A study of the Hilbert-Mumford criterion for the stability of projective varieties. (English) Zbl 1200.14095
In this paper, which can be seen as a companion of their paper [J. Differ. Geom. 72, No. 3, 429–466 (2006; Zbl 1125.53057)], the authors give various results about the notions of slope stability and Chow-slope stability seen as Hilbert-Mumford criteria in the setup of geometric invariant theory.
In the first part of the paper, various notions of algebraic stability for a couple \((X,L)\), where \(X\) is an algebraic variety and \(L\) a polarisation over \(X\), are described very precisely. Essentially, the authors give some partial relations between these different notions of stability (Hilbert stability, asymptotic Hilbert stability, Chow stability, asymptotic Chow stability, \(K\)-stability). \(K\)-stability, which was introduced by G. Tian (and generalized later by S. K. Donaldson), is expected to be related to the existence of a constant scalar curvature Kähler metric in the class of \(c_1(L)\). To define a \(K\)-stable manifold, one looks at its “test configurations” with general fibre \((X,L)\), which means to consider
– a flat projective family of \(\mathbb Q\)-polarized schemes \((\mathcal X,\mathcal L)\rightarrow \mathbb C\);
– an action of \(\mathbb C^*\) on \((\mathcal X,\mathcal L)\) covering the usual action of \(\mathbb C^*\) on \(\mathbb C\) such that the fibre \((\mathcal X_t,\mathcal L_t)\) is isomorphic to \((X,L)\) for all \(t \in \mathbb C^*\).
We refer to [R. P. Thomas, “Notes on GIT and symplectic reduction for bundles and varieties”, preprint, arXiv:math/0512411; also in: Essays in geometry in memory of S. S. Chern. Somerville, MA: International Press. Surveys in Differential Geometry 10, 221–273 (2006; Zbl 1132.14043)] as a survey on this topic. As for geometric invariant theory, one checks the stability of \((X,L)\) by considering the positivity of some weight which, in the case that the fibre at \(t=0\) is normal, reduces to the so-called Futaki invariant. In full generality, it is hard to get a description of all the test configurations and only the product configurations can be described easily. The authors have previously considered the test configurations canonically associated to the subschemes of \(X\) (by deformation to the normal cone of the subscheme) [op. cit.]. They have obtained a weaker notion of stability, called slope stability, which has the great advantage of being more easily computable. Let us recall it. For a subscheme \(Z\) of \(X\), the Seshadri constant \(\varepsilon(Z,L)\) is the supremum of \(c\) such that \(L^k \otimes\mathcal I_Z^{ck}\) is globally generated for large \(k\). Denote by \(\widehat X\) the blowup of \(X\) along the subscheme \(Z\) with exceptional divisor \(E\). Because of the Riemann-Roch formula, one obtains an expansion of \[ h^0(L^k-xkE)=\dim H^0(L^k-xkE) = a_0(x)k^n + a_1(x)k^{n-1}+ \cdots \] where the \(a_i(x)\) can be seen as polynomials over \(\mathbb R\). Hence, one can define \[ \mu_c(\mathcal I_Z)=\frac{\int_0^c (a_1(x)+\frac{a_0'(x)}{2})dx}{\int_0^c a_0(x)dx}, \] and \((X,L)\) is said to be slope semistable with respect to \(Z\) if \[ \mu_c(\mathcal I_Z)\leq \mu(X,L):= \frac{a_1(0)}{a_0(0)} \] for all \(c \in (0,\varepsilon(Z,L)]\).
