The spectral geometry of the Dolbeault Laplacian with coefficients in a holomorphic vector bundle for a Hermitian submersion (Q2735677)

Let \(\pi:(Z,g_Z)\rightarrow(Y,g_Y)\) be a Hermitian submersion. Let \(E_Y\) be a holomorphic vector bundle over \(Y\) and let \(E_Z:=\pi^*E_Y\) be the associated holomorphic pull back bundle over \(Z\). Let \(D_Y^q:=\Delta^{(0,q)}_{E_Y}\) and \(D_Z^q:=\Delta^{(0,q)}_{E_Z}\) be the associated Dolbeault Laplacians with coefficients in \(E_Y\) and \(E_Z\). The author studies when the pull back of an eigensection of \(D_Y^q\) is an eigensection of \(D_Z^q\). NEWLINENEWLINENEWLINEThe author shows if this happens, then the eigenvalue does not change if \(q=0\) and can only increase in general; the author also gives examples where the eigenvalue can actually change. Finally, the author establishes necessary and sufficient conditions to ensure that pullback \(\pi^*\) intertwines the two operators.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00038].