% Addendum received on 9 February 1996 from the author:
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\title
Addendum to paper [5]
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\address
Bronis\l aw Przybylski \newline
Institute of Mathematics, University of \L\'od\'z \newline
ul. Stefana Banacha 22, 90--238 \L\'od\'z, Poland
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Complex Poisson manifolds in the sense of paper [5] were also introduced
independently in paper [3]. In particular, one can see that any complex
Poisson Lie group (see [2] and [4]) is a complex Poisson manifold. The
natural examples of complex Poisson manifolds can be obtained from those
defined in complex algebraic geometry. This follows from paper [1] where
natural Poisson structures defined on some smooth complex algebraic varieties
which can be meant as smooth complex Poisson varieties are considered.
On the other hand, in monograph [2] the authors define a complex Poisson
manifold $(M,J,\Pi)$ to be a complex manifold $(M,J)$ together with a real
Poisson bivector field $\Pi $ on $M$ which is $J$-invariant, i.e. $(J\otimes
J)\Pi = \Pi$. Furthermore, they notice that for such $M$ the symplectic leaves
are complex submanifolds of $M$ equipped with the symplectic structures
(equiv. pseudo-K\"ahler structures) induced by $\Pi$ which have to be
$J$-invariant too. In turn, they remark however that complex Poisson-Lie
groups are not complex Poisson manifolds unless the Poisson bracket is zero
(see [2], 1.3A Remark). This unsatisfactory property suggests that the latter
definition of a complex Poisson manifold is rather not well-adopted.
It turns out that the first definition of a complex Poisson manifold (see [5])
can equivalently be expressed as the complex manifold $(M,J)$ together with a
complex Poisson bivector field $\Pi$ on $M$ which is skew $J$-invariant, i.e.
$(J\otimes J)\Pi = -\Pi$, and holomporphic, i.e. $\Pi$ is subject to the
following conditions:
{\it If} $\alpha$ {\it and} $\beta$ {\it are holomorphic functions defined on
some open subset of} $M$, {\it then so is the function} $\Pi(d\alpha,d\beta)$;
{\it If} $\alpha$ {\it is an antiholomorphic function defined on some open
subset} $U$ {\it of} $M$, {\it then} $\Pi(d\alpha,d\beta)$ {\it vanishes on}
$U$ {\it for any smooth complex function} $\beta$ {\it defined on} $U$.
\Refs
\ref \no1 \by F. Bottacin \pages 391--433 \paper Symplectic geometry on moduli
spaces of stable pairs \yr 1995 \vol 28 \jour Ann. scient. \'Ec. Norm. Sup.
\endref
\ref \no2 \by V. Chari and A. Pressley \book A guide to quantum groups \publ
Cambridge University Press \publaddr Cambridge \yr 1994 \endref
\ref \no3 \by F. Loose, \pages 395--404 \paper Meromorphic Hamiltonian systems
on complex surfaces \yr 1993 \vol 3 \jour Differential Geometry and its
Applications \endref
\ref \no4 \by J-H. Lu and A. Weinstein \pages 501--526 \paper Poisson Lie
groups, dressing transformations and Bruhat decompositions \yr 1990 \vol 31
\jour J. Differential Geometry \endref
\ref \no5 \by B. Przybylski \pages 227--241 \paper Complex Poisson manifolds
\yr 1993 \jour Differential Geometry and its Applications, Proc. Conf. Opava
(Czechoslovakia), August 24-28, 1992, Silesian University, Opava
\endref
\endRefs
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