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We consider compact Kählerian manifolds 
$X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure 
$\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor 
$D(\unicode[STIX]{x1D6F1})$. We prove that 
$(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on 
$H^{2}$ of the open symplectic manifold 
$X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this 
$H^{2}$ provided the Hodge number 
$h_{X}^{2,0}=0$, and finally that the degeneracy locus 
$D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of 
$(X,\unicode[STIX]{x1D6F1})$.
