We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the first author. As direct corollaries, we prove several deformation invariance theorems for Hodge numbers. Moreover, we also study the Gauduchon cone and its relation with the balanced cone in the Kähler case, and show that the limit of the Gauduchon cone in the sense of D. Popovici for a generic fiber in a Kählerian family is contained in the closure of the Gauduchon cone for this fiber.is the identity matrix.Using this calculation and its corollaries, we are able to reprove an important result (Proposition 2.7) in deformation theory of complex structures, which asserts that the holomorphic structure on X t is determined by ϕ(t). Actually, we obtain that for a differentiable function f defined on an open subset of X 0where the differential operator d is decomposed as d = ∂ t + ∂ t with respect to the holomorphic structure on X t and e i ϕ follows the notation (1.1).Motivated by the new proof of Proposition 2.7, we introduce a mapwhich plays an important role in this paper and is given in Definition 2.8. This map is a real linear isomorphism as t is arbitrarily small. Based on this, we achieve:Proposition 1.2 (=Proposition 2.13). For any α ∈ A * , * (X 0 ), ∂ t (e iϕ|iφ (α)) = 0 6

Abstract. We prove several formulas related to Hodge theory and the Kodaira-SpencerKuranishi deformation theory of Kähler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of Calabi-Yau manifolds. Similar constructions are also applied to the deformation spaces of compact Kähler manifolds.

We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of (•, 0)-and (0, •)-Hodge numbers on a compact complex manifold, and obtain the equality for the numbers of the blow-ups and blow-downs in the weak factorization of the bimeromorphic map between two compact complex manifolds with equal (1, 1)-Hodge number or equivalently second Betti number. Many examples of the latter one are listed. Inspired by these, we obtain the bimeromorphic stability for degeneracy of the Frölicher spectral sequences at E1 on compact complex threefolds and fourfolds.

We use a sheaf-theoretic approach to obtain a blow-up formula for Dolbeault cohomology groups with values in the holomorphic vector bundle over a compact complex manifold. As applications, we present several positive (or negative) examples associated to the vanishing theorems of Girbau, Kawamata-Viehweg and Green-Lazarsfeld in a uniform manner and study the blow-up invariance of some classical holomorphic invariants.