on stability criterion of complete intersections
Tóm tắt
For protective varieties, it is known that Chow stable implies N-th Hilbert-Mumford stable for N sufficiently large, which follows from the works of J. Fogarty [2, 6]. In this article, we firstly shall provide a simple criterion for Chow stability of complete intersections. The criterion for Chow stability was previously provided by Mumford [5], but our calculation is different from Mumford’s in that ours is based on the results of Zhang’s article [10]. Applying it, we secondly shall give an elementary proof of the above implications in a complete intersections case.
Tài liệu tham khảo
Bauer, T.R. Stability and Futaki invariants of Fano hypersurfaces, math.AG/0111116.
Fogarty, J. Truncated Hilbert functors,J. Reine Angew. Math.,234, 65–88, (1969).
Gieseker, D. Geometric invariant theory and applications to moduli problems, Invariant Theory, (Montecatini,1982), 45–73,Lecture Notes in Math.,996, Springer, Berlin, (1983).
Lu, Z. On the Futaki invariants of complete intersectons,Duke Math. J.,100(2), 359–372, (1999).
Mumford, D. Stability of projective varieties,Enseignement Math. (2),23(1–2), 39–110, (1977).
Mumford, D., Fogarty, J., and Kirwan, F.Geometric Invariant Theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, 34, Springer-Verlag, Berlin, (1994).
Phong, D.H. and Sturm, J. Stability, energy functional, and Kähler-Einstein metrics,Comm. Anal. Geom.,11, 565–597, (2003).
Phong, D.H. and J. Sturm, J. The Futaki invariant and the Mabuchi energy for complete intersections,Comm. Anal.Geom.,12, 323–345, (2004).
Samuel, P. Méthodes d’algèbre abstraite en géométrie algébrique, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 4, Springer-Verlag, Berlin-Gottingen-Heidelberg, (1955).
Zhang, S. Heights and reductions of semi-stable varieties,Compositio Math.,104(1), 77–105, (1996).