| 研究概要 |
(Logarithmic integrals associated to mixed Tate motives) We studied integrals which appear as periods of mixed Tate motives.
(1) For a semi-algebraic set S in complex n-dimensional space and a differential form having poles along co -ordinate hyperplanes, assuming a certain condition on the dimension of the intersection of S and the pole divisor, we showed that the integral of the form on S absolutely converges. (2) We showed the Cauchy formula for semi-algebraic sets in complex n-space with respect to its intersection with coordinate hyper -planes. (3) We defined a complex of semi-algebraic chains of complex n-space, and showed that it calculates the homology of the n-space.
(Relative algebraic correspondences and quasi DG categories)
(1) Fix an algebraic variety S as a base. We defined the complex of algebraic correspondences between varieties over S; we showed that the class of varieties over S, together with the complex of algebraic correspondences constitutes a quasi DG category (a generalization of a DG category). (2) Given a quasi DG category C, we gave the construction of another quasi DG category C', whose associated homotopy category has a natural structure of a triangulated category.
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