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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then the metric must be Kähler when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c not-equals 0"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo> ≠ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c\neq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and must be Chern (or Levi-Civita) flat when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c equals 0"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c less-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c\leq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) by Chen-Chen-Nie [Sci. China Math. 64 (2021), pp. 763–780], etc. In this short note, we consider compact quotients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash normal upper Gamma"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal"> Γ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G/\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a discrete subgroup. We confirm the conjecture when the Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathfrak g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> either is almost abelian, or contains a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant abelian ideal of codimension 2.