Bott–Chern harmonic forms on complete Hermitian manifolds

Abstract

Let [Formula: see text] be a Hermitian manifold of complex dimension [Formula: see text]. Assume that the torsion of the Chern connection [Formula: see text] is bounded, and that there exists a [Formula: see text]exhausting function [Formula: see text] such that [Formula: see text] are bounded. We characterize [Formula: see text] Bott–Chern harmonic forms, extending the usual result that holds on compact Hermitian manifolds. Finally, if [Formula: see text] is Kähler complete, [Formula: see text], with [Formula: see text] bounded, and the sectional curvature is bounded, then we get a vanishing theorem for [Formula: see text] Bott–Chern harmonic [Formula: see text]-forms, if [Formula: see text].

Aeppli cohomology and Gauduchon metrics

“…In this context, Gromov [7] got a vanishing result for L 2 -Dolbeault cohomology, that is, he proved that if (M, J, g, ω) is a complete Kähler manifold of complex dimension n such that ω is d-bounded, i.e., ω = dη, with η bounded, then H p,q dR (M ) = 0 for p + q = n. In his proof, he made use of the L 2 -Hodge decomposition theorem, and showed the vanishing of L 2 de Rham harmonic forms. In [16] and [17] (see also [10] for the almost Kähler setting), the authors of the present note extend Gromov's result for W 1,2 Bott-Chern harmonic forms, giving a characterization of W 1,2 Bott-Chern harmonic forms on Stein d-bounded manifolds, respectively on complete Hermitian manifolds. Along the same line, we prove the following result (see Theorem 3.3), which implies Theorem 1.1.…”

$W^{1,2}$ Bott-Chern and Dolbeault decompositions on Kähler manifolds

“…During the last years, Tomassini and the author of the present paper studied W 1,2 Bott-Chern harmonic forms, namely smooth forms which are in the kernel of the operator ∆BC with bounded W 1,2 norm, on d-bounded Stein manifolds [15], and on complete Hermitian manifolds [16]. We proved some characterizations of W 1,2 Bott-Chern harmonic forms and vanishing results following Gromov [9].…”

“…Theorem 1.3. [16,Theorem 4.4] Let (M, J, g, ω) be a complete Kähler manifold. Assume that the sectional curvature is bounded.…”