On parametrization, optimization and triviality of test configurations

We give a parametrization of test configurations in the sense of Donaldson via spherical buildings, and show the existence of"optimal"destabilizing test configurations for unstable varieties, in the wake of Mumford and Kempf. We also give an account of the recent slight amendment to definition of K-stability after Li-Xu, from two other viewpoints: from the one parameter subgroups and from the author's blow up formalism.


Introduction
The concept of test configurations of polarized varieties is introduced by Donaldson [Don02] to generalize K-stability after Tian [Tia97] which is a certain version of GIT stability [Mum65] for polarized varieties. Indeed, to each test configuration, certain GIT weights correspond, which is a motivation of its introduction. Precisely speaking, they can be regarded as certain equivalent classes of one parameter subgroups of the general linear group GL(H 0 (X, L ⊗r )) for a given polarized variety (X, L).
The aim of this short notes is twofold. First is to point out that, roughly speaking, dividing sets of test configurations further, all equivalent classes can be parametrized by a certain simplicial complex, the so-called spherical building invented by J. Tits (cf. [Tit74]). Moreover, thanks to a theorem of Rousseau [Rou78] and Kempf [Kem78], originally known as the Tit's center conjecture, we conclude the existence of "optimal" destabilizing test configurations. The outcome is a geometric analogue of the Harder-Narasimhan filtrations of unstable sheaves on a fixed variety. (The analogy becomes clearer after recent work of [GSZ11]. Namely, it claims that the filtration coincides the optimal destabilizing objects in the sense of [Rou78], [Kem78]. ) We remark here that the spherical building has already been discussed in the Mumford's classic [Mum65] to give applications to stability problem although he used the word "flag complex" with completely same meaning. Thus, we should note that our statements are essentially consequences of [Mum65], [Rou78], [Kem78], and only some small pieces of arguments, so that they are not essentially new. However, it seems to the author that the results have not been recognized by the mathematical community of this area, which is one motivation for him of writing this notes.
The second objective of this notes is to explain from our point of view, about the pathological (almost trivial but not trivial) test configurations whose existence is pointed out in [LX11, subsection 2.2]. They addressed that we need to slightly modify the definition of K-stability by regarding them as trivial or "throwing them away". For details, see subsection 3.3. In that subsection, we also explain how the blow up formalism [Od11] works for that amended notion of K-stability. Moreover, the sub-locus of the spherical building which parametrizes those test configurations of specific kind will be described as well.

