##
**Chow-stability and Hilbert-stability in Mumford’s geometric invariant theory.**
*(English)*
Zbl 1156.14039

This article is an important contribution to the theory of Geometric Invariant Theory (GIT) stability of polarized varieties.

Let \(M\) be a compact \(n\)-dimensional complex manifold and \(L\) an ample line bundle over \(M\), and call \(P(r)=\chi(M, L^r)\) the Hilbert polynomial of the pair \((M,L)\) (it has degree \(n\)). Then for each \(r>0\) large, the choice of a basis of global holomorphic sections of \(L^r\) gives an embedding of \(M\) in \(\mathbb{P}^{P(r)-1}\). Different bases correspond to projectively equivalent embeddings, and they are related by an element of \(G=SL(P(r),\mathbb{C})\). To any such embedding a classical construction of Chow associates a point in a large projective space, the Chow point of the embedding. We say that \((M,L^r)\) is Chow stable if this Chow point is GIT stable with respect to the natural action of \(G\) (this is well defined because two different embeddings lie in the same \(G\)-orbit). We say that \((M,L)\) is asymptotically Chow stable if \((M,L^r)\) is Chow stable for all \(r\) large.

Another classical construction of Hilbert associates to a given projective embedding of \(M\) in \(\mathbb{P}^{P(r)-1}\) a sequence of points \(\mathrm{Hilb}_{r,k}\), each lying in a big projective space with a natural \(G\)-action. We say that \((M,L^r)\) is Hilbert stable if the points \(\mathrm{Hilb}_{r,k}\) are GIT stable for all \(k\) large. We say that \((M,L)\) is asymptotically Hilbert stable if \((M,L^r)\) is Hilbert stable for all \(r\) large.

A theorem of J. Fogarty [J. Reine Angew. Math. 234, 65–88 (1969; Zbl 0197.17101)] says that asymptotic Chow stability of \((M,L)\) implies asymptotic Hilbert stability (see J. Ross and R. P. Thomas [J. Algebr. Geom. 16, No. 2, 201–255 (2007; Zbl 1200.14095)] for a modern proof). The main result of this paper is that conversely asymptotic Hilbert stability also implies asymptotic Chow stability.

The proof of this result can be sketched as follows, using the notations of J. Ross and R.P. Thomas [loc. cit.]. It turns out that asymptotic Hilbert stability is equivalent to the fact that for any \(r\) large \(\tilde{w}_{r,k}>0\) for all \(k\) large, where \[ \tilde{w}_{r,k}=w(k)rP(r)-w(r)kP(k), \] and \(w(k)\) is a polynomial of degree \(n+1\) (strictly speaking, this has to be checked for a whole family of polynomials \(w(k)\), but it is enough to check them one by one). One then writes \[ \tilde{w}_{r,k}=\sum_{i=0}^{n+1} e_i(r)k^i, \] where the coefficients \(e_i(r)\) are polynomials of degree \(n+1\). Then asymptotic Chow stability is equivalent to the fact that \(e_{n+1}(r)>0\) for all \(r\) large.

It is clear now that asymptotic Chow stability implies asymptotic Hilbert stability. On the other hand, for any positive number \(k'\) one computes that \[ \frac{\tilde{w}_{r,kk'}}{kk'P(kk')}-\frac{\tilde{w}_{r,k}}{kP(k)}= \tilde{w}_{k,kk'}\frac{rP(r)}{k^2k'P(kk')P(k)}. \] We then define a sequence of numbers \(k_i\) inductively by setting \(k_0=r\) and by choosing \(k_{i+1}=k_i k'\) with \(k'\) large so that \[ \tilde{w}_{k_{i},k_{i}k'}>0, \] which is possible because of the assumption of asymptotic Hilbert stability. It follows from the equation above that the sequence of numbers \[ A_i=\frac{\tilde{w}_{r,k_i}}{k_iP(k_i)} \] is stricly increasing. Since \[ \frac{\tilde{w}_{r,k_0}}{k_0P(k_0)}=0, \] we see that \(A_i>0\) for all \(i>0\). Hence the limit \(\lim_{i\to\infty}A_i\) exists and is positive, but this is equal to \(e_{n+1}(r)\) up to a positive constant, and so we have shown that \((M,L)\) is asymptotically Chow stable.

