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

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**Additional resources for Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes**

**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.