A Kähler structure of the triplectic geometry

We study the geometry of the triplectic quantization of gauge theories. We show that the triplectic geometry is determined by the geometry of a Kähler manifold $\mathcal N$ endowed with a pair of transversal polarizations. The antibrackets can be brought to the canonical form if and only if $\mathcal N$ admits a flat symmetric connection that is compatible with the complex structure and the polarizations.