A rigidity theorem for the deformed Hermitian-Yang-Mills equation (Q2225894)

Let \((X,\omega)\) be an \(n\)-dimensional compact Kähler manifold, \(L\) be a holomorphic line bundle over \(X\), \(F\) be the Chern connection curvature w.r.t. the Hermitian metric \(h\) on \(L\), \(\mathcal{X}\) be a Kähler metric on \(X\), and \(\displaystyle\xi=\frac{(\omega-F)^n}{\omega^n}\). Let \(Z_{L.[\omega]}=\displaystyle\int_X\xi\frac{\omega^n}{n!}\) such that \([\omega]\) belongs to \(H^{1,1}(X,\mathbb R)\) and \(\theta\) (resp. \(\hat\theta\)) represents the argument of \(\xi\) (resp. \(Z_L.[\omega]\)). Then, the main result focuses on the deformed Hermitian-Yang-Mills equation, see Theorem 1. Precisely, the authors state that if \(\theta\equiv\widehat\theta\) \([2\pi]\) and if there is a positive constant \(C\) such that \(-\displaystyle\frac{1}{C}{<}\lambda_l{<}C\) for \(l\in \{1,\ldots,n\}\), then \(\lambda_l\) is a constant where \(\lambda_l\) is an eigenvalue of \(\mathcal{X}^{-1}(\omega+i\partial\overline{\partial}\varphi)\) an endomorphism on \(T^{1,0}X\) such that \(\varphi\) is a smooth function. Furthermore, the authors show that if \(\omega\) is endowed with a positive orthogonal bisectional curvature, then \(iF\) is proportional to \(\omega\). Also, the authors provide an analogous result for the \(J\)-equation, i.e., when \(\displaystyle\sum_{l=1}^n\lambda_j^{-1}\) is a constant (Theorem 2).