Compatibility between non-Kähler structures on complex (nil)manifolds

Abstract

We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds: locally conformally Kähler, k-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a k-Gauduchon metric is necessarily a torus. Combined with the main result in [FV16], this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally Kähler is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyperKähler structure.

Locally conformally balanced metrics on almost abelian Lie algebras

“…The same transversality no longer holds when considering the weaker LCB condition instead of the balanced condition, and the same Hermitian metric can even be SKT and LCB at the same time: in [17], it was proven that every non-Kähler compact homogeneous complex surface admits a compact torus bundle carrying an SKT and LCB metric; moreover, an example of compact nilmanifold in any even dimension admitting a left-invariant metric which is both SKT and LCB with respect to a fixed left-invariant complex structure was exhibited in [26].…”

“…If g is nilpotent, the it admits an LCB structure (J, g) if and only if it is isomorphic to one of the following: (0, 0, 0, 0, 0, f 12 ), (0, 0, 0, f 12 , f 13 , f 14 ). If g is non-nilpotent, then it admits an LCB structure (J, g) if and only if it is isomorphic to one of the following: 26 , pf 36 , qf 46 , qf 56 , 0), pr = 0, p = ±q, l 2 = (f 16 , pf 26 + f 36 , pf 36 , pf 46 + f 56 , pf 56 , 0), p = 0, l 3 = (pf 16 , qf 26 , qf 36 , rf 46 + f 56 , −f 46 + rf 56 , 0), pq = 0, q = ±r, l 4 = (pf 16 , qf 26 + f 36 , −f 26 + qf 36 , rf 46 + sf 56 , −sf 46 + rf 56 , 0), pqs = 0, q = ±r, 26 , pf 36 , 0, 0, 0), p = 0, l 10 = (pf 16 , qf 26 + f 36 , −f 26 + qf 36 , 0, 0, 0), pq = 0, l 11 = (f 16 , f 26 , pf 36 , pf 46 , 0, 0), p = 0, ±1, l 12 = (f 16 , f 26 , f 46 , 0, 0, 0), l 13 = (f 16 , f 26 , qf 36 + rf 46 , −rf 36 + qf 46 , 0, 0), q = ±1, r = 0, l 14 = (pf 16 + f 26 , −f 16 + pf 26 , f 46 , 0, 0, 0), l 15 = (f 16 + f 26 , f 26 , f 36 + f 46 , f 46 , 0, 0), l 16 = (pf 16 + f 26 , −f 16 + pf 26 , qf 36 + rf 46 , −rf 36 + qf 46 , 0, 0), r = 0, p 2 + q 2 = 0, p = ±q, Proof. Let (J, g) be an LCB structure on g. As in Theorem 3.1, we need to examine each matrix A i in (3.2) to see whether they can satisfy the LCB condition A t i v = 0 for some suitable metric and vector v. Of course v = 0 is a sufficient condition and, in this case, after discarding the algebras admitting balanced or LCK structures (including the nilpotent (0, 0, 0, 0, f 12 , f 13 ), which admits balanced structures, by [32]), we have that A 1 yields l 1 , l 6 , l 9 and l 11 , A 2 yields l 3 , l 8 , l 10 and l 13 , A 3 yields l 4 and l 16 , A 4 yields l 2 and l 15 , A 5 yields l 5 and l 17 .…”

“…A further weakening of both the balanced and the LCK conditions is given by the locally conformally balanced (LCB) condition, whose definition is analogous to the one for the LCK condition and which is equivalently defined by dθ = 0. LCB metrics have been studied, for instance, in [5,6,17,24,26,28,30,31,33]. When n = 2, balanced metrics are Kähler and LCB metrics are LCK.…”

“…In [26], the authors investigate the existence of two different types of special Hermitian metrics on a fixed compact complex nilmanifold (namely, the quotient of a simply connected nilpotent Lie group by a lattice): in Section 4, we consider analogous questions for almost abelian solvmanifolds, highlighting similarities and differences with respect to the nilpotent setting.…”

$p$-Kähler and balanced structures on nilmanifolds with nilpotent complex structures

“…[18]). Moreover, by [14] if (X, J) is a 2-step nilmanifold with leftinvariant complex structure and J-invariant center, then J is nilpotent. We now show that p = n − k in Theorem 2.3 is optimal.…”