Invariance of generalized Gauduchon sign sets under modifications

Determine whether, for every compact complex manifold M and every integer 1≤k≤dimℂ(M)−2, the sign set C_k(M)={sgn c_k(ω):ω is a Hermitian metric on M} of generalized Gauduchon constants is invariant under modifications of M.

Background

For a compact complex manifold M, the generalized Gauduchon constant c_k(ω) is associated with a Hermitian metric ω through the equation √−1∂∂̄(e^vω^k)∧ω^{n−k−1}=c_k e^vω^n. The set C_k(M) records which of the signs −1, 0, and 1 occur as c_k(ω) varies over all Hermitian metrics. The cited question asks whether this sign set is unchanged under modifications, such as blow-ups.

The paper establishes only partial invariance: under a finite sequence of blow-ups with smooth centers, occurrence of the signs −1 and 1 is preserved in the forward direction, and the full sign set {−1,0,1} is preserved when it already occurs on the original manifold. These results do not settle invariance under arbitrary modifications or establish the converse direction in general.