PomeB8583A0E17A6Theorem 23.2.15.B8583A0E17A6 If the least eigenvalue of $\kappa _\lambda $ relative to $\omega $ is constant on $X$, then it is zero and $\kappa _\lambda $ is semipositive. Its ordinary Bott–Chern class is nonzero and pseudoeffective.
If all relative eigenvalues are constant, the rank is constant and lies between $1$ and $n-1$. In particular, the conclusion holds on a compact quotient $\Gamma \backslash G$ with left-invariant complex structure and strictly LCK tensors.
Proof. Let $r$ be the constant least relative eigenvalue. If $r<0$, put $t_0=-1/r$. Then $\eta _ t>0$ for $0\leq t<t_0$, whereas $\eta _{t_0}$ is semipositive with a kernel at every point. Thus $V(t_0)=0$. The preceding proposition and polynomial continuity instead give $V(t_0)\geq V(0)>0$. This proves $r\geq 0$ and semipositivity.
The nonnegative smooth form $\kappa _\lambda ^ n$ has integral zero by the same proposition, hence vanishes everywhere. Thus the least eigenvalue is zero at every point. Applying the Lee degree (23.2.8) to a conformal Gauduchon representative, as above, proves that the class is nonzero. The form $\kappa _\lambda $ itself represents it positively.
Constant spectrum gives constant rank. The positive pairing (23.2.8) excludes rank zero, and the vanishing top power excludes rank $n$. On the stated compact quotient, the relative matrix is constant in a left-invariant frame, which proves the last assertion. $\square$
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