On a complex manifold a Hermitian metric which is simultaneously SKT and balanced has to be necessarily Kähler. It has been conjectured that if a compact complex manifold has an SKT metric and a balanced metric both compatible with , then is necessarily Kähler. We show that the conjecture is true for nilmanifolds.

We construct new smooth solutions to the Hull–Strominger system, showing that the Fu–Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $$13 \le k \le 22$$13 ≤ k ≤ 22 and $$14\le r\le 22$$14 ≤ r ≤ 22 , the smooth manifolds $$S^1\times \sharp _k(S^2\times S^3)$$S 1 × ♯ k ( S 2 × S 3 ) and $$\sharp _r (S^2 \times S^4) \sharp _{r+1} (S^3 \times S^3)$$♯ r ( S 2 × S 4 ) ♯ r + 1 ( S 3 × S 3 ) , have a complex structure with trivial canonical bundle and admit a solution to the Hull–Strominger system.