In the second part, the authors give two theorems that we shall describe now. A test configuration of \((X,L)\) is by definition birational to \((X\times \mathbb C,L)\) and thus dominated by a blowup \((\mathcal X,\mathcal L)=(\mathrm{Bl}_{\mathcal I}(X\times \mathbb C),p^*(L\otimes \mathcal I))\) of \(X\times \mathbb C\) in a \(\mathbb C^*\)-invariant ideal \(\mathcal I\) supported on a thickening of the central fibre \(X \times \{0\}\). Note that now \(\mathcal L\) is just semi-ample, which leads the authors to study the notion of semi-test configuration. In particular, they manage to express the weights of a test configuration in terms of the weights of a semi-test configuration that dominates it. The proof is based on the Stein factorisation theorem and the study of some natural exact sequences when \(X\) is normal (in that case, any test configuration is dominated by its normalisation). When \(X\) is not normal, one needs to do inductively some iterative blowups (a finite number of times) in subschemes supported in the scheme theoretic central fibre. On the other hand, suppose that \((\mathcal X,L)\) is a semi-test configuration and \(\mathcal{Z}\subset \mathcal X\) is a \(\mathbb C^*\)-invariant subscheme with general fibre \(Z \subset X\) and central fibre \(Z_0 \subset\mathcal X_0\). By blowing up along \(Z_0\) we obtain \((\mathrm{Bl}_{Z_0}(\mathcal X),\mathcal L_c)\overset{\pi}{\longrightarrow} (\mathcal X,L),\) a semi-test configuration, where \(\mathcal L_c=\pi^* L-cE\) with \(E\) the exceptional divisor and \(c\leq \varepsilon(Z_0,L)\). Now if the thickenings \(j\mathcal{Z}\subset \mathcal X\) are flat over \(\mathbb C\) (for all \(j\in \mathbb N\)) and the \(\mathbb C^*\) action on \(H_{\mathcal X_0}^0(L^k)\) has only weights between \(-\delta K\) and \(0\) (where \(\delta>0\)) then the authors relate the weight of \(H^0((\mathrm{Bl}_{Z_0}(\mathcal X))_0,\mathcal L_c^k)\) with the weight of \(H^0(\mathcal X_0,L^k)\) plus some terms \(w_k(Z)\). Here the weights \(w_k(Z)\) are defined by \[ w_k(Z)=\sum_{j=1}^{ck} h^0(X,L^k\otimes \mathcal I_Z^j) - ck\, h^0(X,L^k). \]
With these two results in hand, the authors prove that for \(X\) normal and an arbitrary test configuration with associated subschemes \((Z_0 \subset \dots \subset Z_r \subset X)\) such that there exists a resolution of singularities of \(X\) preserving the flatness of the thickenings, one can express the weight of the test configuration only in terms of the \(w_k(Z_i)\). Another case is when the divisors that appear in the resolution of the singularities of \((X,Z_i)\) are all reduced with normal crossings, for which it turns out that again the same type of formula holds. This gives a partial converse to the fact that \(K\)-stability implies slope stability. In particular, the method is completely successful for curves for which \(K\)-stability and slope stability can be identified in this way.
In another section, the authors define the notion of Chow-slope stability for a couple \((X,\mathcal O_X(1))\subset \mathbb P^N\). For any subscheme \(Z \subset X\) and \(c\in \mathbb N\) with \(c\leq\epsilon(Z,L)\), one defines \[ \mathrm{Ch}_{c}(\mathcal I_Z)=\frac{\sum_{i=1}^c h^0(X,\mathcal I_Z(1))}{\int_0^c a_0(x)dx}, \] where \(a_0(x)\) is given by the Riemann-Roch formula for \(\mathcal I_Z^{ck}(k)\) as before. Then \(X\) is said to be Chow-slope stable if for any subscheme \(Z\subset X\) one has \[ \mathrm{Ch}_c(\mathcal I_Z)<\frac{h^0(X,\mathcal O_X(1))}{a_0(0)}. \] The authors prove that if \(X\) is Chow stable then it is Chow-slope stable. A partial converse is given for smooth curves.
Finally a section is dedicated to applications of the previous results. A striking example is the following: Suppose that the polarised variety \((X,L)\) has at worst canonical singularities, and \(K_X\) is numerically trivial or \(K_X\) is big and nef with \(L\) sufficiently close to \(K_X\). Then \((X,L)\) is slope \(K\)-stable. This is particularly interesting since very recently, using complex analysis or Ricci-flow methods, the existence of Kähler-Einstein currents on minimal varieties of general type has been proved. The case of curves of genus \(g \geq 1\) is also investigated, and the authors prove their \(K\)-stability by a completely algebraic proof.
