Positivity of the CM line bundle for families of K-stable klt Fano varieties

The Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.


Introduction
Throughout the article, the base field is an algebraically closed field k of characteristic zero.

Main theorem
The interest in the moduli space of singular K -polystable Fano varieties stems from the classification theory of algebraic varieties. The birational part of the classification theory, also called the Minimal Model Program [2,3,[12][13][14]17,45,58,59,68], predicts that up to specific birational equivalences, each projective variety decomposes into iterated fibrations with general fibers of 3 basic types: Fano, weak Calabi-Yau, and general type To be precise, one here needs to allow pairs, see Sect. 1.2, but the boundary free case is a good first approximation.
The above 3 types are defined by having a specific class of mild singularities and negative/numerically trivial/positive canonical bundles. Then the moduli part of the classification theory is supposed to construct a projective, compactified moduli spaces for the above 3 basic types of varieties. According to our current understanding, the moduli part seems to be doable only in the presence of a singular Kähler-Einstein metric, e.g., [114,Conj 8.11, and the following 2 paragraphs], which is predicted to be equivalent to the algebraic notion of K -polystability [11,[28][29][30]82,88,89,94,115]. We refer the reader to Definition 4.8 and to Corollary 4.9 for the precise definition and for a characterization of K -semistability used in the present article. Additionally, see Sect. 1.6 for an explanation on K -polystability.
In particular, on the Fano side, for the moduli part of the classification theory one should construct algebraically the following two spaces: • the stack M K-ss n,v of K -semistable Fano varieties of dimension n and anticanonical volume v, as well as, • the projective good moduli space M K-ps n,v of M K-ss n,v parametrizing Kpolystable Fano varieties of dimension n and anti-canonical volume v.
M u-K-s n,v has finite quotient singularities, and hence the normalization in Theorem 1.1.(c) can be dropped from the statement.
We know that M u-K-s n,v is smooth at the points corresponding to smooth Fano varieties [67,105], and to terminal Fano 3-folds [106,Thm 1.7]. Unfortunately, these unobstructedness statements do not hold for all Fano varieties, for example, [106,Rem 2.13] gives a counterexample. However, the counterexample is a cone over a del Pezzo surface of degree 6, which is not uniformly K -stable, as its automorphism groups is not finite. This leads to the following question. Question 1.7 Is the deformation space of uniformly K -stable Fano varieties unobstructed?

Technical statements
Our most general statements implying Theorem 1.1, just as points (a) and (b) of Theorem 1.1, come in two flavors: semi-positivity and positivity statements. We start with the semi-positivity statements, which we are able to show also in the logarithmic case. Let us first present the precise definition of the CM-line bundle in this setting.
If f : (X, ) → T is a flat morphism of relative dimension n from a projective normal pair to a normal projective variety such that −(K X/T + ) is Q-Cartier and f -ample. Then we define the CM line bundle by λ f, := − f * ((−(K X/T + )) n+1 ). (1.7.a) This cycle, up to multiplying with a positive rational number, is the first Chern class of the functorial line bundle on T defined in [101,102], see also Proposition 3.7 and [43,44,103]. In particular, one defines λ to be the unique Q-line bundle λ on M K-ss n,v such that for every ν : T → M K-ss n,v , if f : X → T is the associated family, then ν * λ = λ f := λ f,0 .
Our most general semi-positivity statements then are the following. We note that by a general geometric fiber we mean a fiber over any geometric point Spec L → U , where U ⊆ T is a fixed non-empty open set. Theorem 1.8 Let f : X → T be a flat morphism of relative dimension n with connected fibers between normal projective varieties and let be an effective Q-divisor on X such that −(K X/T + ) is Q-Cartier and f -ample. Let λ f, be the CM line bundle on T as defined in (1.7.a).
(a) Pseudo-effectivity: If T is smooth and (X t , t ) is K-semi-stable for general geometric fibers X t , then λ f, is pseudo-effective. (b) Nefness: If all fibers X t are normal, does not contain any fibers (so that we may restrict on the fibers), and (X t , t ) is K -semi-stable for all geometric fibers X t , then λ f, is nef.
Next we state our positivity statements. These pertain to families with maximal variation. Here, a family f : X → T of Fano varieties as in Theorem 1.9 has maximal variation if there is a non-empty open set of T over which the isomorphism equivalence classes of the fibers are finite. In the logarithmic case one faces considerable extra difficulties when the variation comes partially also from the variation of the boundary, as it was also the case for the KSBA stable moduli [76]. Hence, to keep the length of the article under control, we address here only the question of positivity in the boundary free case. The logarithmic version was addressed after the initial submission of the present article in [104].  Remark 1. 10 We note that both, K -semistability [ We also remark that in Theorem 1.9 we carefully said "geometric fiber" instead of just "fiber". The reason is that we use the δ-invariant description of K -stability, and the δ-invariant of a variety is not invariant under base extension to the algebraic closure, see Remark 4.16. So, for scheme theoretic fibers over non algebraically closed fields the δ-invariant can have non semi-continuous behavior.
Remark 1. 11 We proved Theorem 1.8 and Theorem 1.9 in the stated generality, as in this setting the relative canonical divisor exists and admits reasonable base-change properties on very general curves in moving families of curves on the base, see Sect. 2.4 for details. Nevertheless, in situations where this base-change is automatic, Theorem 1.9 directly implies statements over nonnormal, non-projective, and even non-scheme bases. This is made precise for example in the following statement: Corollary 1.12 Let f : X → T be a flat, projective morphism with connected fibers to a proper algebraic space, such that there is an integer m > 0 for which ω [m] X

/T is a line bundle and all the geometric fibers are K -semi-stable klt Fano varieties. Let N be the CM-line bundle associated to the polarization ω [−m]
X/T . Then, N is nef, and if the variation of f is maximal and the very general geometric fiber is uniformly K -stable, then N is big.
The CM line bundle over a general base is defined in Notation 3.6, following [102]. Remark 1.13 Note that over C the positivity properties of Theorem 1.9 (nefness, pseudo-effectivity, bigness, ampleness) can be also characterized analytically, e.g., [37,Prop 4.2] Remark 1.14 Negativity of −K X/T point of view. Unwinding definition (1.7.a), we obtain that, in the case of one dimensional base, Theorem 1.9 states that (−K X/T ) n+1 is at most zero/smaller than 0. Using this in conjunction with the base-change property of the CM line bundle proved in Proposition 3.8 we obtain that Theorem 1.9, especially the last 3 points, prove strong negativity properties of −K X/T for families of klt Fano varieties. For example, one obtains that if C → T is a general enough curve, then the top self intersection of (−K X/T )| f −1 C is negative.
There do exist birational geometry statements claiming that −K X/T is not nef, e.g., [124,Prop 1]. Our negativity statements point in this direction but go further. However, it is not a coincidence that strong negativity statements on −K X/T did not show up earlier, as in fact Theorem 1.9 is not true for every family of klt Fano varieties. Indeed, Example 12.1 shows that in Theorem 1.9 one cannot relax the K -semi-stable Fano assumption to just assuming klt Fano. The development of the notions of K -stability in the past decade was essential for creating the chance of proving negativity statements for −K X/T of the above type.
We also note that as −K X/T is not nef usually in the situation of Theorem 1.9, c.f., Theorem 1.20 and Example 12.2, the negativity of (−K X/T ) n+1 is independent of the negativity of κ(−K X/T ). In fact, assuming the former, κ(−K X/T ) can be −∞ (Example 12.4), dim X (Example 12.2), and also something in between −∞ and dim X (Example 12.4).

