| Co-Investigator(Kenkyū-buntansha) |
MABUCHI Toshiki Grad. Sch. of Sci., Osaka univ. Ass. Professor, 大学院・理学研究科, 教授 (80116102)
NAGASE Michihiro Grad. Sch. of Sci., Osaka univ. Ass. Professor, 大学院・理学研究科, 教授 (70034733)
USUI Sanpei Grad. Sch. of Sci., Osaka univ. Ass. Professor, 大学院・理学研究科, 教授 (90117002)
TAKAHASHI Satoru Grad. Sch. of Sci., Osaka univ. Lecturer, 大学院・理学研究科, 講師 (70226835)
SAKUMA Makoto Grad. Sch. of Sci., Osaka univ. Ass. Prof, 大学院・理学研究科, 助教授 (30178602)
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| Research Abstract |
Since the advent of the Kodaira Embedding Theorem and the theory on deformations of complex structures, in particular, the stability theorem of Kahler structure, that is, since 1960's it has been a major question that very compact Kahler manifold can be deformed to a projective algebraic manifold. By the Kodaira Embedding Theorem, if a compact Kahler manifold has a Kahler class whose periods are all rational, it is a projective algebraic manifold. By the stability theorem of Kahler structure, any small deformation of a compact Kahler manifold is again Kahler. Moreover, Kahler classes can be chosen to deform continuously, and hence their periods vary also continuously. Therefore, many people conjectured that, one can deform nicely so that a Kahler class on the deformed manifold has rational periods, i.e., every compact Kahler manifold can be deformed to an algebraic manifold. This approach, examining variations of periods of Kahler classes under deformations of complex structures, howev
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er, has not yielded particular resuls on this question yet.
Taking a more direct approach, we succeeded to answer this question affirmatively. First, we construct a C∞embedding, which is sufficiently near to be holomorphic, of a compact Kahler manifold to a complex projective space, by making use of the kernel of C∞ line bundle coefficient Dirac-type operator. Next, we show that the image of this embedding is a complex submanifold of the complex projective space. This emebedding mapping is the higher dimensional quasi-conformal mapping in the title of our research project. Finally, we show that the image of this embedding is a small deformation of the original Kahler manifold.
Moreover, our proof give a sufficient condition for a sequence of rational cohomology classes which converges to a Kahler class can be obtained as a sequence of pull-back via diffeomorphisms of rational polarizations on a sequence of projective algebraic manifolds. In particular, rigid compact Kahler manifolds are projective algebraic manifolds whose Picard numbers are maximal. Less
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