RESEARCH OF ARITHMETIC VARIETIES AND ALGEBRAIC/ANALYTIC STACKS

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We construct Kummer-Kawamata coverings as algebraic stacks, which play roles such as the curvatures of Hermitian line bundies in complex geometries or the Frobenius maps in the category of varieties over fields of positive characteristics. Applying it, we can translate Esnault-Viehweg vanishing theorems into those in the form of Kawamata vanishings. We obtain vanishing theorems for vector bundles with rational coefficients. We find a general method for enlarging vanishing theorems into those with logarithmic poles. We find divisors on Kummer coverings as algebraic stacks which are assumed to exist in the paper with title Kaehler analogue of Weil conjecture by Serre. However we are not satisfied with it since the theory is not purely algebraic. We prepare more algebro-analytic theory for it. We prove one of Fujita conjecture ; if an ample divisor has the self-intersection number greater than one, the divisor adding a canonical divisor with multiple of the ample divisor more than the dim … More ension of a variety becomes very ample. If the multiple is equal to the dimension, it becomes free. We make use of Kummer coverings and the part of the actions of the automorphism group and induction argument. Fujita found the counter example such that when an ample divisor has the self-intersection of one, the divisor such that the multiple is the dimension plus one does not become free. Studing Esnault-Vehweg type vanishing theorem we find divisors of numerically equivalent to zero are essential instead of those of linearly equivalent to zero if the supports of fractional parts are normal crossing. Hence we define numerical litaka dimension is the maximal Iitaka dimension among the numerical equivalence class. We propose an analogue of Iitaka-Viehweg conjecture for this new dimension. Assuming logarithmic Iitaka-Viehweg conjecture, we obtain the numerical cone theorem as an analogue of Kawamata-Shokulov's Cone theorem and then we have a Zariski decomposition theorem. We define log-stacks to be log-algebraic if there exists a log-scheme dominating the log-stack by surjective log-smooth morphism. We hope this concept is useful in the research of minimal models. We obtain Esnault-Viehweg type vanishing theorems without the resolution of singularities conjecture in the category of varieties over fields of positive characteristics. Also we obtain a vanishing type theorem which works in the category of non Fujiki manifolds. Less