Statements
We work over an arbitrary algebraically closed field k. We use the terminologies of line bundles and invertible sheaves interchangeably. By a polarized variety (X, L), we mean that X is a projective equidimensional variety and L is an ample line bundle on it. At the beginning, we recall the definition of (semi) test configurations and introduce certain (equivalence) relations for test configurations as follows.
Definition 2.1. A test configuration (resp. semi test configuration) for a polarized variety (X, L) is a quasi-projective variety X with an invertible sheaf L on it with: (i) a G m action on (X , L) (ii) a projective flat morphism p : X → A 1 such that p is G m -equivariant for the usual action on A 1 : L is relatively ample (resp. relatively semi ample), and (X , L)| p −1 (A 1 \{0}) is G mequivariantly isomorphic to (X, L ⊗r ) × (A 1 \ {0}) for some positive integer r, called exponent, with the natural action of G m on the latter and the trivial action on the former.
Definition 2.2. For a given polarized variety (X, L), we introduce the following three pre-equivalent conditions. We call two test configurations (X i , L i )(i = 1, 2) • P-pre-equivalent if (X 2 , L 2 ) is a base change of (X 1 , L 1 ) by the power mor- To say in other (but a little rough) words, it is exactly when they are isomorphic "except for the linearizations". Consider an equivalent relation among test configurations which is generated by P-pre-equivariance relation and R-equivariance relation (resp. P-pre-equivariance relation, Q-pre-equivariance relation and R-equivalence relation) and call it Tequivalence relation (resp. U-equivalence relation).
A T-equivalent class of test configuration (X , L) will be called a test degeneration and we write it as (X , L). Note that, by the base change, central fiber does not change scheme-theoritically (including polarization), which is the origin of this name. Note that, exponents are well-defined for each test degeneration.
A U-equivalent class of test configuration will be called a test class. Of course, a test degeneration gives a test class.
Let us assume the polarization L attached to X is very ample. Then, we have the following. We explain the rigorous meaning in subsections 3.1, 3.2.
Here, the spherical building ∆(GL(H 0 (X, L)) is a certain simplicial complex. Its geometric realization |∆(GL(H 0 (X, L))|, which we donote as ∆ r , with a natural topology is known to have a certain natural compatible metric structure (cf. [AB08, Chapter 12]). We will give the definition of the building in subsection 3.1. We will also see that although it is non-compact with respect to the usual topology, it will be compact with certain "much stuck" topology to be introduced later. Without fixing exponents, we thus also have: Here, the inductive system is given by taking power of polarization of test configurations i.e., (X , L) (X , L ⊗l ) with l ∈ Z >0 . Note that these ∆ r and ∆ ∞ can be expected to correspond to space of geodesics of (finite-dimensional or infinitedimensional) symmetric spaces.
For the basic of stability, we refer to [Mum65], [Od11]. Based on the continuous (and piecewise-linear) behavior of GIT weights [Mum65, Prop 2.14], we see the continuity of normalized Chow weights: Proposition 2.5. The normalized Chow weights form continous function on |∆(GL(H 0 (X, L ⊗r )))| Q with respect to the usual topology. On the other hand, it is a lower semi-continuous function with regards to the "stuck" topology.
For the definitions of stuck topology and normalized Chow weights, see subsection 3.1 and section 4 respectively. We also have the following abstract existence theorems of "optimal" destabilizing objects, thanks to [Rou78], [Kem78].
Corollary 2.6 (of Theorem 2.3). If [X ⊂ P] is Chow-unstable, it has a unique maximally destabilizing test degeneration.
Corollary 2.7. If (X, L) is K-unstable, it has a unique maximally destabilizing series of test degenerations.
For the definitions of these "maximally destabilizing" objects, we will explain in section 4 but they are essentially with regard to the restricted action of SL(H 0 (X, L ⊗r )). Note that, since if (X, L) is K-unstable, X ֒→ P(H 0 (X, L ⊗r )) 1 is Chow unstable for sufficiently large r ∈ Z >0 (cf. e.g. [RT07, section 3]), Corollary 2.7 simply follows from Corollary 2.6. It would be interesting to know the behavior of Donaldson-Futaki invariants on ∆ ∞ or to compare Corollary 2.7 with the frameworks of newer stability notions of [Don10] and [Sze11].
It would be also interesting to discuss relation with the similar existence statement in [Sze08,Theorem4] which was for toric case. He used a certain compactness property of the space of convex functions on the Delzant polytope (for which he refers to [Don02, Proposition 5.2.6]).

Parameter space
3.1. Review of spherical building. See [Tit74], [AB08] for more detailed accounts for the materials presented in this subsection. In this subsection, G is assumed to be an arbitrary reductive algebraic group.
Definition 3.1. The spherical building ∆(G) of G is an abstract simplicial complex whose vertices represent maximal (proper) parabolic subgroups and faces represent finite set of maximal parabolic subgroups whose intersection is again parabolic.
Then, it is known that the following holds.
Proposition 3.2 (cf. [Mum65, Chapter 2, section 2]). The rational points of spherical building of G has natural correspondence with classes of one parameter subgroups as follows. We denote the center subgroup of G as Z and "1-ps" stands for one parameter subgroup.
Note that by the relation (iii), and not thinking one parameter subgroup with image in Z, we are essentially considering semisimple part. Of course, fixing a maximal torus T , we have a natural map |∆(T )| → |∆(G)| which is indeed injective. Thus, |∆(G)| is a union of S(T, Q) := (Hom alg.grp (G m , T )⊗ Z Q)/Q >0 which is dense in S(T, R) := (Hom alg.grp (G m , T )⊗ Z R)/R >0 (apartments) naturally homeomorphic to sphere.
It is not compact in the usual topology (see [AB08, Chapter 12] for compatible metric structure and its developments) but it is compact if we introduce a new topology as follows.
Note that a variety G/N (T ) parametrizes set of maximal tori where N (T ) denotes the normalizer (algebraic) subgroup of a fixed maximal torus T of G. Thus, we have a natural map φ : S(T, R) × (G/N (T )) k → |∆(G)|. Using this, we can define our new topology as its associated quotient (compact) topology of the product of the Euclidean topology on S(T, R) and the Zariski topology on (G/N (T )) k , via φ. We call it stuck topology of |∆(G)| (or |∆(G)| Q , defined by restriction) in this notes. We fix the above notation from now on.

