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

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We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any suciently large unramied covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings. Let L be an ample line bundle on an algebraic manifold M. From Kodaira's Embedding Theorem, we know that mL is very ample if m is suciently large so that the sections of mL give rise to an embedding of M into some projective space. A natural question is on the estimates of such m so that mL is very ample. In particular, one may ask the same question for the canonical line bundle of a manifold with negative rst Chern class, or of general type. Of particular interest among such manifolds is the class of locally Hermitian symmetric spaces. The main purpose of this article is to show that for compact locally Hermitian symmetric manifolds of non-compact type, K is very ample if the injectivity radius of the manifold is greater than some eective constant which can be estimated in terms of some geometric data. Eective estimates for mK = K +( m 1)K with m 2 in terms of injectivity radius have been obtained in (HT) and (Y2). The diculty in the case of m =1c an be seen from the fact that in constructing holomorphic sections from the Bochner- Kodaira technique, one always needK +L for some extra positivity fromL .I n (Y2), it is shown that for a tower of coverings of compact Hermitian locally symmetric manifolds Mi, KMi becomes very ample when i approaches to1. To compensate for the extra positivity required for the Bochner-Kodaira technique, we prove our result by showing that the limit of the Bergman kernel on Mi approaches the corresponding one on the universal covering. However, the limiting process makes the whole approach highly non-eective and requires an innite sequence of normal coverings of the original manifold. The main purpose of this article is to show the very ampleness of KM for any unramied covering M of Mo with injectivity radius or degree of the coverings greater than some eectively estimable constant. For this purpose, an alternate approach using heat kernel estimates combined with Atiyah's