Remark 1.15
There are two main reasons why our positivity statements (a), (b) and (c) of Theorem 1.9 work in the uniformly K -stable case, but not in the K -polystable case: (a) We rely on the characterization of K -semistability and uniform K -stability via the δ invariant given by [18,52]. Such characterization is not available for the K -polystable case.
(b) Our Theorem 1.20 about the nef threshold, on which the above 3 points of Theorem 1.9 rely, fails in the K -polystable case according to Example 12.3. Hence, one would need a significantly different approach to extend points (a), (b) and (c) of Theorem 1.9 to the K -polystable case.
Remark 1. 16 One could make definition (1.7.a) also without requiring flatness. We do not know if Theorem 1.9 holds in this situation. Nevertheless, we note that it would be interesting to pursue this direction for example for applications to Mori-fiber spaces with higher dimensional bases, see Corollary 1.17. Also we expect that the reduced fiber assumption of point (a) of Theorem 1.9 can be removed, as we need it for technical reasons, namely we want the base changes over normalizations of general elements of movable families of curves to be nice, and also because the conjectured K -semi-stable reduction should eliminate it. Here, K -semi-stable reduction means the conjecture that K -semsitable families of Fano varieties over function fields of DVR's can be extended over the DVR after a finite base-change.

Boundedness of the volume
Fujita showed in [48,Thm 1.1] that vol(−K X ) ≤ (n + 1) n for every Ksemistable Fano variety X of dimension n, see [83,Thm 3] for better bounds in the presence of quotient singularities. Using Theorem 1.8 we can show similar bounds for Fano varieties X admitting a Fano fibration structure with K -semi-stable general fiber.

Corollary 1.17
If (X, ) is a normal Fano pair, and f : (X, ) → P 1 is a fibration with K -semi-stable general geometric fibers (F, F ), then vol(−(K X + )) ≤ 2 dim (X ) vol(−(K F + F )) ≤ 2 dim (X ) dim(X ) . Remark 1.18 Corollary 1.17 is sharp for surfaces and threefolds. Indeed, a del Pezzo surface of degree 8 and the blow-up of P 3 at a line, whose anti-canonical volume is 54, can be fibred over P 1 with K -semis-table fibres. Remark 1. 19 Classification of (uniform) K -(semi/poly)stable Fano varieties: to explain which varieties Corollary 1.17 pertains to, we provide a short list on Fano varieties that are either known to be K -semi-stable or not K -semi-stable. In fact, one typically wants to figure out for a given Fano variety the behavior with respect to all four K -stability properties, see Sect. 1.6. This has been an active area of research recently. To start with, let us recall that K -semi-stable Fano varieties are always klt.
A Del-Pezzo surface is K-polystable if and only if it is not of degree 8 or 7 [112,116]. Smooth Fano surfaces with discrete automorphism groups are even uniformly K-stable, and their delta invariant, see Sect. 4, is bounded away from 1 in an effective way [96]. Smoothable singular K-stable Del-Pezzo surfaces are classified in [93].
K-stable proper intersection of two quadrics in an odd dimensional projective space are classified in [108], see also [7]; in particular, smooth varieties of these types are always K-stable. Cubic 3-folds are studied in [84], where again smooth ones are K -stable, and so are the ones containing only A k singularities for k ≤ 4. Under adequate hypotheses, in [38], it is shown that Galois covers of K-semistable Fano varieties are K-stable. This can be applied for instance to double solids. Furthermore, birational superigid Fano varieties are K-stable under some addition mild hypothesis [91,110,126]. However, according to the best knowledge of the authors, there is not a complete classification of K-stable smooth Fano threefolds.
If one wants to study klt Fano varieties from the point of view of the MMP, it is particularly relevant to see if one can apply Corollary 1.17 to the case of Mori Fibre Spaces with one dimensional bases. In [32,Corollary 1.11], it is shown that if a smooth Fano surface or a smooth toric variety can appear as a fibre of MFS, then it is K-semistable. We do not know if the analogous result holds in dimension 3. However, there are examples of smooth Fano fourfolds with Picard number one, which then can be general fibers of MFS's, that are not K-semistable [47], see also [33].

Byproduct statements
As a byproduct of our technique for proving Theorem 1.8, we obtain the following bound on the nef threshold of −(K X/T + ) with respect to λ f, in the uniformly K -stable case. Theorem 1.20 Let f : X → T be a flat morphism with connected fibers from a normal projective variety of dimension n + 1 to a smooth curve and let be an effective Q-divisor on X such that • −(K X/T + ) is Q-Cartier and f -ample, and • X t , t is uniformly K -stable for fibers X t over general geometric points t ∈ T . Set • set δ := δ X t , t for t very general geometric point, and • let v := (−K X/T − ) t n for any t ∈ T .
Recall that the uniformly K -stable assumption in Theorem 1.20 is equivalent to assuming δ > 1, see Definition 4.8 and Corollary 4.9. In particular, δ−1 > 0 in the last line of the statement.

Remark 1.21
The reason for assuming in Theorem 1.20 that X t , t is uniformly K -stable for general geometric fibers, but setting δ to be δ X t , t only for very general geometric fibers is technical. On one hand, uniform Kstability is known to be an open property by [21,Thm A], and hence one may assume it on the general geometric fiber without imposing any additional assumption. On the other hand, only the function t → min{1, δ X t , t }, but not t → δ X t , t itself, is known to be constructible [22,Prop 4.3]. For t → δ X t , t , it is only known that it is constant on the complement of countably many closed sets by Proposition 4.15.

Remark 1.22
One cannot have a nef threshold statement, as in Theorem 1.20, for K -polystable Fano varieties instead of uniformly K -stable ones. Indeed, take the family f : X → T given by Example 12.3. It has K -polystable fibers, deg λ f = 0, but −K X/T is not nef. In particular, for any a ∈ Q, −K X/T + a f * λ f ≡ −K X/T , and hence for any a ∈ Q, −K X/T + a f * λ f is not nef.
We also recover a structure theorem when the CM line bundle λ f is not positive: Theorem 1.23 Let f : X → T be a flat morphism of relative dimension n with connected fibers between normal projective varieties and let be an effective Q-divisor on X such that −(K X/T + ) is Q-Cartier and f -ample. Assume that X t , t is uniformly K -stable for fibers X t over general geometric points t ∈ T . If H is an ample divisor on T , such that λ f, · H dim T −1 = 0, then for every integer q > 0 divisible enough, f * O X (q(−K X/T − )) is an H -semistable vector bundle of slope 0.

Corollary 1.24
Assume k = C, and let f : X → T be a surjective morphism from a normal projective variety of dimension n + 1 to a smooth, projective curve such that −K X/T is Q-Cartier and f -ample, and the general fiber of f is uniformly K -stable. Then, deg λ f = 0 if and only if, f is analytically locally a fiber bundle.

Similar results in other contexts
Roughly, there are three types of statements above: (semi-)positivity results, moduli applications, inequality of volumes of fibrations. Although in the realm of K -stability ours are the first general algebraic results, statements of these types were abundant in other, somewhat related, contexts: KSBA stability, GIT stability, and just general algebraic geometry. Our setup and our methods are different from these results, still we briefly list some of them for completeness of background. We note that KSBA stability is related to our framework as it is shown to be exactly the canonically polarized K -stable situation [88,89,94]. Also, GIT stability is related, as K -stability originates from an infinite dimensional GIT, although it is shown that it cannot be reproduced using GIT, e.g., [120]. General

Overview of K-stability for Fano varieties
In the present article we define K -semi-stability and uniform K -stability using valuations, see Definition 4.8, which is equivalent then to the δ-invariant definition given in Corollary 4.9. These definitions were shown to be equivalent to the more traditional ones that use test configurations [18,Theorem B]. However, this approach has a considerable disadvantage: there is no known delta invariant type definition of K -stability and K -polystability. While we do not use these notions in any of the statements or proof of our results, we believe that they are important notions in the study of Fano varieties. Hence, for completeness we state the classical definitions involving test configurations for all the four notions of K -stability. We refer the reader to [40,41] or more recent papers such as [25,39] for more details. For a Fano variety X we mention the following notions of K -stability: We also note that the Yau-Tian-Donaldson (in short, YTD) conjecture asserts that a klt Fano variety admits a singular Kähler-Einstein metric if and only if it is K-polystable. This is known to hold for smooth [28][29][30]115] and smoothable Fano varieties [81], and independently [109] in the finite automorphism case, and for singular ones admitting a crepant resolution [82]. In the literature, there are also many proposed strenghtenings of the notion of K-stability; they should be crucial to extend the YTD conjecture to the case of constant scalar curvature Kähler metrics. In this paper we are interested in uniform K-stability [10,25,39], which at least for smooth Fano manifold is known to be equivalent to K-stability (we should stress that the proof is via the equivalence with the existence of a Kähler-Einstein metric). One can also strenghten the notion of K-stability by possibly looking at non-finitely generated filtrations of the coordinate ring, see [31,111,121].

