Symplectic Structures Integrated without Degeneracies

We shall show that a sufficiently nontrivial symplectic structure does not admit an integrating family without degeneracies. We can impart an exact meaning to this assertion and prove it only for Kahlerian symplectic manifolds of dimension not exceeding four (that is, 2 or 4). But this statement has evidently a more general character and is valid in a much more general situation. 

Definition 3.4.6 A symplectic structure M w) will be called isotrivial if there exists a finite nonbranching covering x M — M, such that M splits into the direct product of an n-dimensional complex torus, completely isotropic with respect to the form 

A symplectic structure will be called locally trivial if it is integrable and if there exists a locally trivial integrating family. 

We shall similarly define local triviality and isotriviality for an arbitrary morphism (a morphism is isotrivial if, after a finite-sheeted nonbranching change of the base, it becomes a projection onto a multiplier of the direct product). 

Theorem 3.4.4. Let M be a Kahlerian manifold of dimension m = 2n 4 with a symplectic form w. Suppose that the symplectic structure is integrable 

---

An example of a manifold which is completely integrable (and even integrable without degeneracies) is given by the product of two nonalgebraic complex tori of equal dimension with a symplectic structure, in which multipliers are isotropic. But this manifold is not meromorphic ally integrable.. 

---