Minimizing CM degree and slope stability of projective varieties

We discuss a minimization problem of the degree of the CM line bundle among all possible fillings of a polarized family with fixed general fibers. We show that such minimization implies the slope semistability of the fiber if the central fiber is smooth.

Theorem 1.2. Let (X , L) → C be a polarized family and (X 0 , L 0 ) be the fiber over a closed point 0 ∈ C. Assume that X 0 is a smooth variety and that the inequality CM (X , L) ≤ CM (X ′ , L ′ ) holds for any polarized family (X ′ , L ′ ) → C isomorphic to (X , L) over C \{0}. Then, the fiber (X 0 , L 0 ) is slope semistable.
The notion of the slope stability of polarized varieties is introduced in [RT] as a weak version of the K-stability, that is, the K-stability for a special class of test configurations obtained by a deformation to the normal cone. In comparison with Conjecture 1.1, we note that Theorem 1.2 holds for not only Fano families, but also any polarized families, although we assume the smoothness of the central fiber. Also, note that the minimization assumption in Theorem 1.2 is weaker than that in Conjecture 1.1 in the sense that we do not need the minimization over base changes in Theorem 1.2.
Sketch of the proof of the main theorem.
Let Z ⊂ X 0 be a proper closed subscheme and c ∈ (0, ǫ(Z, L 0 )) be a rational number, where ǫ(Z, L 0 ) is the Seshadri constant of Z with respect to L 0 . Take the deformation to normal cone over Z π Z : T Z = Bl Z×{0} (X 0 × A 1 ) → X 0 × A 1 polarized by a relatively ample Q-line bundle L Z,c = π * Z p * 1 L 0 (−cE Z ), where p 1 : X 0 × A 1 → X 0 is the first projection and E Z is the Cartier exceptional divisor. Let T Z → P 1 be the natural compactification of T Z → A 1 . We need to show the inequality DF (T Z , L Z,c ) ≥ 0 in order to prove slope semistability of the central fiber (X 0 , L 0 ).
To show the inequality, we define another polarized family (B, M) → C by π : B = Bl Z X → X , M = π * L(−cE) where E is the Cartier exceptional divisor. We relate the difference of the CM degree CM (B, M) − CM (X , L) to DF (T Z , L Z,c ) by making use of a degeneration technique as follows. First we take the deformation of X to the normal cone Then, the blow-up of the total family along Z gives a deformation of B to T Z ∪ X 0 X . Although there may exist exceptional divisors of the blow-up contained in the central fiber of the deformation in general, we show that the smoothness of X 0 ensures there are no such exceptional divisors. Then we have the equality by the flatness. Thus, by the using minimizing assumption, we get the inequality DF (T Z , L Z,c ) ≥ 0 to reach the conclusion.
A postscript note: Soon after the first version of this article had been posted on the arXiv, the author was informed that Blum and Xu [BX] proved the separatedness of the K-moduli, which had been the original motivation for our study. Moreover, they told the author that they proved one direction of Conjecture 1.1 and that C.Li and X. Wang independently obtained the same result; the K-semistable filling implies the CM-minimization.
Acknowledgements: The author would like to express great gratitude to his supervisor Yoshinori Gongyo for his continuous encouragement and valuable advice. He is also deeply grateful to Yuji Odaka for sharing his problem, for teaching him the backgrounds and the related notions, and for his warm encouragement. The author is supported by the FMSP program at the University of Tokyo.