Outline of the proof
Our proof for the semi-postivity and the positivity statements for the CM line bundle are different. Hence, we discuss the corresponding outlines separately in Sect. 1.7.1 and in Sect. 1.7.3, respectively. Additionally, as it is an indispensable link between semi-positivity and positivity, we present the ideas behind the nefness threshold statement of Theorem 1.20 in Sect. 1.7.2. For simplicity, we restrict in all cases to the non-logarithmic situation, that is, to statements about −K X/T instead of −(K X/T + ). As all the assumptions and consequences are invariant under base-extension to another algebraically closed field, we may also assume that k is uncountable. In particular, the very general geometric fibers whose existence is assumed in the statements also show up as closed fibers.

Semi-positivity statements.
As nefness and pseudo-effectivity can be checked via non-negative intersection with effective or moving 1-cycles, respectively, points (a) and (b) of Theorem 1.8 can be reduced to the case of one dimensional base. Hence, we assume that the base of our fibration f : X → T is a curve, in which case pseudoeffectivity and nefness are both equal to the degree being at least zero. So, we are supposed to prove that deg λ f ≥ 0 or equivalently that (−K X/T ) n+1 ≤ 0, see (1.7.a). We argue by contradiction, so we assume that (−K X/T ) n+1 > 0. If we fix a Q-divisor H on T of small enough positive degree, then by the continuity of the intersection product (−K X/T − f * H ) n+1 > 0 also holds. As X is normal and fibered over the curve T over which −K X/T is ample, this implies via a Riemann-Roch computation that the Q-linear system |−K X/T − f * H | Q is nonempty, see Remark A.3. Our initial idea is to obtain a contradiction from this fact: in fact, Proposition 7.2 shows that there are no is not klt for general t ∈ T . Indeed, every family with negative CM line bundle has to satisfy the conditions stated in the previous sentence, according to Proposition 7.2. An explicit example is given in Example 12.1.
Our second idea is that maybe the K -stable assumption leads us to a as above that also satisfies the klt condition when restricted to a general fiber. According to the delta invariant description of K -semi-stability (Corollary 4.9), if X t is K -semi-stable, then up to a little perturbation one can obtain klt divisors the following way: for q 0, let D 1 , . . . , D l be divisors corresponding to any basis of H 0 X t , −q K X t ; then the divisor To this end, it is enough to lift for q 0, every element of a basis of H 0 X t , −q K X t to elements of H 0 (X, q(−K X/T − f * H )). Using some perturbation argument, it suffices to show the existence of linearly independent sections s 1 , . . . , s l ∈ H 0 X t , −q K X t such that s i lifts, and is close enough to 1. This in turn would be implied by the following: let E q be the subsheaf of f * O X (−q K X/T ) spanned by the global sections, then For the readers more familiar with the language of volumes and restricted volumes, we note that (1.24.a) is equivalent to showing that the restricted volume of −K X/T over a general fiber is equal to the anti-canonical volume of the fibers. Unfortunately, (1.24.a) still does not hold. For example, if one takes the isotrivial family of P n 's over T := P 1 (as in Example 12.1 for n = 2), then In this situation E q is the direct sum of the factors with degree greater than q deg H ∼ qε (here 1 ε > 0). Then one can compute that (1.24.a) does not hold. For example, in the case of n = 1, So, we see that the limit of (1.24.a) is 1 2 − ε. The idea that saves the day at this point is the product trick, which was pioneered in the case of semi-posivity questions by Viewheg [117]. The precise idea is to replace X by an m-times self fiber product X (m) over T . Let f (m) : X (m) → T be the induced morphism, Sect. 2.2. Then, one can replace the initial goal with showing that there exists t , t is klt for t ∈ T general. Running through the previous arguments for X (m) instead of X , this would boil down to showing that where E q,m is a subsheaf given by certain condition specified below of the subsheaf generated by global sections of The extra condition in the definition of E q,m is due to the need that has to be klt on a general fiber. This would be automatic if the conjecture that products of K -semi-stable klt Fano varieties are K -semi-stable was known. Unfortunately, this is a surprisingly hard unsolved conjecture in the theory of K -stability 1 . Hence, we elude it by considering only bases of H 0 X t , −q K X t that are induced from bases of H 0 X t , −q K X t . As log canonical thresholds are known to behave well under taking products, see Proposition 4.14, if the restriction | X (m) t to a general fiber is a divisor corresponding to such basis, the K -stability of X t implies that X is klt. Hence, the additional condition in the definition of E q,m is that it is the biggest subsheaf as above such that E q,m t is spanned by simple tensors for a basis t 1 , . . . , t l of f * O X (q(−K X/T − f * H )) t to be specified soon. So, we are left to specify a basis of (1.24.b) holds. For that we use the Harder- Let the basis v 1 , . . . , v l be any basis adapted to the restriction of this filtration over t, that is, to 0 = F 0 t ⊆ · · · ⊆ F r t . The lower part of the filtration, until the graded pieces reach slope 2g, where g is the genus of T , is globally generated. Furthermore, there is an induced Harder-Narasimhan filtration on the sheaf in (1.24.c). The part of slope at least 2g in the last filtration that we defined is globally generated such that its restriction over t ∈ T is generated by simple tensors in v i , Proposition 5.9. Hence, if E q,m is this part of the Harder-Narasimhan filtraton, then it is enough to prove that The final trick of the semi-positivity part is then that (1.24.d) can be translated to a probability limit, which then is implied by the central limit theorem of probability theory, see Theorem 5.11. We explain here the probability theory argument via the example of The claim then is that as m goes to infinity the rank of the non-negative degree part of F m over the rank of F m converges to 1. It is easy to see that this is the limit of the left hand side of the following equation as m goes to infinity: use author coding: 2 dx by the classical De Moivre-Laplace theorem, a special case of the central limit theorem. We obtain (1.24.d) by taking A → ∞ limit, and using that the above integral integrates the density function of the standard Gaussian normal distribution.

Nefness threshold, that is, Theorem 1.20.
This part uses the same ideas as the above semi-positivity part, but in a different logical framework. That is, the argument is not a proof by contradiction.
Instead, the starting point is that Hence, again up to a little perturbation and by using the ideas of the previous point, there is an integer m > 0 such that there exists a t , t is klt for t ∈ T general. Then standard semi-positivity argument (Proposition 6.4) shows that is nef. Lastly, one divides by δ − 1, converges to 0 with H , and lastly by a standard lemma (Lemma 8.1) removes the (_) (m) .

Positivity.
The rough idea here is to use a twisted version of the ampleness lemma, c.f., [71, 3.9 Ampleness Lemma] and the slight modification in [76, Thm 5.1]. We need a twisted version of the ampleness lemma as the techniques developed until this point in the article do not work directly over higher dimensional bases. The main idea here is that to get bigness of λ f it is enough to show positivity of λ f over a very general element C of each moving family of curves of T in a bounded way. Below we explain how we do this.
The main benefit of proving the result on the nefness threshold, Theorem 1.20, is the following: one can prove, again using standard semi-positivity arguments, see Proposition 6.4, that Q := f * O X (−r K X/T + α f * λ f ) is nef, for some constants r and α. Furthermore, these constants r and α can be cho-sen to be uniform, as f runs through all families obtained by base-changing on a very general element C of a moving family of curves on T . Then, the ampleness lemma (Theorem 9.8) gives an ample line bundle B on T such that for all curves C as above, C · B ≤ C · det Q. Then one can use another trick from (semi-)positivity theory, already contained in Viehweg's work, which shows that for q := rk Q there is an embedding Using the adjunction of f (q) * and f (q) * , we obtain the inequality of divisors which survives the restriction over C by the genericity assumption in the choice of C. From here, a simple intersection computation shows that C · B bounds deg λ f | C from below up to some uniform constants, not depending on the choice of C, see the end of the proof of point (a) of Theorem 1.9. Before the argument starts, in Sects. 2, 3 and 4 we present notation and background, as well as, simpler statements. The division of this part between the above 3 sections is based on topics. Section 2 contains general topics, Sect. 3 contains the definition of the CM line bundle and the related statements, and Sect. 4 contains the definition and the basics about the δ-invariant and K -stability.