Let \(M\) be a compact \(n\)-dimensional complex manifold and \(L\) an ample line bundle over \(M\), and call \(P(r)=\chi(M, L^r)\) the Hilbert polynomial of the pair \((M,L)\) (it has degree \(n\)). Then for each \(r>0\) large, the choice of a basis of global holomorphic sections of \(L^r\) gives an embedding of \(M\) in \(\mathbb{P}^{P(r)-1}\). Different bases correspond to projectively equivalent embeddings, and they are related by an element of \(G=SL(P(r),\mathbb{C})\). To any such embedding a classical construction of Chow associates a point in a large projective space, the Chow point of the embedding. We say that \((M,L^r)\) is Chow stable if this Chow point is GIT stable with respect to the natural action of \(G\) (this is well defined because two different embeddings lie in the same \(G\)-orbit). We say that \((M,L)\) is asymptotically Chow stable if \((M,L^r)\) is Chow stable for all \(r\) large.

Another classical construction of Hilbert associates to a given projective embedding of \(M\) in \(\mathbb{P}^{P(r)-1}\) a sequence of points \(\mathrm{Hilb}_{r,k}\), each lying in a big projective space with a natural \(G\)-action. We say that \((M,L^r)\) is Hilbert stable if the points \(\mathrm{Hilb}_{r,k}\) are GIT stable for all \(k\) large. We say that \((M,L)\) is asymptotically Hilbert stable if \((M,L^r)\) is Hilbert stable for all \(r\) large.

A theorem of J. Fogarty [J. Reine Angew. Math. 234, 65–88 (1969; Zbl 0197.17101)] says that asymptotic Chow stability of \((M,L)\) implies asymptotic Hilbert stability (see J. Ross and R. P. Thomas [J. Algebr. Geom. 16, No. 2, 201–255 (2007; Zbl 1200.14095)] for a modern proof). The main result of this paper is that conversely asymptotic Hilbert stability also implies asymptotic Chow stability.

The proof of this result can be sketched as follows, using the notations of J. Ross and R.P. Thomas [loc. cit.]. It turns out that asymptotic Hilbert stability is equivalent to the fact that for any \(r\) large \(\tilde{w}_{r,k}>0\) for all \(k\) large, where \[ \tilde{w}_{r,k}=w(k)rP(r)-w(r)kP(k), \] and \(w(k)\) is a polynomial of degree \(n+1\) (strictly speaking, this has to be checked for a whole family of polynomials \(w(k)\), but it is enough to check them one by one). One then writes \[ \tilde{w}_{r,k}=\sum_{i=0}^{n+1} e_i(r)k^i, \] where the coefficients \(e_i(r)\) are polynomials of degree \(n+1\). Then asymptotic Chow stability is equivalent to the fact that \(e_{n+1}(r)>0\) for all \(r\) large.

It is clear now that asymptotic Chow stability implies asymptotic Hilbert stability. On the other hand, for any positive number \(k'\) one computes that \[ \frac{\tilde{w}_{r,kk'}}{kk'P(kk')}-\frac{\tilde{w}_{r,k}}{kP(k)}= \tilde{w}_{k,kk'}\frac{rP(r)}{k^2k'P(kk')P(k)}. \] We then define a sequence of numbers \(k_i\) inductively by setting \(k_0=r\) and by choosing \(k_{i+1}=k_i k'\) with \(k'\) large so that \[ \tilde{w}_{k_{i},k_{i}k'}>0, \] which is possible because of the assumption of asymptotic Hilbert stability. It follows from the equation above that the sequence of numbers \[ A_i=\frac{\tilde{w}_{r,k_i}}{k_iP(k_i)} \] is stricly increasing. Since \[ \frac{\tilde{w}_{r,k_0}}{k_0P(k_0)}=0, \] we see that \(A_i>0\) for all \(i>0\). Hence the limit \(\lim_{i\to\infty}A_i\) exists and is positive, but this is equal to \(e_{n+1}(r)\) up to a positive constant, and so we have shown that \((M,L)\) is asymptotically Chow stable.

Reviewer: Valentino Tosatti (Cambridge)

### MSC:

14L24 | Geometric invariant theory |

32Q15 | Kähler manifolds |

32Q20 | Kähler-Einstein manifolds |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

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