A note on almost abelian groups with constant holomorphic sectional curvature

A long-standing conjecture in non-Kähler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant c c , then the metric must be Kähler when c ≠ 0 c≠ 0 and must be Chern (or Levi-Civita) flat when c = 0 c=0 . The conjecture is known to be true in dimension 2 by the work of Balas-Gauduchon [Math. Z. 190 (1985), pp. 39–43], Sato-Sekigawa [Math. Z. 205 (1990), pp. 659–668], and Apostolov-Davidov-Muskarov [Trans. Amer. Math. Soc. 348 (1996), pp. 3051–3063] in the 1980s and 1990s. In dimension 3 or higher, the conjecture is still open except in some special cases, such as for all twistor spaces by Davidov-Grantcharov-Muskarov [Rocky Mountain J. Math. 39 (2009), pp. 27–48], for locally conformally Kähler manifolds (when c ≤ 0 c≤ 0 ) by Chen-Chen-Nie [Sci. China Math. 64 (2021), pp. 763–780], etc. In this short note, we consider compact quotients G / Γ G/Γ where G G is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and Γ Γ is a discrete subgroup. We confirm the conjecture when the Lie algebra g { g} of G G either is almost abelian, or contains a J J -invariant abelian ideal of codimension 2.