Organization of the paper
We also include a table on the location of the proofs of the theorems stated in the introduction.

Statements of the introduction
Their proofs

Base-changes
All base-changes are denoted by lower index. For example, if f : X → T is a family, F is a coherent sheaf on X and S → T is a base-change, then F S := h * F, where h : S × T X → X is the projection morphism.

Fiber product notation
The most important particular notation used in the article is that of fiber products. That is, for a family f : X → T of varieties we denote the m-times fiber product of X with itself over T by X (m) . As in our situation the base is always clear, we omit it from the notation. Hence, X (m) denotes the fiber product over T of m copies of X , and for a point t ∈ T , X (m) t denotes the fiber product over t of m copies of X t . In this situation, p i : X (m) → X denotes the projection onto the i-th factor, and we set for any divisor D or line bundle L:

General further notation
A variety is an integral, separated scheme of finite type over k. A vector bundle is a locally free sheaf of finite rank.
The Q-linear system of a Q-divisor D on a normal variety is |D| Q := { L is an effective Q -divisor | ∃m ∈ Z, m > 0 : m L ∼ m D }.
A geometric fiber of a morphism f : X → T is a fiber over a geometric point, that is over a morphism Spec K → T , where K is an algebraically closed field extension of the base field k. We say that a condition holds for a very general geometric point/fiber, if there are countably many proper closed sets, outside of which it holds for all geometric points/fibers. General point/fiber is defined the same way but excluding only finitely many proper closed subsets. The (geometric) generic point/fiber on the other hand denotes the scheme theoretic (geometric) generic point/generic fiber.

Relative canonical divisor
For a flat family f : X → T the relative dualizing complex is defined by Grothendieck upper shriek functor as defined in [61]. If f is also a family of pure dimension n, then the relative canonical sheaf is the lowest non-zero cohomology sheaf ω X/T := h −n (ω • X/T ) of the relative dualizing complex. To obtain the absolute versions of these notions one uses the above definition for T = Spec k. The important facts regarding the relative dualizing sheaf that we use in the present section are the following: Lemma 2.4], and then as ω X is S 2 [75,Cor 5.69], ω X/T is also reflexive in this case [63]. (c) By the previous two points, if f is flat, X is normal and either T is smooth or the fibers are normal, then ω X/T is reflexive, and hence it corresponds to a linear equivalence class of Weil divisors which we denote by K X/T . (d) On the relative Cohen-Macaulay locus U ⊆ X (that is, on the open set where the fibers are Cohen- In particular, by the above we always have the following assumptions on our families: f : X → T is flat with fibers being of pure dimension n, and either T is smooth, or the fibers of f are normal. In both cases we discuss base-change properties of the relative canonical divisor below.

Base-change of the relative log-canonical divisor when the fibers are normal
Let us assume that f : X → T is a projective, flat morphism to a normal projective variety with normal, connected fibers. In particular then X is also normal. Assume additionally that there is an effective Q-divisor given on X , such that does not contain any fiber, and K X/T + is a Q-Cartier divisor. Let U ⊆ X be the smooth locus of f , which is an open set, and by the normality assumption on the fibers, U ∩ X t is a big open set on each fiber X t , see Sect. 2.3 for the definition of a big open set. Let S → T be a morphism from another normal projective variety. Then, we may define a pullback S as the unique extension of the pullback of | U to U S ; the key here is that | U is Q-Cartier. Moreover, if σ : X S → X is the induced morphism, then as Q-Cartier divisors Indeed, it is enough to verify this isomorphism on U , as U is big in X and U S is big in X S . However, over U the linear equivalence (2.0.e) holds by the definition of S and by the base-change property of point (d) above.
In particular, f S : X S → S and S satisfies all the assumptions we had for f : X → T and .

Base-change of the relative log-canonical divisor when the base is smooth
Let f : X → T be a flat morphism from a normal projective variety to a smooth, projective variety with connected fibers. Let be an effective Qdivisor on X such K X/T + is Q-Cartier. Let T norm ⊆ T be the open set over which the fibers of X are normal.
Note that by the smoothness assumption on T , at a point x ∈ X , the fiber X f (x) is Gorenstein if and only if X is relatively Gorenstein if and only if X is Gorenstein. Let U ⊆ X be the open set of relatively Gorenstein points over T . Let ι : C → T be a finite morphism from a smooth, projective curve such that ι(C) ∩ T norm = ∅, and denote by σ : X C → X the natural morphism.
We claim that σ −1 U is big in X C . This is equivalent to showing that for each c ∈ C, X c is Gorenstein at some point, and that for general c ∈ C, there is a big open set of X c where X c is Gorenstein. The former is true for all schemes of finite type over k, hence also for X c . The latter is true by the ι(C) ∩ T norm = ∅ assumption. This concludes our claim. Now, let π : Z → X C be the normalization of X C , ρ : Z → X and g : Z → C the induced morphisms and set W := ρ −1 U . The notations are summarized in the following diagram: Then, [76,Lem 9.13] tells us that there is a natural injection To be precise, [76,Lem 9.13] assumes σ −1 U to be normal, but as the proof does not use it, this is an unnecessary assumption. Combining 0.f) which is an isomorphism over the locus T red over which the fibers of f are reduced. Indeed, over T red the fibers of X C → C are all reduced, and by the ι(C) ∩ T norm = ∅ assumption the general fiber of X C → C is normal. In particular, over T red , X C is R 1 and S 2 , and hence normal. So, π is the identity over T red .
Let m > 0 be then an integer such that m(K X/T + ) is Cartier. That is, L := O X (m(K X/T + )) is a line bundle, and furthermore, m yields an embedding ω ⊗m U/T → L| U . Composing this with the m-th power of the homomorphism of (2.0.f) we obtain: Indeed, for the last remark, the main thing to note is that the regular locus of X , over which m is necessarily Cartier, pulls back to a big open set of g −1 ι −1 T red , as general fiber of f C is normal and special fiber of f C over T red are reduced. Hence π is an isomorphism over g −1 ι −1 T red and also the pullback ρ| g −1 ι −1 T red * m is sensible the usual way: restricting to the regular locus, performing the pullback there, and then taking divisorial extension using bigness of the open set. Lastly, the map (2.0.g) is given by an effective divisor D. If we set Z := D m , using that W is big in Z , we obtain:

Proposition 2.1 Consider the following situation:
• let f : X → T be a flat morphism from a normal projective variety to a smooth, projective variety with connected fibers, • let be an effective Q-divisor on X such K X/T + is Q-Cartier, • let T norm ⊆ T and T red ⊆ T be the open set over which the fibers of X are normal or reduced, respectively, • let ι : C → T be a finite morphism from a smooth, projective curve such that ι(C) ∩ T norm = ∅, and • let π : Z → X C be the normalization, and ρ : Z → X and g : Z → C be the induced morphisms.
Then, there is an effective Q-divisor Z on Z such that:

The definition of the CM line bundle
Here we present the definition of the CM line bundle in two cases: (a) in the non logarithmic case for arbitrary polarizations, and (b) in the logarithmic case for the anti-log-canonical polarization.
In the first case, we also connect it to the other existing definitions in the literature. In the second case, we are not able to present such connections, because the lack of literature would force us to work out many details about the Paul-Tian type definition [101,102], and then prove the equivalence with that: this would be beyond the scope of the present article. In any case, it is important to stress that the definitions are different in the two cases: One does not obtain the logarithmic version by simply plugging in the logarithmic relative anti-canonical divisor into the polarization of the nonlogarithmic case. The reason for the difference is that that in the logarithmic case the CM line bundle has to take into account also the variation of the boundary, see the paragraph before Theorem 1.9.