The paper is long and contains many engrossing results. It highlights the complexity of building a Kobayashi-Hitchin correspondence for algebraic manifolds and, at the same time, the beauty of this subject.
In the first part of the paper, various notions of algebraic stability for a couple \((X,L)\), where \(X\) is an algebraic variety and \(L\) a polarisation over \(X\), are described very precisely. Essentially, the authors give some partial relations between these different notions of stability (Hilbert stability, asymptotic Hilbert stability, Chow stability, asymptotic Chow stability, \(K\)-stability). \(K\)-stability, which was introduced by G. Tian (and generalized later by S. K. Donaldson), is expected to be related to the existence of a constant scalar curvature Kähler metric in the class of \(c_1(L)\). To define a \(K\)-stable manifold, one looks at its “test configurations” with general fibre \((X,L)\), which means to consider
– a flat projective family of \(\mathbb Q\)-polarized schemes \((\mathcal X,\mathcal L)\rightarrow \mathbb C\);
– an action of \(\mathbb C^*\) on \((\mathcal X,\mathcal L)\) covering the usual action of \(\mathbb C^*\) on \(\mathbb C\) such that the fibre \((\mathcal X_t,\mathcal L_t)\) is isomorphic to \((X,L)\) for all \(t \in \mathbb C^*\).
We refer to [R. P. Thomas, “Notes on GIT and symplectic reduction for bundles and varieties”, preprint, arXiv:math/0512411; also in: Essays in geometry in memory of S. S. Chern. Somerville, MA: International Press. Surveys in Differential Geometry 10, 221–273 (2006; Zbl 1132.14043)] as a survey on this topic. As for geometric invariant theory, one checks the stability of \((X,L)\) by considering the positivity of some weight which, in the case that the fibre at \(t=0\) is normal, reduces to the so-called Futaki invariant. In full generality, it is hard to get a description of all the test configurations and only the product configurations can be described easily. The authors have previously considered the test configurations canonically associated to the subschemes of \(X\) (by deformation to the normal cone of the subscheme) [op. cit.]. They have obtained a weaker notion of stability, called slope stability, which has the great advantage of being more easily computable. Let us recall it. For a subscheme \(Z\) of \(X\), the Seshadri constant \(\varepsilon(Z,L)\) is the supremum of \(c\) such that \(L^k \otimes\mathcal I_Z^{ck}\) is globally generated for large \(k\). Denote by \(\widehat X\) the blowup of \(X\) along the subscheme \(Z\) with exceptional divisor \(E\). Because of the Riemann-Roch formula, one obtains an expansion of \[ h^0(L^k-xkE)=\dim H^0(L^k-xkE) = a_0(x)k^n + a_1(x)k^{n-1}+ \cdots \] where the \(a_i(x)\) can be seen as polynomials over \(\mathbb R\). Hence, one can define \[ \mu_c(\mathcal I_Z)=\frac{\int_0^c (a_1(x)+\frac{a_0'(x)}{2})dx}{\int_0^c a_0(x)dx}, \] and \((X,L)\) is said to be slope semistable with respect to \(Z\) if \[ \mu_c(\mathcal I_Z)\leq \mu(X,L):= \frac{a_1(0)}{a_0(0)} \] for all \(c \in (0,\varepsilon(Z,L)]\).
