A Graded Domain Is Determined at Its Vertex. Applications to Invariant Theory

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 319)


We will prove that a positively graded domain/\(\mathbf{C}\) is uniquely determined by its completion at the irrelevant maximal ideal. As an application we will prove that the logarithmic Kodaira dimension of the smooth locus of a quotient of an affine space modulo a reductive algebraic group is \(-\infty \).


Graded ring Logarithmic Kodaira dimension Invariant theory 

Mathematics Subject Classification (2000)

14L24 14L30 13A02 


  1. 1.
    Gurjar, R.V.: On a conjecture of C.T.C. Wall. J. Kyoto Univ. 31, 1121–1124 (1991)Google Scholar
  2. 2.
    Gurjar, R.V., Simha, R.R.: Some results on the topology of varieties dominated by \({ C}^n\). Math. Zeit. 211, 333–340 (1992)CrossRefGoogle Scholar
  3. 3.
    Hochschild, G.: Basic Theory of Algebraic Groups and Lie Algebras. Graduate Texts in Mathematics. Springer, Berlin (1981)Google Scholar
  4. 4.
    Hauser, H., Müller, G.: Algebraic singularities have maximal reductive automorphism groups. Nagoya Math. J. 113, 181–186 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Iitaka, S.: Algebraic Geometry. An Introduction to Birational Geometry of Algebraic Varieties. Graduate Texts in Mathematics, vol. 76. Springer, Berlin (1982)Google Scholar
  6. 6.
    Müller, G.: Reduktive automorphismengruppen von analytischer \({ C}\)-algebren. J. Reine Angew. Math. 364, 26–34 (1986)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia

Personalised recommendations