| 研究実績の概要 |
1. Integrals of logarithmic forms on semi-algebraic sets. We studied the integral of a logarithmic differential form on a compact semi-algebraic set of complex affine space. Such integrals appear naturally as periods of algebraic varieties. We investigated geometric conditions under which the integral converges. We proved the convergence under a simple assumption (called ``admissibility'') on the dimension of the intersection of the semi-algebraic set with the pole divisors of the form.
2. Generalized Cauchy formula for semi-algebraic sets. We formulated and proved a generalization of
the Cauchy residue formula. Here, instead of a simple curve in complex plane, we take a semi-algebraic set in complex affine space and a differential form with logarithmic singularities. Such a formula may be easily verified for examples, and has been widely used. Our theorem is a general statement, assuming only the admissibility of the pertinent semi-algebraic set.
3. Quasi DG categories and mixed motivic sheaves. We introduced the notion of quasi DG category,
generalizing that of DG category. For a quasi DG category, we defined the quasi DG category of ``C-diagrams" with values in the original quasi DG category. We proved that the homotopy category of the quasi DG category of ``C-diagrams" has a canonical structure of a triangulated category.
If we start with the quasi DG category of varieties (over a base variety), then this construction gives us the triangulated category of mixed motivic sheaves.
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