Definition 3.1 CM line bundle in the non-logarithmic setting.
Let f : X → T be a flat morphism of normal projective varieties of relative dimension n, and L an f -ample Q-Cartier divisor on X . For every integer q divisible enough, the Hilbert polynomial of a (equivalently any) fiber X t is We define the Chow-Mumford line bundle as the pushforward cycle which is an abuse of language as it is not a line bundle but rather a Q-Cartier divisor class, according to Proposition 3.7. We would also like to stress that λ f,L is a divisor class (in the Weil group, or equivalently the first Chow group), as opposed to a fixed divisor.
If L is not indicated, then we take L = −K X/T , which we assume to be an f -ample Q-Cartier divisor, and we use the notation λ f := λ f,L .
Remark 3.2 Note that in the L = −K X/T case: As X in Definition 3.1 is assumed to be normal, so is X t for t a general closed point. In particular, Lemma A.2 implies that In particular if L = −K X/T we obtain that μ L = n. Hence, we obtain the definition we used in (1.7.a): We only define the logarithmic version of the CM line bundle in the antilog-canonically polarized case. If = 0, this definition agrees with the case of L = −K X/T of the non-logarithmic definition, according to the final formula of Remark 3.2.
where M i are uniquely determined line bundles on T .
For future reference, we note that as the left side of (3.4.a) is invariant under base-change for q 0, the above unicity of M i implies that: where μ s L is the number defined in Definition 3.1. For simplicity we regard L C M, f,s L as a Cartier divisor. As we explained earlier in the case of Definition 3.3 a definition as above is not worked out in the literature to such an extent, and hence we do not consider it here.
The proof of the following proposition will be given in "13".

The delta invariant and K -stability
Here we give the definitions and the properties used in the present article of δ-invariants, as well as we present the definition of K -semi-stability and uniform K -stability in Definition 4.8. In the rest of the article we will use the characterizations of K -semi-stability and K -stability via δ-invariants given in Corollary 4.9. We also prove in the present section that the δ-invariant is constant at the very general fibers of a log-Fano family, see Proposition 4.15.

Definitions
Basis-type divisors and the delta invariant have been introduced by K. Fujita and Y. Odaka in [52], see also [18]; in this section we recall their definitions. • Z is a variety over k, • L is a Q-Cartier divisor on Z , and • q > 0 is an integer for which q L is Cartier.
, for which the corresponding s i ∈ H 0 (Z , q L) form a k-basis of H 0 (Z , q L), and D can be expressed as Let be a fixed effective Q-divisor on Z such that (Z , ) is a klt pair. Given a Q-Cartier effective divisor D on Z , we define its log canonical threeshold as Remark that since (Z , ) is klt, the above threshold is a positive number. Let us recall the definition of the α invariant. We write α(Z , ) for α(Z , ; −K Z − ).
The α invariant has been introduced by Tian in relation with the existence problem for Kähler-Einstein metrics. The delta invariant is a variation on the alpha invariant. The main difference is that in the case of α invariant one considers the log canonical threshold of all divisors in the Q-linear system, while in the δ invariant is defined using only basis type divisors. In particular, while α(X ) ≥ dim X dim X +1 only implies K -semi-stability [92,113], δ(X ) ≥ 1 happens to be equivalent to it [18,Theorem B], see also Corollary 4.9. The delta invariant was introduced in [52, Definition 0.2]. In [18], although it was also denoted by δ, it is called the stability threshold.
Remark 4. 4 We note the following subtleties of Definition 4.3: • According to [76,Lem 8.8], the infimum of point (a) is in fact a minimum.
• According to Corollary 4.7, the definition of point (b) does not depend on the choice of s, and the limsup in point (b) is in fact a limit.

Relation to K-stability
In this section we follow closely [18], as we want to adapt some of their result from Fano varieties over C to Fano pairs over k. Similar adaptation was done also in [19]. Consider the situation: Let v be a non-trivial divisorial valuation on Z associated to a prime divisor E over Z , we consider the filtration Denote by B q the set of qs-basis type divisors with respect to qs L. As observed for instance in [52, proof of Lemma 2.2], The next statement is a logarithmic version of [18,Theorem 4.4], following very closely the arguments given there.
where A(v) is the log-discrepancy of v with respect to the klt pair (Z , ), and the inf is taken over all non-trivial divisorial valuations. In particular, δ(Z , ; L) is independent of the choice of s. (b) Assuming furthermore that L is ample, the following bounds hold Proof Point (a). Set δ q := δ qs (Z , ; L) and δ := δ(Z , ; L). We first prove the inequality Thanks to Eqs. (4.5.a) and (4.5.b), we can write use author coding: We now prove the inequality This inequality follows from the key uniform convergence result [18, Corollary 3.6]: for every ε > 0 there exists a q 0 = q 0 (ε) such that for all q > q 0 and all divisorial valuations v we have taking the liminf on q on the right hand side, and then letting ε go to zero, we get the requested inequality. We obtain point (a) combining Equations 4.6.a and 4.6.b.
. Given a divisorial valuation v, we define its q-th pseudo-effective threshold as When L is ample, [18,Prop. 3.11] gives the following bounds which imply point (b) (again, the proof in [18] is over the complex numbers, but it works also over k). We give the following definition of K-stability, which is equivalent to the more classical one by [87, Theorem 6.1 (ii)] and [50, Theorem 1.5].
Here A(v) denotes the log-discrepancy of ν with respect to the pair (Z , ).
The following corollary is now an immediate consequence of the above definition and Theorem 4.6

Products
The following conjecture is motivated by the equivalence between K-stability and Kähler-Einstein metrics in the Fano setting, it has been already proposed in [96,Conjecture 1.11].
The analogue result for the alpha invariant and any polarization appeared for example in [76,Proposition 8.11], but used to be present much earlier in a smaller generality, i..e, in the smooth non-log case, for example in Viehweg's works. See also [96,Thm. 1.10] and [27,Lemma 2.29] for the Fano case. We can prove a weaker result for the delta invariant in Proposition 4.14.
Definition 4.11 (Product basis type divisor) Let (W, W ) and (Z , Z ) be two klt pairs, let L W and L Z Q-Cartier divisors on W and Z , respectively, and let q > 0 be an integer such that both q L W and q L Z are Cartier and where p W and p Z are the projections.

Remark 4.12
In Definition 4.11, if D W is associated to a basis s i and D Z to a basis t i , then D is associated to the basis s i t j .
Proof As we work in characteristic zero, we may take the product of a log resolution of (W, W ) and of (Z , Z ). This will be a log-resolution for (W × Z , W Z ), with the union of the discrepancies of the original two logresolutions, so the claim. Proposition 4.14 With the notations of Definition 4.11, let D be a q-product basis type divisor. Then, and both (W, W + t D W ) and (Z , Z + t D Z ) are log canonical because of the hypothesis on t, so the claim follows from Lemma 4.13 The full Conjecture 4.10 has been proved in the preprint [125], published after the first version of this paper has appeared.

