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Пишет Misha Verbitsky (![[info]](http://lj.rossia.org/img/userinfo.gif){kind=small} tiphareth) |
The conjecture is in fact very easy for all
codimensions except one.
\theorem
Let $M, \Omega$ be a compact holomorphic symplectic Kahler manifold,
$Z\subset M$ a subvariety of codimension $\geq 2$.
Then any non-zero exterior power of $\Omega$ is non-exact
on $M \backslash Z$.
\proof
Suppose that $\Omega = d\alpha$ on $M \backslash Z$.
Then $\alpha$ is a holomorphic form, hence it can be
extended to $M$ by Hartogs. However, on $M$ all
holomorphic forms are closed, as follows from Hodge theory.
\endproof
codimensions except one.
\theorem
Let $M, \Omega$ be a compact holomorphic symplectic Kahler manifold,
$Z\subset M$ a subvariety of codimension $\geq 2$.
Then any non-zero exterior power of $\Omega$ is non-exact
on $M \backslash Z$.
\proof
Suppose that $\Omega = d\alpha$ on $M \backslash Z$.
Then $\alpha$ is a holomorphic form, hence it can be
extended to $M$ by Hartogs. However, on $M$ all
holomorphic forms are closed, as follows from Hodge theory.
\endproof
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