Geometry of Maurer-Cartan elements on complex manifolds (Q981721)

This paper studies Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology in the generalized context of extended Poisson manifolds which are determined by Maurer-Cartan elements on complex manifolds. Recall that a complex vector bundle \(A\) over a manifold \(M\) is called a complex Lie algebroid if it is equipped with a bundle map \(a:A\rightarrow T_{\mathbb{C}}M=TM\otimes\mathbb{C}\), called the anchor map, and the space \(\Gamma\left( A\right) \) of sections is equipped with a Lie bracket \(\left[ \cdot,\cdot\right] \) such that \(a\) induces a Lie algebra homomorphism from \(\Gamma\left( A\right) \) to \(\Gamma\left( T_{\mathbb{C}}M\right) \) and \(\left[ u,\cdot\right] \) is a derivation on the \(C^{\infty}\left( M,\mathbb{C}\right) \)-module \(\Gamma\left( A\right) \) for any \(u\in \Gamma\left( A\right) \), where \(u\) acts on \(C^{\infty}\left( M,\mathbb{C} \right) \) as \(a\circ u\). When the dual bundle \(A^{\ast}\) is also a complex Lie algebroid with anchor map \(a_{\ast}\), the pair \(\left( A,A^{\ast}\right) \) is called a complex Lie bialgebroid if the degree-one square-zero differential \(d\) or equivalently \(d_{\ast}\) on the graded commutative algebra \(\left( \Gamma\left( \wedge^{\cdot}A^{\ast}\right) ,\wedge\right) \) or \(\left( \Gamma\left( \wedge^{\cdot}A\right) ,\wedge\right) \), canonically induced by the Lie algebroid structure on \(A\) or \(A^{\ast}\) respectively, is a derivation with respect to both \(\wedge\) and \(\left[ ,\right] \). Furthermore any Hamiltonian operator \(H\in\Gamma\left( \wedge^{2}A\right) \), satisfying the Maurer-Cartan equation \(d_{\ast}H+\frac{1}{2}\left[ H,H\right] =0\) by definition, determines a new Lie bialgebroid \(\left( A,A_{H}^{\ast}\right) \) with the anchor map \(a_{\ast}^{H}=a_{\ast}+a\circ H^{\sharp}\) on the dual \(A_{H}^{\ast}\) of \(A\). For a complex manifold \(X\), by considering the subbundle \(A:=T^{1,0} X\oplus\left( T^{0,1}X\right) ^{\ast}\) of \(T_{\mathbb{C}}X\) and its dual \(A^{\ast}\) which form a Lie bialgebroid such that \(a\) is the projection from \(A\) onto \(T^{1,0}X\) and locally all sections in \(\left\{ \frac{\partial }{\partial z_{i}},d\overline{z_{i}}\right\} _{i}\) mutually \(\left[ ,\right] \)-commute for complex coordinates \(\left\{ z_{i}\right\} _{i}\), the authors define an extended Poisson manifold as a complex manifold \(X\) equipped with a Hamiltonian operator \(H\) for this particular Lie bialgebroid \(\left( A,A^{\ast}\right) \). Such an extended Poisson structure \(H\) on \(X\) generalizes the notion of a holomorphic Poisson structure on a complex manifold where \(H=\pi\) is a holomorphic Poisson bivector field. In this paper, the Lichnerowicz-Poisson cohomology \(H^{\cdot}\left( X,H\right) \) and the Koszul-Brylinski Poisson homology \(H_{\cdot}\left( X,H\right) \) are defined for extended Poisson manifolds \(\left( X,H\right) \), and a sufficient condition for their finite-dimensionality in the case of a compact complex manifold \(X\) is given. A duality \(H^{2n-k}\left( X,H\right) \cong H_{k}\left( X,H\right) \), generalizing the Serre duality of Dolbeault cohomology, is established for extended Poisson manifolds \(\left( X,H\right) \) of complex dimension \(n\) with a vanishing Evens-Lu-Weinstein modular class of the associated complex Lie algebroid \(A_{H}^{\ast}\).