On the spectrum of the twisted Dolbeault Laplacian on line bundles over Kähler manifolds (Q605286)

Let \(M\) be a compact Kähler manifold of complex dimension \(n\) and let \(E\) be a holomorphic line bundle over \(M\). Let \(\nabla\) be a connection on \(E\) which is compatible with the holomorphic structure on \(E\), let \(h\) be the associated Hermitian metric on \(E\), and let \(\Delta_E\) be the associated twisted Dolbeault Laplacian on \(E\). Let \(i\Lambda F\) be the (normalized) contraction of the curvature of the connection \(\nabla\) with respect to the Kähler form and let \(F_0=\max\{i\Lambda F\}\). The authors use Dirac operator techniques to establish a lower bound for the first eigenvalue \(\lambda_1\) of \(\Delta_E\) showing: Main Theorem: Adopt the notation established above. The first eigenvalue satisfies \(\lambda_1\geq-{n\over 2n-1}F_0\). If equality is satisfied, then \(i\Lambda F=2\pi\)\textrm degree\((E)/(n-1)!\)vol\((M)\) is constant. If \(\psi\) is an associated eigensection, then \(\psi\) is in the kernel of the twistor operator. In particular, the connection is the unique, up to gauge, Hermitian-Einstein connection on \(E\). The authors also show: Corollary. Let \(M\) be a Riemann surface. Then the first eigenvalue satisfies \(\lambda_1\geq\sqrt{-2F_0}\). The main component in the proof is the relation between the twisted complex Dirac operator and the twisted Dolbeault Laplacian and relies upon the Weitzenböck formula.