Preliminaries
The aim of this section is to recall some definitions and related results used in the proof of the main theorem.
2.1. Test configurations and the DF invariant. In this subsection, we recall the definition of test configurations and the DF invariant, which appear in the definition of K-stability.
Definition 2.1. A test configuration (X , L) for a polarized variety (V, L) consists of the following data: (1) A variety X admitting a projective flat morphism f : X → A 1 , (2) An f -ample Q-line bundle L on X , (3) A C * -action on (X , L) compatible with the natural C * -action on A 1 via f , such that the restriction (X , L)| C * over C * is C * -equivariantly isomorphic to (V, L) × C * .
If we only assume that L is f -semiample instead of f -ample, then (X , L) is called a semi-test configuration. A test configuration (X , L) is said to be trivial if X is equivariantly isomorphic to the trivial family V × A 1 with the trivial action on the first factor V .
Given a test configuration (X , L) for an n-dimensional polarized variety (V, L), there is an C * -action on H 0 (X 0 , L k 0 ) for a sufficiently divisible positive integer k induced by that on the central fiber (X 0 , L 0 ). If we decompose the C-vector space H 0 (X 0 , L k 0 ) into eigenspaces with respect to the action of C * , the eigenvalues can be written as some power of t ∈ C * . We call the exponent as the weight of the action on each eigenvector. The total weight w(k) is the sum of the weight over the eigenbasis. By the equivariant Riemann-Roch theorem, we have an expansion Also we write an expansion of χ(V, L) χ(V, L k ) = a 0 k n + a 1 k n−1 + O(k n−2 ) for sufficiently divisible k.
Definition 2.2. ([Don]) In the above notation, the Donaldson-Futaki invariant for a test configuration (X , L) is defined as Note that we can naturally extend the definition of the Donaldson-Futaki invariant to arbitrary semi -test configurations (see [RT]).
We do not use the following definition of K-stability in the proof of Theorem 1.2, but we introduce it to clarify the motivation of our study.
Definition 2.3. ( [Don], see also [Sto] with the natural C * -action outside some closed subset of codimension at least 2. Note that we assume non-triviality in codimension 1 of test configurations in the definition of K-(poly)stability [LX1,Sto]. If V is normal, we only need to consider non-trivial normal test configurations for K-(semi)stability since the Donaldson-Futaki invariant does not increase by normalization [RT,Remark 5.2].
2.2. the CM degree. In this paper, we only need to treat the degree of the CM line bundle over a curve, which we define a priori as follows. For more details, we refer to [FR].
for sufficiently divisible positive integer k. The coefficient a i is independent of the choice of a fiber since χ is constant over a flat family. Let g(C) denote the genus of C. Then the CM degree is defined as 0 . This value is nothing but the degree of the CM line bundle λ CM [PT, FS] of L on C.
Remark 2.5. Given a normal test configuration (X , L) → A 1 for a normal polirized variety (V, L), let (X , L) → P 1 denote the natural C *equivariant compactification, that is, we add the trivial fiber (V, L) × {∞} over ∞ ∈ P 1 . Then it is well known (see for example [Oda1,BHJ]) that the total weight w(k) on H 0 (X 0 , L 0 ) can be written as . Using the asymptotic Riemann-Roch formula, we get the equalities using the notation in (2.1). Thus, the CM degree of (X , L) coincides with the Donaldson-Futaki invariant of (X , L). In this viewpoint, the CM degree is often called the Donaldson-Futaki invariant, too.
2.3. Slope stability. In this subsection, we recall the notion of the slope semistability of polarized varieties introduced in [RT].
Let (V, L) be an n-dimensional polarized variety. Write for sufficiently divisible positive integer k. Then the slope of (V, L) is defined as Let Z ⊂ V be a proper closed subscheme defined by an ideal I Z and take the blow-up along Z Let E be the Cartier exceptional divisor defined by I Z onV then the Seshadri constant ǫ(Z, L) of Z with respect to L is defined as For a rational number x ∈ (0, ǫ(Z, L)], write for a sufficiently divisible k. Here, a 0 (x) and a 1 (x) are polynomials of x. Then the slope along Z with respect to c ∈ (0, ǫ] ∩ Q is defined as The slope semistability is a (strictly) weaker notion than the Ksemistability as in Theorem 2.8. To see this, first take a deformation to the normal cone over Z π : and let F be the Cartier exceptional divisor. We define a Q-line bundle L Z,c := π * p * 1 L(−cF ) for c ∈ (0, ǫ] ∩ Q. Lemma 2.7. In the above setting, L Z,c is ample over Proof. See [RT,Proposition 4.1]. Thus, we can see (T Z , L Z,c ) as a (semi-)test configuration of (V, L) for c ∈ (0, ǫ) and for c = ǫ if σ * L(−ǫE) is semiample.
Theorem 2.8. In the above notation, the Donaldson-Futaki invariant DF (T Z , L Z,c ) is a positive multiple of µ(V, L) − µ c (I Z , L) for any rational number c ∈ (0, ǫ) and for c Proof. See [RT,Section 4].
Remark 2.9. As in [PR], a blow-up of P 2 at two points is slope semistable, although it is not K-semistable. So this example shows that the slope semistability is indeed strictly weaker than the K-semistability.

