Rational Curves on Algebraic Varieties

点击翻译按钮获取中文摘要

1. The head investigator, Masayoshi Miyanishi, investigated jointly with Kayo Masuda, an automorphism of infinite order of the affine space such that the automorpism fixes pointwise a subvariety and that the codimension of the subvariety is small. As a by-product, they obtained an algebro-geometric characterization of the affine 3-space in terms of GィイD2mィエD2-actions. The head investigator also worked out, jointly with R.V.Gurjar of the Tata Institute, a generalization of the Jacobian Conjecture to the Q-homology planes and proved that the conjecture holds true if the logarihmic Kodaira dimension is l and in the most cases if the logarithmic Kodaira dimension is 0 or -∞. A major work of the head investigator during this period of research under the above title is to have written down a book on open algebraic surfaces which is to be published from the American Mathematical Society.2. Takayuki Hibi investigate finite graphs via the algebras associated to such graphs and the ideals of the … More algebras. Study of the Betti numbers of the algebras associated to various simplical complexes provides us with abundant imformations and has become an effective tool which combines combinatorics, commutative algebras and algebraic geometry.3. Akira Fujiki obtained interesting results via an investigation of algebraic reduction of the twistor spaces on Hopf surfaces. Twistor space is a notion which is close to projective space bundle and expected to clarify differences between algebraic varieties and complex manifolds.4. Yoshinori Namikawa investigated Calabi-Yau manifolds via deformation theory. He also reconstructed flops via deformation theory which appear in the study of birational mappings of higher dimensional algebraic varieties.5. Jun Murakami constructed a new invariant for three-dimensional manifolds which is based on the Kontsevich invariant for the knots and studied its properties. He constructed a topological quantum theory by making use of this invariant, and a family of mapping groups of surfaces as its application.6. Koji Yanagawa made a joint research with E.Ballico on h-vectors of the Poincare series which are defined by Cohen-Macaulay domains over a field of characteristic ρ. Less