In the second part, the authors give two theorems that we shall describe now. A test configuration of \((X,L)\) is by definition birational to \((X\times \mathbb C,L)\) and thus dominated by a blowup \((\mathcal X,\mathcal L)=(\mathrm{Bl}_{\mathcal I}(X\times \mathbb C),p^*(L\otimes \mathcal I))\) of \(X\times \mathbb C\) in a \(\mathbb C^*\)-invariant ideal \(\mathcal I\) supported on a thickening of the central fibre \(X \times \{0\}\). Note that now \(\mathcal L\) is just semi-ample, which leads the authors to study the notion of semi-test configuration. In particular, they manage to express the weights of a test configuration in terms of the weights of a semi-test configuration that dominates it. The proof is based on the Stein factorisation theorem and the study of some natural exact sequences when \(X\) is normal (in that case, any test configuration is dominated by its normalisation). When \(X\) is not normal, one needs to do inductively some iterative blowups (a finite number of times) in subschemes supported in the scheme theoretic central fibre. On the other hand, suppose that \((\mathcal X,L)\) is a semi-test configuration and \(\mathcal{Z}\subset \mathcal X\) is a \(\mathbb C^*\)-invariant subscheme with general fibre \(Z \subset X\) and central fibre \(Z_0 \subset\mathcal X_0\). By blowing up along \(Z_0\) we obtain \((\mathrm{Bl}_{Z_0}(\mathcal X),\mathcal L_c)\overset{\pi}{\longrightarrow} (\mathcal X,L),\) a semi-test configuration, where \(\mathcal L_c=\pi^* L-cE\) with \(E\) the exceptional divisor and \(c\leq \varepsilon(Z_0,L)\). Now if the thickenings \(j\mathcal{Z}\subset \mathcal X\) are flat over \(\mathbb C\) (for all \(j\in \mathbb N\)) and the \(\mathbb C^*\) action on \(H_{\mathcal X_0}^0(L^k)\) has only weights between \(-\delta K\) and \(0\) (where \(\delta>0\)) then the authors relate the weight of \(H^0((\mathrm{Bl}_{Z_0}(\mathcal X))_0,\mathcal L_c^k)\) with the weight of \(H^0(\mathcal X_0,L^k)\) plus some terms \(w_k(Z)\). Here the weights \(w_k(Z)\) are defined by \[ w_k(Z)=\sum_{j=1}^{ck} h^0(X,L^k\otimes \mathcal I_Z^j) - ck\, h^0(X,L^k). \]
With these two results in hand, the authors prove that for \(X\) normal and an arbitrary test configuration with associated subschemes \((Z_0 \subset \dots \subset Z_r \subset X)\) such that there exists a resolution of singularities of \(X\) preserving the flatness of the thickenings, one can express the weight of the test configuration only in terms of the \(w_k(Z_i)\). Another case is when the divisors that appear in the resolution of the singularities of \((X,Z_i)\) are all reduced with normal crossings, for which it turns out that again the same type of formula holds. This gives a partial converse to the fact that \(K\)-stability implies slope stability. In particular, the method is completely successful for curves for which \(K\)-stability and slope stability can be identified in this way.
In another section, the authors define the notion of Chow-slope stability for a couple \((X,\mathcal O_X(1))\subset \mathbb P^N\). For any subscheme \(Z \subset X\) and \(c\in \mathbb N\) with \(c\leq\epsilon(Z,L)\), one defines \[ \mathrm{Ch}_{c}(\mathcal I_Z)=\frac{\sum_{i=1}^c h^0(X,\mathcal I_Z(1))}{\int_0^c a_0(x)dx}, \] where \(a_0(x)\) is given by the Riemann-Roch formula for \(\mathcal I_Z^{ck}(k)\) as before. Then \(X\) is said to be Chow-slope stable if for any subscheme \(Z\subset X\) one has \[ \mathrm{Ch}_c(\mathcal I_Z)<\frac{h^0(X,\mathcal O_X(1))}{a_0(0)}. \] The authors prove that if \(X\) is Chow stable then it is Chow-slope stable. A partial converse is given for smooth curves.
Finally a section is dedicated to applications of the previous results. A striking example is the following: Suppose that the polarised variety \((X,L)\) has at worst canonical singularities, and \(K_X\) is numerically trivial or \(K_X\) is big and nef with \(L\) sufficiently close to \(K_X\). Then \((X,L)\) is slope \(K\)-stable. This is particularly interesting since very recently, using complex analysis or Ricci-flow methods, the existence of Kähler-Einstein currents on minimal varieties of general type has been proved. The case of curves of genus \(g \geq 1\) is also investigated, and the authors prove their \(K\)-stability by a completely algebraic proof.
The paper is long and contains many engrossing results. It highlights the complexity of building a Kobayashi-Hitchin correspondence for algebraic manifolds and, at the same time, the beauty of this subject.
Reviewer: Julien Keller (Marseille) (MR2274514)
References:
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