Proper Modifications of Generalized p-Kähler Manifolds

Abstract

In this paper, we consider a proper modification f :M → M between complex manifolds, and study when a generalized p−Kähler property goes back from M toM . When f is the blow-up at a point, every generalized p−Kähler property is conserved, while when f is the blow-up along a submanifold, the same is true for p = 1. For p = n−1, we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We get some partial results also in the non-compact case; finally, we end the paper with some examples of generalized p−Kähler manifolds.2010 Mathematics Subject Classification. Primary 53C55; Secondary 32J27, 32L05.

Forms and currents defining generalized p-Kähler structures

“…Thus we get T ∈ (Imσ 2p−1 ∩ Kerσ 2q+1 ) ⊥ : here we use the closure of Imσ 2p−1 to go further (see f.i. [31] p. 127), so we get, as required, Case (2). Let us consider the l.c.s.…”

“…Notice that: pK =⇒ pW K =⇒ pS =⇒ pP L; as regards examples and differences under these classes of manifolds, see [1], [2], [3].…”

“…Case (2). Let us consider the Frechet space E 2p (M) R = (⊕ a+b=2p E a,b (M)) R , and denote by π the projection on the addendum E p,p (M) R .…”

“…In [2] we use closed real (p, p)−forms, which are positive on a fixed compact set, to give the definition of locally p−Kähler manifold (see [2], Definition 6.1) and to study when, in a proper modification f : X → X with compact center, the property of being locally (n − 1)−Kähler comes back from X to X. Nevertheless, there are "exact"(p, p)−forms on X with Ω > 0 on Y for every p > m: in fact, if not, by Theorem 8.1(1) there would exist a pluriharmonic (i.e.…”

On local stabilities of -Kähler structures

“…admitting a Kähler modification) is a -manifold. The converse is an open question as in [Ale17, Introduction]: Is the modification of a -manifold still a -manifold? A recent result [YY17, Theorem 1.3] of S. Yang and X. Yang, by means of a blow-up formula for Bott–Chern cohomologies and the characterizations by Angella and Tomassini [AT13] and Angella and Tardini [AT17] of -manifolds, and also [RYY17, Main Theorem 1.1] by S. Yang, X. Yang and the first author confirm this question in dimension three, that is, the modification of a -threefold is still a -threefold.…”