Deformation to test configurations
We fix a polarized family (X , L) → C such that the fiber (X 0 , L 0 ) over a fixed closed point 0 ∈ C is a variety. The aim of this section is to construct a deformation of another polarized family over X to a test configuration of the central fiber (X 0 , L 0 ), and compare their CM degrees.
3.1. Construction. We refer to [Ful] for a detailed description of a deformation to the normal cone, which we use for the construction.
First we take a deformation to the normal cone over X 0 Then the central fiber V 0 of V → A 1 can be written as a union V 0 =X X 0 P glued along X 0 . HereX ∼ = X is the strict transform of X ×{0} and P is the exceptional divisor. Note that since the normal bundle of X 0 × {0} is trivial, P is isomorphic to X 0 × P 1 and so has a natural C * -action induced by that on P 1 . P is glued toX along one of the C * -invariant fiber X 0 ×{∞} ⊂ X 0 ×P 1 ∼ = P . Also we remark that V admits a natural flat morphism to a surface Bl (0,0) (C × A 1 ) by the universal property of blow-ups. Consider a closed subscheme Z ⊂ X set-theoretically supported in X 0 . Let Z be the strict transform of Z × A 1 ⊂ X × A 1 on V. Then Z gives a flat degeneration of Z ⊂ X to a C * -invariant closed subscheme Z 0 ⊂ P by Lemma 3.1 below. We take the blow-up along Z Π : W = Bl Z V → V (3.1) and let G be the Cartier exceptional divisor. Identify the general fiber of V → A 1 with X and let π 0 : T = Bl Z 0 P → P, π : B = Bl Z X → X denote the strict transform of P and V t ∼ = X on W respectively, and be each Cartier exceptional divisor. We have the following diagram: We fix a positive rational number c and define a Q-line bundle F := (Π * σ * q * 1 L)(−cG) on W where q 1 : X × A 1 → X is the first projection. Then, by taking restriction to each component of fibers, we have where p 1 : X 0 × P 1 → X 0 is the first projection. Thus, when c is sufficiently small, (T , N ) can be seen as a test configuration for (X 0 , L 0 ) and the general fiber (B, M) of (W, F ) → A 1 is a polarized family.
Next we show how we can treat the above deformation algebraically (see also [LX2,LZ]). Let be the extended Rees algebra (see [Eis,6.5]) of the ideal I X 0 ⊂ O X defining X 0 . Then, as in [LZ,Lemma 4.1] we have isomorphisms of so that we can discribe the above deformation algebraically as be a sheaf of ideals which defines a subscheme supported in (the thickening of) X 0 . For a non-zero local section f of I defined around the generic point of X 0 , let k = ord X 0 (f ) be the minimum integer such that f ∈ I k X 0 and definẽ f : as local sections of R and O X 0 [s] respectively. Here, [f ] denotes the image of f ∈ I k X 0 in I k X 0 /I k+1 X 0 . Moreover, we define the sheafĨ on W to be the sheaf of ideals locally generated by {f | f ∈ I} in R and the sheaf in(I) on X 0 × A 1 s to be the sheaf of ideals locally generated by Lemma 3.1. In the above setting, the following hold: (1) We have the equalities (3) R/Ĩ is flat as a sheef of C[t]-modules, and so Z is flat over A 1 .
Proof. (1)Ĩ ⊂ I[t, t −1 ] ∩R is clear by the definition. In order to see the opposite inclusion, it suffices to show that f t −k ∈Ĩ for any f ∈ I ∩ I k X 0 . If ord X 0 (f ) = k, this follows from the definition ofĨ. If ord X 0 (f ) > k, take any g ∈ I such that ord X 0 (g) = k, then we get f t −k = (f + g)−g ∈ I. Thus we obtain the first equality. The last equality follows since the (2) By the first equality in (1),Ĩ is the largest ideal in R among ideals which coincide with I[t, t −1 ] when they are extended to O X [t, t −1 ]. Sõ I defines the scheme theoretic closure of Z × C * in V, which is nothing but Z. It also follows that in(I) defines Z 0 from the last equality in (1).
(3) is in [LZ,Lemma 4.1] and can be proved exactly in the same way as [LX2,Lemma 4.1], but here we provide a direct proof. The flatness is clear outside 0 ∈ A 1 s . To show the flatness over 0 ∈ A 1 s , we only need to check that t is a non-zero divisor in R/Ĩ, since (t) is the only non-trivial ideal in the base C[t] (t) . Take any s ∈ R such that st ∈Ĩ.
X 0 for i ≥ 1 and f i ∈ I for i ≤ 0. Combining the above, we get f i ∈ I ∩ I i X 0 for i ≥ 1 and f i ∈ I for i ≤ 0, which shows s ∈Ĩ.

