## Journal of the Mathematical Society of Japan

### A necessary condition for Chow semistability of polarized toric manifolds

Hajime ONO

#### Abstract

Let $\Delta \subset R^n$ be an $n$-dimensional Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_\Delta$ and the very ample $(C^\times)^n$-equivariant line bundle $L_\Delta$ on $X_\Delta$ associated with $\Delta$. In the present paper, we show that if $(X_\Delta,L_\Delta^i)$ is Chow semistable then the sum of integer points in $i\Delta$ is the constant multiple of the barycenter of $\Delta$. Using this result we get a necessary condition for the polarized toric manifold $(X_\Delta,L_\Delta)$ being asymptotically Chow semistable. Moreover we can generalize the result in [4] to the case when $X_\Delta$ is not necessarily Fano.

#### Article information

Source
J. Math. Soc. Japan Volume 63, Number 4 (2011), 1377-1389.

Dates
First available in Project Euclid: 27 October 2011

http://projecteuclid.org/euclid.jmsj/1319721144

Digital Object Identifier
doi:10.2969/jmsj/06341377

Mathematical Reviews number (MathSciNet)
MR2855816

Zentralblatt MATH identifier
1230.14069

#### Citation

ONO, Hajime. A necessary condition for Chow semistability of polarized toric manifolds. J. Math. Soc. Japan 63 (2011), no. 4, 1377--1389. doi:10.2969/jmsj/06341377. http://projecteuclid.org/euclid.jmsj/1319721144.

#### References

• S. K. Donaldson, Scalar curvature and projective embeddings, I, J. Differential Geom., 59 (2001), 479–522.
• W. Fulton, Introduction to Toric Varieties, Number 131, In: Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1993.
• A. Futaki, Asymptotic Chow semi-stability and integral invariants, Internat. J. Math., 15 (2004), 967–979.
• A. Futaki, H. Ono and Y. Sano, Hilbert series and obstructions to asymptotic semistability, Adv. Math., 226 (2011), 254–284.
• I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1994.
• M. M. Kapranov, B. Sturmfels and A. V. Zelevinsky, Chow polytopes and general resultants, Duke Math. J., 67 (1992), 189–218.
• T. Mabuchi, An obstruction to asymptotic semistability and approximate critical metrics, Osaka J. Math., 41 (2004), 463–472.
• T. Mabuchi, An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, I, Invent. Math., 159 (2005), 225–243.
• D. Mumford, Stability of projective varieties, Enseign. Math. (2), 23 (1977), 39–110.
• D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, Third edition, Ergeb. Math. Grenzgeb. (2), 34, Springer-Verlag, Berlin, 1994.
• B. Nill, Gorenstein toric Fano varieties, Dissertation, Universität Tübingen, 2005.
• B. Nill and A. Paffenholz, Examples of non-symmetric Kähler-Einstein toric Fano manifolds, arXiv:0905.2054.
• T. Oda, Convex bodies and algebraic geometry, Ergeb. Math. Grenzgeb. (3), 15, Springer-Verlag, Berlin, 1988.
• H. Ono, Y. Sano and N. Yotsutani, An example of asymptotically Chow unstable manifolds with constant scalar curvature, to appear in Ann. Inst. Fourier, arXiv:0906.3836
• E. Viehweg, Quasi-Projective Moduli for Polarized Manifolds, Ergeb. Math. Grenzgeb. (3), 30, Springer-Verlag, Berlin, 1995.
• X.-J. Wang and X. Zhu, Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math., 188 (2004), 87–103.