In this paper, we compute the Poisson cohomology groups for any Poisson Del Pezzo surface.

In this paper, we give a description of holomorphic multi-vector fields on smooth compact toric varieties, which generalizes Demazure's result of holomorphic vector fields on toric varieties. Based on the result, we compute the Poisson cohomology groups of holomorphic toric Poisson manifolds, ı.e., toric varieties endowed with T -invariant holomorphic Poisson structures.Key words and phrases. toric varieties, holomorphic multi-vector fields, holomorphic Poisson manifolds, Poisson cohomology.We give some comments for Theorem A:• In the case of k = 1, the above theorem was proved by Demazure [7]. The setis called the root system for (N, ∆). • In the case of k = n, we have the well-known results• The special case of X = CP n was proved in [15]. We would like to point out a sign mistake in the Theorem 3.3 of [15], where V k I should be the weight space corresponding to the character −I. • If X is a toric Fano manifold, Equation (1.3) can be obtained by Theorem 3.6 in [20] since ∧ kThe second part of this paper is devoted to the study of holomorphic Poisson manifolds, especially, the computation of Poisson cohomology groups of T -invariant holomorphic Poisson structures on toric varieties.Recall that a holomorphic Poisson manifold is a complex manifold X equipped with a holomorphic bivector field π such that [π, π] = 0, where [·, ·] is the Schouten bracket. Holomorphic Poisson manifolds are studied by many mathematicians from different viewpoints. The algebraic geometry of Poisson manifolds was first studied by Bondal [1] and Polishchuk [23]. Deformation quantization of Poisson varieties was studied by Kontsevich [17]. Hitchin [12-14] and Gualtieri [11] investigated holomorphic Poisson manifolds as a special case of generalized complex manifolds. The relation of holomorphic Poisson manifolds and Lie algebroids were revealed in [18]. And Poisson structures on flag varieties were studied in [2, 10].The Poisson cohomology groups H • π (X) of a holomorphic Poisson manifold (X, π) is the cohomology group of the complex of sheaves:where d π = [π, ·] and dim X = n. The Poisson cohomology groups of holomorphic Poisson manifolds are computed in various situations [5,15,16,21].

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Abstract. The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex triples and holomorphic symplectic 2-forms on manifolds respectively. Basic properties of such structures are established.