Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces (Q2701662)

For a projective algebraic manifold \(M\) with ample canonical bundle \(K\) no estimates are in general known for \(m\in \mathbb N\) to guarantee that \(mK\) is very ample. But in this paper it is shown that for \(M\) a compact locally Hermitian symmetric space of non-compact type such estimates can be effectively given. NEWLINENEWLINENEWLINEThe author proves that a finite unramified covering \(M'\) of \(M\) has very ample canonical bundle if the injectivity radius of \(M'\) is bounded from below by a constant which can be effectively estimated. It depends on the diameter of \(M\) and on the rate of convergence of the heat kernels \(k_{i}(t,x,y)\) of the Laplacian operators on \((0,\infty)\times M'\times M'\) to the Bergman kernel for \(t\rightarrow\infty\). A central part of the proof consists in calculating estimates for the heat kernels and for the Bergman kernel when the injectivity radius is greater than an effectively estimable constant. If \(M\) is the quotient of a bounded symmetric domain by an arithmetic lattice and \(M'\) is an unramified covering of \(M\), then \(M'\) has very ample canonical bundle if the order of the covering is greater than some estimate which can be effectively given in terms of the defining number field of the lattice. The main arguments also work for Kähler manifolds whose Riemannian sectional curvature is bounded between two negative constants.