A version of the twisted Poincar{e} duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by twisting the Poisson module structure in a canonical way, which is constructed from the modular derivation. In the case that the Poisson structure is unimodular, the twisted Poincar{e} duality reduces to the Poincar{e} duality in the usual sense. The main result generalizes the work of Launois-Richard cite{LR} for the quadratic Poisson structures and Zhu cite{Zhu} for the linear Poisson structures.
The Type I, II and hybrid (I+II) seesaw mechanism, which explain why neutrinos are especially light, are consequences of the left-right symmetric model (LRSM). They can be classified by the ranges of parameters of LRSM. We show that a nearly cancellation in general Type-(I+II) seesaw is more natural than other types of seesaw in the LRSM if we consider their stability against radiative correction. In this scenario the small neutrino masses are due to the structure cancellation, and the masses of the right handed neutrino can be of order of O(10)TeV. The realistic model for non-zero neutrino masses, charged lepton masses and lepton tribimaximal mixing can be implemented by embedding $A_4$ flavor symmetry in the model with perturbations to the textures.
