%           Addendum received on 9 February 1996 from the author:


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\title 
Addendum to paper [5] 
\endtitle 

\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 
\endaddress 

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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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