Integrability on Complex Symplectic Manifolds

1 Different Notions of Complex Integrability and Their Interrelation 

Fomenko has formulated [143] the following general problem how can one algorithmically find a maximal linear commutative algebra of functions on a symplectic manifold and establish how many parameters describe the set of all such algebras Above we have discussed the real version of this problem, now we shall briefly treat the compact version. 

The specific feature of the compelx analytic case is that on complex manifolds there are few holomorphic functions. For instance, if a manifold is compact then, by the well-known Liouville theorem, there is not a single non-constant holomorphic function on this manifold. Therefore there is no point in literally transferring the standard definition of the completely integrable symplectic structure from the smooth case. We shall further analyze several distinct notions of Liouville integrability. This question has been examined by Markushevich, and the results are presented below. 

A symplectic manifold will be understood as a complex manifold M on which there exists a closed holomorphic 2-form w r(M, nondegerate at all points of M. The nondegeneracy condition automatically implies that the dimension of M equals 2n, i.e., is even (we mean the complex dimension) and that the canonical fibre bundle of holomorphic differential forms whose highest degree equals 2n is trivial. 

Indeed, the form = a A A a (n times) is its basis. The latter condition will be denoted as Km = 0, where Km the so-called canonical class of divisors on A/, and if this condition is fulfilled, we say that the canonical class M is trivial. The pair (M,a ), in which the form oj is specified up to proportionality, will be called a symplectic structure. 

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