind is pleasing for college kids and divulges the wonderful thing about mathematics." --Acta Scientiarum Mathematicarum

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Example text

For example, take a closed symplectic submanifold V of dimension 2 n - 2 in the standard space (CP", coo) such that V is homologous to C P n-1CCP". The space II~P" has "many" J-curves for every almost complex structure J tamed by coo. In particular this is true for the structures J which make the submanifold V J-complex. C~. Cp a rational curve in V through every two points in V. ) form cool V is isomorphic to the standard one. C. Let (V, co) =(S 2, col)O(S 2, co2). 4. But if S col 9 ~ co2, then one needs an additional genericity assumption on J to insure a S2 S2 proper behaviour of J-curves.

B3. The above discussion extends up to certain degree to closed symplectic manifolds V of dimension 2n>6. For example, let (V, co) admit a symplectic embedding of the space ( ~ P " - 1, 09o) for 2n = dim V, such that the normal bundle of C P " - 1 C V is trivial. Consider a co-tame almost complex structure J on V whose restriction to ~ P " - 1 equals the standard structure on ~ P " - 1. Then, if the class of [-~p1] is J-simple, the family of the projective lines in C P " - 1 extends to all of V, thus imposing certain restriction on the topology of V.

G denotes the reflection of S 2 in the equator S x C S 2. Observe that the fixed point set of this I is the torus W = S ~ x S 1 and that W necessarily is totally real in (V, E), that is no tangent plane to W belongs to E. E'2. I f v~ W, then the E-curves Sl(V) and S2(v) are I-invariant and their intersections with W are simple closed (1-dimensional) curves which are homologous in W to the circles S 1 x s and s x S ~, s ~ S 1, correspondingly. Furthermore, every E-curve C = C(vl, v2, v3) for vi~ W, i = 1,2, 3 also is I-invariant and the intersection C n W is a simple closed curve in W.

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