In this paper, we present two kinds of total Chern forms [Formula: see text] and [Formula: see text] as well as a total Segre form [Formula: see text] of a holomorphic Finsler vector bundle [Formula: see text] expressed by the Finsler metric [Formula: see text], which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133–144] to some extent. As some applications, we show that the signed Segre forms [Formula: see text] are positive [Formula: see text]-forms on [Formula: see text] when [Formula: see text] is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler–Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou’s one [Finsler geometry on complex vector bundles, in A Sampler of Riemann–Finsler Geometry, MSRI Publications, Vol. 50 (Cambridge University Press, 2004), pp. 83–105] and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.
By use of a natural extension map and a power series method, we obtain a local stability theorem for p-Kähler structures with the (p, p + 1)-th mild ∂∂-lemma under small differentiable deformations.
By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the
$(n-1,n)$
th mild
$\partial \overline {\partial }$
-lemma by power series method and the other one on p-Kähler structures with the deformation invariance of
$(p,p)$
-Bott–Chern numbers.
We present a new method to solve certain∂-equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a ∂∂-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at E 1 -level, as well as certain injectivity theorem on compact Kähler manifolds.Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic (n, q)-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by differential geometric method.Corollary 0.3 (=Corollary 3.1). With the same notations as in Theorem 0.1, if α ∈ A 0,0 (X, Ω p X (log D)) with∂∂α = 0, then ∂α = 0. It is well-known that Deligne's degeneracy of logarithmic Hodge to de Rham spectral sequences at E 1 -level [8] is a fundamental result and has great impact in algebraic geometry, such as vanishing and injectivity theorems. P. Deligne and L. Illusie [9] also proved this degeneracy by using a purely algebraic positive characteristic method. For compact Kähler manifolds, as the second application of Theorem 0.1, we can give a geometric and simpler proof of Deligne's degeneracy theorem.Theorem 0.4 (=Theorem 3.2). With the same notations as in Theorem 0.
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