Abstract
We give here a necessary and sufficient condition for a subvariety of a projective non-singular variety to be contracted in an algebraic variety which is again non-singular projective, and study some geometric properties of the contraction.
Similar content being viewed by others
References
ARTIN, M.:Some numerical criteria for contractability of curves on algebraic surfaces. Amer. J. Math.84, 485–496 (1962)
BĂDESCU, L.: Contractions rationelles des variétés algébriques. Ann. Sc. Norm. Sup. Pisa27, 743–768 (1973)
—: On certain isolated normal singularities. Nagoya Math. ' J.61, 205–220 (1976)
—, FIORENTINI, M.:Criteri di semifattorialità e di fattorialità per gli anelli locali con applicazioni geometriche. Ann. Mat. Pura Appl.103, 211–222 (1975)
GIESEKER, D.: Contrbutions to the theory of positive embeddings in algebraic geometry. Thesis, Harvard Univ. Cambridge (1969)
GRIFFITHS, P. A.: The extension problem in complex analysis II: Embedding with positive normal bundle. Amer. J. Math.88, 366–446 (1966)
GROTHENDIECK, A., DIEUDONNÉ, J.:Éléments de Géométrie Algébrique. Publ. Math. IHES4,8,11 (1960, 1961, 1961)
HARTSHORNE, R.: Ample Vector Bundles. Publ. Math. IHES29, 63–94 (1966)
—: Ample Subvarieties of Algebraic Varieties. Lecture Notes in Mathematics156, Berlin-Heidelberg-New York, Springer 1970
—: Ample vector bundles on curves. Nagoya Math. J.43, 73–89 (1971)
HIRONAKA, H.: Resolution of singularlities of an algebraic variety over a field of characteristic zero:I. Ann. of Math.79., 109–203 (1964)
KODAIRA, K.: On Kähler varieties of restricted type. Ann. Math.60, 28–48 (1954)
LASCU, A. T.: Sous-variétés régulirement contractibles d'une variété algébrique. Ann. Sc. Norm. Sup. Pisa23, 675–696 (1969)
MOIŠEZON, B.: On n-dimensional compact complex varieties with n-algebraically independent meromorphic functions. A.M.S. Translations63, 51–177 (1967)
NAGATA, M.: Existence theorem for nonprojective complete algebraic varieties. Illinois J. Math.2, 490–498 (1958)
NAKANO, S.: On the inverse of monoidal transformation. Publ. R.I.M.S., Kyoto Univ.6, 483–502 (1970–71)
ŠAFAREVIČ, I.R.: Lectures on Minimal Models and Birational Transformations of Two Dimensional Schemes. Tata Lecture Notes37, Bombay (1966)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ishii, S. Some projective contraction theorems. Manuscripta Math 22, 343–358 (1977). https://doi.org/10.1007/BF01168221
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01168221