Behavior in families
Here we prove that the δ-invariant is constant on very general geometric points. Recall that a geometric point of T is a map from the spectrum of an algebraically closed field to T . Key examples are the closed points and the geometric generic point (i.e. the algebraic closure of the function fields) of T .
Proof We may fix an integer s > 0 such that s L is Cartier and f * O X (qs L) is non-empty and commutes with base-change for any integer q > 0. In particular, then for all t ∈ T , s L t is Cartier and h 0 (X t , qs L t ) is positive and independent of t for any integer q > 0.
We claim that for each integer q > 0 there is a real number d > 0 and a non-empty Zariski open set U q ⊆ T such that for each geometric point t ∈ U , δ qs X t , t ; L t = d. Assuming this claim, by setting T q := T \ U q we obtain the statement of the proposition.
So, we fix an integer q > 0, and in the rest of the proof we show the above claim. We also set r := h 0 (X t , qs L t ) and l := qsr, where the former is independent of t ∈ T by the above choice of s.
Set W := P(( f * O X (qs L)) * ). Then, for any geometric point t ∈ T we have natural bijections: where k t is the residue field of t, and X K and X t are the corresponding base-changes, as explained in Sect.
According to [76,Lem 8.8] , the log canonical threshold function y → lct y ; Z y , y , which takes values on the geometric points of Y , is lower semi-continuous. Furthermore, the second paragraph of [76,Lem 8.8] shows that there is a dense open set Y 0 ⊆ Y such that lct y ; Z y , y is the same for every y ∈ Y 0 . Applying this iteratedly to the complement of Y 0 , we obtain that y → lct y ; Z y , y takes only finitely many values on Y , say r 1 > r 2 > · · · > r l , and the level sets are constructible subsets of Y . Hence, are open sets, and for any geometric point y : It follows that for any geometric point t ∈ T , After the above discussion, our claim follows immediately. Indeed, we just need to choose a to be the smallest integer such that L a contains the generic fiber of π . Then there is a non-empty open set U ⊆ T contained in In particular, for any geometric point t ∈ U : Therefore, by setting d := r a , (4.15.c) implies that δ qs X t , t = d for all geometric points t ∈ U .
Remark 4. 16 We note that one could define the δ-invariant also over over non algebraically closed base fields, with verbatim the same definition as Definition 4.3. If (Y K , K ) is a projective klt pair and N K is a Q-Cartier divisor defined over a non-closed field K , and furthermore we choose a basis type divisor D = In particular, if one takes a conic bundle f : X → T without a section, and η is the generic point of T , then for the generic fiber we have δ X η = 2, but for all geometric fiber (including the geometric generic fiber) outside of the discriminant locus we have δ X t = 1. So, the δ-invariant is not the same for a general and for the generic point (in general). In particular, one cannot replace "any geometric point t ∈ T " in Proposition 4.15 with just "any point t ∈ T ".

Remark 4.17
The special case of Proposition 4.15 when d = 1 and = 0 (so for K -semi-stability via [18]) was shown in [20,Thm 3] with other methods.

Remark 4.18 Proposition 4.15 is very weak version of what is expected to hold.
It is conjectured, cf., [20], that δ is lower semi-continuous, and furthermore the δ ≥ 1 set is also open. Some of this has been proven in [81, Thm 1.1(i)] and [20]. The full conjecture has been proven recently in [22], after the first version of this paper was published.

Growth of sections of vector bundles over curves
In this section, we present results about the growth of the number of sections of vector bundles over curves. We apply these in Sects.

Proposition 5.3 In the situation of Notation
Proof We prove the h 0 vanishing, and then the h 1 vanishing follows by Serreduality. The bundle ω T ⊗ E * is also semi-stable and μ(ω as a section would give a subbundle of E of slope 0.

Proposition 5.4 In the situation of Notation
Proof Fix a closed point t ∈ T , and let G be either E or E(−t). Riemann-Roch tells us that In particular, h 0 (E) = h 0 (E(−t)) + rk(G). So, by looking at the usual exact sequence: we see that E is in fact generated at t. As t was chosen arbitrarily, E is globally generated. Remark 5.6 When T = P 1 , we have a canonical decomposition with a j < a j+1 . In this case, the Harder-Narasimhan filtration turns out to be In the study of K-stability, a key situation is when T = P 1 is the base of a test configuration f : (X , L) → T trivially compactified at infinity, and E = f * (qL) for some q > 0. The classical localization formula, see for instance [119,Example 1], implies that the Harder-Narasimhan filtration of E is equal to the weight filtration with respect to the G m -action induced by the test configuration. We can thus think at the Harder-Narasimhan filtration as a generalization of the weight filtration.
On the other hand, we also note that the Harder-Harasimhan filtration is much more general than the weight filtration as it exists for any family not only for test configuartions, in particular for non-isotrivial families over arbitrary curve bases. This is a crucial point for our argument. Proof We prove both statements at once, by induction on the length of the Harder-Narasimhan filtration of E. If = 1, both statements were shown in Proposition 5.3. So, we may assume that > 1. However, then we may include E in an exact sequence 0 where G is semi-stable of rank at least 2g and E also satisfies the assumption of the proposition, but with replaced by − 1. Hence we know both statements for E replaced by E . Applying now long exact sequence of cohomology to (5.7.a) yields: where the two vanishings are due to Proposition 5.3 and induction, respectively. This proves our cohomology vanishing statement.
For the global generation statement, we just use that both G and E are globally generated again by Proposition 5.7 and induction respectively. Hence, according to (5.7.b), the sections generating these two bundles at a given t give a section generating E at t.
After the above basic statements, we work towards Theorem 5.11. This is a statement about tensor powers of vector bundles of positive degree. In particular, we need to understand the Harder-Narasimhan filtration of a tensor power, in terms of the Harder-Narasimham filtration of the original vector bundle. The necessary notation is introduced in Notation 5.8.

Notation 5.8
In the situation of Notation 5.5, fix also a closed point t ∈ T , which will be the point at which the global sections we are interested in would need to become simple tensors. Then the Harder Narasimhan filtration induces Let {e i } be a basis of E t adapted to this filtration. By this, we mean that the intersection of F j t and {e i } gives a basis of F j t for every j. Fix an integer m > 0, which will be the power of the tensor-power of E that we are examining. We will parametrize subsheaves of E ⊗m that are tensor products of the Note that S m is closed in the downwards direction, that is, whenever s ∈ S m , and s < s, then s ∈ S m . We also assign a minimal slope ν(s) to s ∈ {1, . . . , } m given by m j=1 μ s j , which will be the actual minimal slope of the corresponding sheaf in the Harder-Narasimham filtration of E ⊗m . Note that ν(s) ≥ 2g if and only if s ∈ S m .
After the above, we arrange the elements of {1, . . . , } m in a decreasing order with respect to ν(_), such that S m consists of the first d elements In fact, it is not clear from the definition that H c is a subbundle as oppoed to just a coherent subsheaf. We prove in Proposition 5.9 that it is indeed a subbundle. For simplicity we also define After the above notational preparation, it is quite straight-forward to state and prove the description of the Harder-Narasimham filtration of E ⊗m that we need:

Proposition 5.9
In the situation of Notation 5.8:  Hence, we have equality in the middle of (5.9.b), and hence the last homomorphism in (5.9.a) is an isomorphism for all c.
In particular, for all 1 ≤ c ≤ e, there is an exact sequence: This concludes (a), as both H c and E ⊗m /H c are iterated extensions of vector bundles, hence they are both vector bundles. Point (c) also follows immediately from the definition of H c as it is a sum of product type subbundles. Lastly, (b) also follows directly from (5.9.c).

Notation 5.10
In the situation of Notation 5.8 (in fact, for introducing the following notation we only need the first two paragraphs of Notation 5.8), let G m,t be the k-linear subspace of (5.11.a) As i=1 p i = 1, we may define a discrete probability space X on elements {1, . . . , l} with measures p 1 , . . . , p l respectively. Let X j be a sequence of independent identically distributed random variables of X that take value μ i on i, and let Z m := m i= j X j . With this language, (5.11.a) tells us that where P(. . . ) denotes the probability of the given condition. Hence we are left to show that lim m→∞ P (Z m ≥ 2g) = 1. . Note that as X is a finite metric space both the expected value μ and the variance σ 2 of X j are finite. Then the central limit theorem states that the random variable Z m −mμ √ m weakly converges to a normal distribution with expected value 0 and covariance σ 2 . In particular, this induces a convergence on the level of distribution functions, or more precisely we would like to use the following convergence, which holds for each real number A and it is shown for example in [ Proof of the claim For this, note first that We continue the proof of Theorem 5.11: Combining our claim and (5.11.c) we obtain that lim inf As this is true for all real numbers A, and lim A→−∞ P( ≥ A) = 1, we obtain that This is exactly the statement of (5.11.b), which was our goal to prove.
Remark 5. 12 We note that in the proof of Theorem 5.11, one can replace the Central Limit Theorem by the weaker statement of Chebyshev's inequality. Indeed, using the notation of the proof, as the variance of Z m is mσ 2 :

Ancillary statements
Here we gather smaller statements that are used multiple times in Sects. 7, 8 and 9.

