Complex balanced spaces

Abstract

In this paper, the concept of balanced manifolds are generalized to reduced complex spaces: the class B and balanced spaces. Compared with the case of Kählerian, the class B is similar to the Fujiki class C and the balanced space is similar to the Kähler space. Some properties about these complex spaces are obtained, and the relations between the balanced spaces and the class B are studied. keywords: class B, balanced space, balanced metric, Fujiki class C AMSC: 32C15, 32C10, 53C55, Proposition 2.2. The class B is invariant under bimeromorphic maps. According to Definition 2.1 and Proposition 2.2, compact balanced manifolds, reduced compact complex spaces in the Fujiki class C and the normalizations of complex spaces in B are all in B. Obviously, if X ∈ B is nonsingular, then X is a balanced manifold.Proposition 2.3. If X and Y are reduced compact complex spaces, then X × Y is in the class B if and only if X and Y are both in the class B.

Strongly Gauduchon spaces

“…where C is the Fujiki class and the first " " is proved in [8], Section 2 . If X is a reduced compact complex space of pure dimension, then X ∈ S G if and only if every irreducible component of X is in S G .…”

“…By Proposition 3.3, Y × Z ∈ S G . Assume Y × Z ∈ B, by [8], Proposition 2.3, we know Z ∈ B. Since Z is nonsingular, Z is balanced, which contradicts the choice of Z.…”

“…Obviously, strongly Gauduchon manifolds and the normalizations of complex spaces in class S G are in class S G . Recall that a reduced compact complex space X is called in class B, if it has a desingulariztion X which is a balanced manifold, referring to [8]. Then complex spaces in class B are in class S G .…”

“…, q is the projection to X, and f is the projection to Y . We can prove that f is a submersion of complex manifolds and q is a modification, referring to [8] We generalize Theorem 3.5 (2) and Theorem 3.9 (2) in [1] as follows.…”

Holomorphic Gromov's partial order

“…Recall a proposition of A. Fujiki in [10] about modifications, whose original proof uses the method of local cohomology. For a convenience, we give a simpler proof as in [9]. Proposition 2.5 ([10], Proposition 1.1).…”