ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 14 Maehara, Kazuhisa, Tokyo Institute of Polytechnics, Japan Deformation of function fields Our function fields are equivalent to the birational equivalence classes of complete algebraic varieties defined over the complex number field. Let f : X → S be a proper smooth connected morphism between smooth varieties with relative divisors A, B on X/S. Assume A_{red} + B_{red} is normal crossing. The dual notion of Ω_{X/S}(A, B) denotes Θ_{X/S}(A, B). There exists a variation of deformation theory of Kodaira-Spencer-Kawamata’s. If Kodaira-Spencer map ρ_{X/S}(A, B) : Θ_{S} → R^{1}f_{∗}Θ_{X/S}(A, B) is zero, then X \ B is locally S-biregular outside A. This will be applied to Viehweg conjecture: κ(det f_{∗}ω_{X/S}^{⊗m}) ≥ var(X/S) for any m > 0 unless f_{∗}ω_{X/S}^{⊗m} ≠ 0. This statement says deformation theory of function fields for algebraic varieties of non negative Kodaira dimension. The torsion freeness of R^{1}f_{∗}ω_{X/S}^{⊗m} is difficult to show.