Normality of total spaces
In the next sections we work mostly in the following setup: Notation 6.1 Let f : (X, ) → T satisfy the following assumptions: (a) T is a smooth, projective curve, (b) X is a normal, projective variety of dimension n + 1, (c) f is a projective and surjective morphism with connected fibers, (d) is an effective Q-divisor on X , (e) −(K X + ) is an f -ample Q-Cartier divisor. (f) (X t , t ) is klt for general t ∈ T . Lemma 6.2 In the situation of Notation 6.1, there exists a finite morphism from a smooth projective curve τ : S → T such that if g : Y → S is the normalization of the pullback of f , and π : Y → X the induced morphism, then g has reduced fibers, and there is an effective Q-divisor on Y such that Proof Let τ be any finite cover such that at the closed points t ∈ T over which the fiber X t is non reduced, the ramification order of τ is divisible by all the multiplicities of all the components of τ . Then, g will have reduced fibers, and Sect.

Lemma 6.3 If f : X → T is a surjective morphism from a normal variety to a smooth projective curves with reduced fibers, m > 0 is an integer and τ : S → T is a finite morphism from another smooth curve, then (a) X × T S is normal, and (b) X (m) is normal, see Sect. 2.2 for the product notation.
Proof First we note that f is flat and hence so is f (m) : X (m) → T by induction on m and the stability of flatness under base-change.
We know that a variety Z is normal if and only if it is S 2 and R 1 . In the particular case, when Z maps to a smooth curve U via a flat morphism g, then Z is S 2 if and only if the general fibers of g are S 2 and the special ones are S 1 (so without embedded points) [ It is immediate then that this characterization of S 2 and R 1 propagates both to fiber powers and to base-changes.

Semi-positivity engine Proposition 6.4 Let f : (X, ) → T be a surjective morphism from a normal, projective pair to a smooth curve such that (X t , t ) is klt for general t ∈ T (recall that X t is normal for t ∈ T general, is Q-Cartier at the codimension 1 points of X t , and hence t makes sense), and let L be a Cartier divisor on X such that L − K X/T − is an f -ample and nef Q-Cartier divisor. Then, f * O X (L) is a nef vector bundle.
Proof According to Lemma 6.2 we may assume that the fibers of f are reduced. According to [97,Lem 3.4], it is enough to prove that for all integers m > 0, the following vector bundle is generated at a general t ∈ T by global sections: [76,Lem 3.6], and see Sect. 2.2 for the fiber product notation For that it is enough to prove that the natural restriction homomorphism We note here that according to Lemma 6.3, X (m) is normal. Furthermore, We also note that the only generality property of t that we use below is that X t is normal, X t Supp t and (X t , t ) is klt. Hence, at this point, we fix a t with such properties.
Set I := J (X (m) , (m) ) , where J denotes the multiplier ideal of the corresponding pair. Then for the above surjectivity the next diagram, the top row of which is exact, shows that it is enough to prove the vanishing of We note that here we used that X t . This then implies that I is trivial in a neighborhood of X (m) t . We conclude by noting that the above cohomology vanishing is given by Nadel-vanishing as

is globally generated for t ∈ T general, and (c) L − K X/T − is an f -ample and nef Q-Cartier Q-divisor.
Then L is nef.
Proof According to Lemma 6.2 we may assume that the fibers of f are reduced, and by further pullback using Lemma 6.3.(a), we may also assume that N is Cartier, whence L is also Cartier. Then, we may apply Proposition 6.4 yielding that f * O X (L) is nef.
Note now that cohomology and base change always holds over a dense open set. So, for general t ∈ T we have a commutative diagram as follows: Assumption (b) tells us that the bottom arrow of diagram (6.5.a) is surjective. Hence, so is the top arrow, and then the natural homomorphism As L is f -nef, we only have to show that if C is a horizontal curve, then C ·L ≥ 0. However, by the previous paragraph,

Semi-positivity
In this section we prove our semi-positivity results. Here, and also in Sect. 9 we use extensively the fiber product notation explained in Sect. 2.2.

Framework and results
The main result of the section is the following, from which the statements of the introduction will follow in a quite straightforward manner. Theorem 7.1 In the situation of Notation 6.1, if δ X t , t ≥ 1 for a very general geometric point t ∈ T , then deg λ f, ≥ 0.

Proofs
The proof of Theorem 7.1 will be by contradiction with the next proposition.

Proposition 7.2
In the situation of Notation 6.1, let H be an ample Q-divisor on T . Then, there do not exist Q-Cartier divisors and on X such that: Proof Assume that there exist and as above. Let a > 0 be a rational number such that −K X/T − + a f * H is ample. Fix a rational number ε > 0 such that εa − (1 − ε) < 0. Apply then Corollary 6.5 by setting the L, N and of Corollary 6.5 to be respectively (εa − (1 − ε)) f * H , −(εa − (1 − ε))H , and + (1 − ε) . These choices satisfy the assumptions of Corollary 6.5 by the right hand side of Eq. (7.2.a), and it yields a contradiction as indicated under the left hand side term of (7.2.a).
Proof of Theorem 7.1 As both the consequences and the conditions of the theorem are invariant under base-extension to another algebraically closed field, we may assume that k is uncountable. In particular, whenever a property is true for very general geometric fibers, it is also true for some closed fibers. That is, by removing countably many proper closed sets from a variety over k, there are some closed points left. The reason is the following: by cutting down with hyperplanes, this statement can be reduced to curves, where it is true because removing countably many closed points from a curve over an uncountable field leaves uncountably many points of the curve intact.
First, according to Lemma 6.2 we may assume that all fibers of f are reduced. This is to guarantee that the m-times iterated fiber product X (m) is normal for any integer m > 0, according to Lemma 6.3.
We argue by contradiction, so assume that deg λ f, < 0. For m big enough, we are going to produce divisors and on X (m) whose existence contradicts Proposition 7.2.
Fix a closed point t in T such that X t is normal, X t Supp t , (X t , t ) is klt and δ(X t , t ) ≥ 1, using Proposition 4.15. Let H be an ample line bundle on T . Fix rational numbers a, ε > 0 and 0 < c < 1 and an integer q > 0, such that: This is possible because Definition 3.3 and the assumption deg From now on, let E := f * O X (q M). Remark that according to [76,Lemma 3.6] for every integer m > 0, and by item (e), the following base change holds In general, it is not possible to lift a basis of E t to sections of E. However, thanks to Theorem 5.11, we can choose a basis e i of E t , an integer m > 0, and global sections s i of E ⊗m so that the sections s i , when restricted over t, are linearly independent pure tensor in the e i , and furthermore We are now ready to construct and on X (m) as in Proposition 7.2. We let To complete the proof of Theorem 7.1, we have to prove that and are as in Proposition 7.2, with f replaced by f (m) . To check item (c), remark that To prove of item (c) of Proposition 7.2, we compute the log canonical threshold. We first remark that, since the sections s i restricted to X (m) t are linearly independent pure tensors in the e i , we have that for the q-product basis type divisor P on X (m) t associated to {e i }, as in Definition 4.11 and Remark 4.12. Using Proposition 4.14, we obtain that lct X Hence, all assumptions of Proposition 7.2 are verified, implying that and cannot exist. Therefore, we obtained a contradiction with our initial assumption that deg λ f, < 0.

Proof of Theorem 1.8 The proof of point (a):
As at the beginning of the proof of Theorem 7.1, we may assume that k is uncountable. According to [24,Thm 0.2], it is enough to show that λ f, ·C ≥ 0 for every morphism ι : C → X from a smooth projective curve such that C → ι(C) is the normalization and ι(C) is a very general curve in a family covering T . In particular, for a very general closed point t ∈ ι(C), X t is normal, (X t , t ) is klt and δ (X t , t ) ≥ 1. Let Z → X C be the normalization, g : Z → C the induced morphism and C · λ f, .
Proof of points (a) and (b) of Theorem 1.8 These are special cases of Theorem 1.8.

