An eigenvalue estimate for the \(\overline{\partial}\)-Laplacian (Q1423997)

Let \(F\) be a holomorphic line bundle over an \(n\)-dimensional compact holomorphic manifold \(X\). Fix a Hermitian metric on \(X\) and a line bundle metric on \(F\). Let \(\square=\bar\partial \bar\partial^*+\bar\partial^*\bar\partial\) be the associated Laplacian. Let \(h_{\leq\lambda}^{(p,q)}(F)\) be the number of eigenvalues (counted with multiplicity) of \(\square\) on \((p,q)\) forms with values in \(F\) which are less than or equal to \(\lambda\). The author gives an asymptotic estimate for the distribution of eigenvalues: Theorem 1.1: Assume \(L\) is a line bundle with a metric of semipositive curvature. Let \(E\) be an arbitrary line bundle. Take \(q\geq1\). If \(0\leq\lambda\leq k\), then \(h_{\leq\lambda}^{n,q}(L^k\otimes E)\leq C(\lambda+1)^qk^{n-q}\). If \(1\leq k\leq\lambda\), then \(h_{\leq\lambda}^{n,q}(L^k\otimes E)\leq C\lambda^n\). Note: By replacing \(E\) by \(E\otimes K^{-1}\), a similar estimate holds for \(h_{\leq\lambda}^{0,q}\). One obtains thereby the estimate \(h^{0,q}_{\leq 0}(L^k\otimes E)\leq Ck^{n-q}\) for the dimensions of the Dolbeault cohomology groups.