3.2.
Comparision of the CM degree. In this subsection, we show equality of the CM degree of the polarized families under a certain assumption and then discuss when the assumption is satisfied. We keep the notation in Subsection 3.1. Proof. By flatness and the assumption, we have the equality Comparing the coefficient of k n+1 and k n , we get i are the coefficients of the expansion (2.3) in Definition 2.4 for each family. Notice that the coefficient a i of the expansion (2.2) in Definition 2.4 is the same for each family. Thus, We give a sufficient condition for the assumption in Proposition 3.2. Geometrically, the assumption says that any thickening of Z is still flat over A 1 .

Proof. Define
t so that we need to prove that the central fiber W • 0 coincides with the restriction T | P 1 \{∞} = Bl Z 0 ×{0} (X 0 × A 1 s ). It is enough to show an isomorphism of R-algebra From the assumption and the flatness, we have Indeed, the first and the third equalities follow from the assumption and the second follows from the flatness of R/ I m . Thus we get the assertion by taking the direct sum.

Proof of the main theorem
The following lemma is needed to ensure that the assumption in Lemma 3.3 is satisfied in the setting of Theorem 1.2.
Lemma 4.1. Let A be a regular ring essentially of finite type over a field k. Assume (h) ⊂ A is a prime ideal such that A/(h) is also a regular ring and an ideal I ⊂ A contains h. Then, for positive integers j < m, I m ∩ (h j ) = h j I m−j holds.
Proof. The inclusion h j I m−j ⊂ I m ∩ (h j ) is clear, so we prove the opposite inclusion. First we may assume A is complete by taking completion with respect to its maximal ideal. Let {x 2 , · · · , x n } denote the lift of regular sequence of parameter of A/(h) to A, then {x 1 , x 2 , · · · , x n } is a regular sequence of parameter of A where we define x 1 = h, which induces the isomorphism A ∼ = k[[x 1 , x 2 , · · · , x n ]] (see [Mat, §28 the proof of Lemma 1]). So we replace A by a formal power series ring k[[x 1 , x 2 , · · · , x n ]] and h by x 1 . Then we can write I = (x 1 , f 1 , · · · , f s ), where each f i is a formal power series of x 2 , · · · , x n . Let B = k[[x 2 , · · · , x n ]] be a subalgebra of A, and J be an ideal in A generated by f 1 , · · · , f s . Let f ∈ A be an element of I m ∩ (x j 1 ). Since f ∈ I m , we can write f = x m 1 g 0 + x m−1 1 g 1 + · · · + x 1 g m−1 + g m , g i ∈ J i .
We may take each g i from B for i > 0. Indeed, we can write where k denotes a pair (k 1 , · · · , k i ). By decomposing as we get The first term is an element of B ∩J i since f 1 , · · · , f s are elements in B.
Replace g i by f k 1 · · · f k i G k ∈ B and g i−1 by g i−1 + f k 1 · · · f k i H k ∈ J i , and repeat this for i = m, m−1, · · · , 1. Thus we can assume g i ∈ B for i > 0. Then we have g i = 0 for i > m − j since f ∈ (x j 1 ). So we get f = x j 1 (x m−j 1 g 0 + · · · + g m−j ) ∈ x j 1 I m−j as desired.
We now prove Theorem 1.2.
Proof of Theorem 1.2. Let Z ⊂ X 0 be a proper closed subscheme and c ∈ (0, ǫ(Z, L 0 )] be a rational number. First we assume c ∈ (0, ǫ(Z, L 0 )). where the first equality follows from Remark 2.5. Thus we can get the desired slope inequality by Theorem 2.8.
For c = ǫ(Z, L 0 ), the slope inequality follows from the above argument and the continuity of slope of Z with respect to c.