3.2.
Proof of Theorem 2.3. Note that we are interested in the case G = GL(H 0 (X, L)) which we assume in this subsection.
As Proposition 3.2 shows, the rational points of the spherical building |∆(G)| Q represent the classes of one parameter subgroups of GL(H 0 (X, L)) with respect to the equivalence relation generated by (i), (ii), (iii) which are nontrivial.
On the other hand, let us recall that each test configuration (X , L) with very ample L of exponent 1 and projection morphism p : X → A 1 is isomorphic to the DeConcini-Procesi family of a one parameter subgroup ([RT07, Proposition 3.7]). More precisely, if we fix G m -equivariant trivialization f : p * L ∼ = O ⊕(M+1) of p * L then we can take a one parameter subgroup λ f whose corresponding DeConcini-Procesi family is isomorphic to the original (X , L).
Let us fix a G m -equivariant trivialization f and take associated eigenspace decomposition of H 0 (X, L) as H 0 (X, L) = ⊕ i V i where the eigenvalue of the action on V i is t wi . Then, as explained in [Don05, a few lines after Lemma 2], any other choice is given by p(t) • f where p ∈ (P (λ)) k and p(t) is defined by Thus, each test configuration is an equivalence class of one parameter subgroups divided by the relation (ii). And it is also easy to see that the relation (i) (resp. (iii)) corresponds to the base change, i.e. P-pre-equivalence (resp. R-equivalence) introduced in Definition 2.2. Hence, test degenerations with very ample polarization of exponent 1 correspond to one parameter subgroups of GL(H 0 (X, L)) modulo (i), (ii), (iii) so that they correspond to rational points of the building |∆(G)| Q .
3.3. Almost trivial test configurations. In [LX11], it is recently pointed out that in the definition of K-stability, we should allow some non-trivial but "almost trivial" test configurations having GIT weights to vanish, or simply dismiss those test configurations of a specific type from our mind. Rigorously speaking, it is characterized by the following condition. Indeed, those should have vanishing Donaldson-Futaki invariants by the formula [Wan08], [Od11].
Definition 3.3 ([Stp11, Definition 1]). A test configuration (X , L) is said to be almost trivial 2 if X is G m -equivariantly isomorphic to the product test configuration, away from a closed subscheme of codimension at least 2.
By taking the fact that they are not necessarily trivial (cf. [LX11, Example 1]) into account, K-stability is slightly reformulated as follows. Now, we give a partial characterization of "almost triviality" from different points of view as follows. We note that the converse does not hold in general.
Proposition 3.5. Assume X is embedded into a projective space P(H 0 (X, L ⊗r )) by the complete linear system |L ⊗r | with some r ∈ Z >0 . Take a diagonalization of the G m -action λ on H 0 (X, L ⊗r ) by the eigenvectors X 0 , · · · , X c , X c+1 , · · · , X N where G m acts on X i by t ai . Assume that a 0 = · · · = a c < a c+1 ≤ · · · ≤ a N .
Proof. Let us assume V (X 0 , · · · , X c ) ∩ X = ∅ and prove that the associated test configuration X λ is not almost trivial. Take a closed point x of V (X 0 , · · · , X c ) ∩ X, then linear projection from x is not birational, i.e. degree should be at least 2 as x ∈ X. Thus, the resulting test configuration should not have a central fiber which is birational to original X.
On the other hand, if c ≤ dim(X), the linear projection from V (X 0 , · · · , X c ) to the corresponding projective space is surjective. If it is generically finite, c = dim(X) follows from it by comparing dimensions. However, in that case, it should not be birational because X is non-degenerate in P(H 0 (X, L ⊗r )). This completes the proof.
In other words, if a test configuration (X , M) is almost trivial then its associated flag ideal J ⊂ O X×A 1 (cf. [Od11]) is of the form (t N ) and the natural dominating morphism from B := Bl J (X × A 1 )( ∼ = (X × A 1 )) to X is isomorphic away from a closed subscheme of X with codimension at least 2. Of course, this is a characterization of the almost triviality. Thus, given a test configuration which is not almost trivial, the associated flag ideal J is not of the form (t N ) or, if it is of such a form, B → X is not isomorphic in codimension 1. For the latter case, we have DF(X , M) > 0 by the arguments of [RT07, section 5] or the formula for the Donaldson-Futaki invariant [Od11].
On the other hand, once a semi test configuration of blow up type (B := Bl J (X × A 1 ), L ⊗r (−E)) is given, we can construct a contracted (ample) test configuration (Proj ⊕ k≥0 H 0 (X × A 1 , J k p * 1 L ⊗rk ), O(1)) with obviously the same Donaldson-Futaki invariant.
In sum, we have the following, which is a corrected proposition of [Od11]. Originally, the author claimed the corresponding statement for original K-stability [Don02] which does not work since he ignored the case J = (t N ) so that the blow up morphism B → X × A 1 is trivial. The proof works for proving the above statement. I apologize for this inaccuracy. Thus, with this slightly modified K-stability notion and Corollary 3.6, the results of [Od11] the sequels are justified in the naturally corrected form, or partially work without change.
We explain a locus of the spherical building, inside which these pathologies occur.