Bounding the nef threshold Lemma 8.1 If f : X → T is a morphism between projective varieties, m > 0 is an integer and M is a Q-Cartier divisor on X , then M is nef if and only if M (m) is nef.
Proof If M is nef, M (m) is nef by definition. For the other direction, assume that M (m) is nef. Let ι : C → X be a morphism from a smooth, projective curve. Take then the diagonal morphism : C → X (m) , which is defined by the equality p i • = ι for each i. Then: Hence, M · C ≥ 0. As this works for any curve C in X we see that M is nef.
Proof of Theorem 1.20 As both the consequences and the conditions of the theorem are invariant under base-extension to another algebraically closed field, we may assume that k is uncountable. In particular, as at the beginning of the proof of Theorem 7.1, whenever a property is true for very general geometric fibers, it is also true for some closed fibers. According to Lemma 6.2 we may assume that all fibers of f are reduced. In particular then for all integers m > 0, X (m) is normal according to Lemma 6.3. Set λ := λ f, .
Fix the following: (a) let H be an ample divisor on T of degree 1, (b) let t ∈ T be a closed point such that X t is normal, X t Supp , (X t , t ) is klt and δ(X t , t ) = δ, (c) let 0 < ε < δ − 1 be an arbitrary rational number, and (d) let 0 < ε ε be another rational number.
It is enough to prove that is nef, as we may converge with ε and ε to δ − 1 and to 0, respectively. Set Note that We now fix a positive integer q so that the following hold: and, by item (g), the following base change holds According to Theorem 5.11, we may find a basis {e i } of E t , an integer m > 0, and global sections s 1 , . . . , s of E ⊗m so that the sections s j , when restricted over t, are linearly independent pure tensor in the e i , and furthermore (8.1.d) Define as Note that according to (8.1.b), So, to show (8.1.a), according to Lemma 8.1 it is enough to prove that K X (m) /T + (m) + is nef, and for that according to [46,Thm 1.13] it is (m) t + t is klt for some (equivalently, a general) t ∈ T . For this we compute the log canonical threshold. We first remark that, since the sections s i restricted to X have the same dimension, say d. This dimension is the dimension of a general isomorphism equivalence class of the fibers of f . As these isomorphism equivalence classes (at least general ones) would be exactly the fibers of any reasonable moduli map, one defines the variation of f as This is a contradiction.

Notation 9.4
In the situation of Notation 9.2, (a) let q 0 > 0 be an integer such that q let g be the genus of T .

Lemma 9.5
In the situation of Notation 9.4, for every positive integer q 0 |q, Proof Assume the contrary, that is, μ(F 1 q ) > 2g, and let t ∈ T be an arbitrary closed point. According to Proposition 5.4, F 1 q (−t) is globally generated. So, there is a ∈ |q(−K X/T − )− f * L |, where L is the divisor determined by t on T . Hence, for := q and L := L q we have ∈ | − K X/T − − f * L| Q . This contradicts Theorem 9.3.
Proof Assume that μ F 1 q > 0, and let H be the image of for some m 0. We claim that H is not zero because of the following: Let η be the generic point of T . Then any x ∈ F 1 q η can be identified with somẽ In particular, the following implications conclude our claim: Let then j be the smallest integer such that F This contradicts Lemma 9.5.

Theorem 9.7
In the situation of Notation 9.2, if q > 0 is an integer such that −q(K X/T + ) is Cartier, then f * O X (−q(K X/T + )) is a semi-stable vector bundle of slope 0.
Proof First, Theorem 1.20 yields that −K X/T − is nef. Then, f * O X (q (−K X/T − )) is also nef, by Proposition 6.4 taking into account the Q-linear equivalence Finally, Proposition 9.6, concludes our proof.

Ampleness lemma
Theorem 9.8 is an extract of the argument of the ampleness lemma of [71] (one assumption removed in [76]). It will be one of the main technical ingredients for the proof of items (a) and (b) of Theorem 1.9 given in Sect. 9.4. We denote by Gr(w, w − q, k) the Grassmanian parameterizing linear subspaces of dimension w − q in the w dimensional k-vectorspace k ⊕w .

v, k) induced by φ is finite to one. Then, for each ample Cartier divisor B on T there is an integer m > 0 and a non-zero homomorphism
Proof We explain how to turn the proof of [76, Thm 5.5] into a proof of the above statement.
First, specialize [76,Thm 5.5] to the case of projective base and, thanks to [76,Rem 5.3], to the special choices of W = Sym d (V ) and G = GL(w, w − q, k). At this point the assumptions of [76,Thm 5.5] become identical to ours, except that in [76,Thm 5.5] was assumed to be weakly positive.
The first, and actually main, step of the proof of [76,Thm 5.5] is to construct a non-zero homomorphism as the one whose existence we have claimed. This homomorphism is displayed in [76,Equation (5.5.5)]. In the remaining part of the proof, which is actually just the the few lines after Equation (5.5.5), the authors use the weakly positivity assumption to deduce weakly positivity of the domain and bigness of the codomain of the morphism, and this is the only place where weakly positivity is used.
We conclude that the argument given in the proof of [76,Thm 5.5] to show the existence of the homomorphism displayed in [76,Equation (5.5.5)] also proves our claim.

Arbitrary base
Proof of point (a) of Theorem 1.9 As in the proofs of Theorem 7.1 and Theorem 1.20, we may assume that k is uncountable. Let η be the generic point of T .
Throughout the proof ι : C → T denotes the normalization of a very general member of an arbitrary covering curve family of T . Very general here means that it is not contained in countably many divisors S i , which we will specify during the proof explicitly. Set: • η C to be the generic point of C, • Z := X C (note that as the fibers of f are reduced, and the general ones are normal, Z is normal), • σ : Z → X and g : Z → C be the induced morphisms, Then the following holds: 1.(a), and λ = λ f | C by Proposition 3.8.(b). • a Q-Cartier divisor L is pseudo-effective if and only if L · C ≥ 0 (for any such C), • according to Proposition 4.15, δ X η C = δ (assuming we add the countably many divisors to S i , over which δ(X t ) < δ, which are given by Proposition 4.15). In particular, as δ > 1 the very general fibers of g are uniformly K -stable, and hence klt, see [89,Theorem 1.3]. • in particular, by Theorem 7.1, deg λ ≥ 0, • by Theorem 1.20, −K Z /C + αg * λ is nef and g-ample.
It is important that throughout the proof all constants, so all rational numbers, will be fixed independently of the particular choice of C (for which there are two choices, first one choses a covering family, and then a very general member of that). For this reason, whenever such a constant is fixed, we do it in a numbered list item, see points above and below.
Choose integers r ≥ 2 and d > 0 such that (c) r K X/T and r αλ f are Cartier, (d) h i (X t , − jr K X t ) = 0 for all integers i, j > 0 and all t ∈ T , (e) −r K X t is very ample for all t ∈ T , (f) the multiplication maps W : up to the action of GL(v, k) on the target. Note that this is achievable because I form a flat family as t varies.
In particular, if we set w := rk W and q := rk Q, then for every t ∈ T (k), K t ⊆ W t determines X t → P v−1 up to the action of GL(v, k), which then means that the orbit of K t in Gr(w, q)/ GL(v, k) determines X t up to isomorphism. Therefore if we apply Theorem 9.8 for W → Q, then the fibers of the classifying map T (k) → Gr(w, q)/ GL(v, k) are contained in the isomorphism classes of the fibers of f and hence, by the maximal variation assumption, there is an open set where these fibers are finite. As, by Proposition 6.4, g * O Z (r (−K Z /C + 2αg * λ)) is a nef vector bundle. Set As the target of ξ is a line bundle, there exists a divisor, on the complement of which ξ is surjective. Let us add this divisor to S i . Then ξ | C is a non-zero homomorphism as follows: and let A be a divisor corresponding to A. As g * O Z (M) is nef and hence so is every bundle that admits a generically surjective map from the left side of ξ C , we obtain that Consider now, the natural embedding: given by the embedding of representations det → q i=1 of GL(q, k). Hence, by adjunction of f where D is an effective divisior on X (q) . We claim that, since O X (d M) is compatible with base change, D does not contain any fiber. Indeed, assuming by contradiction that D contains a fiber X t , we obtain a basis , and this is a contradiction as the s i are linearly independent.
By the continuity of log canonical threshold, there is a 0 < ε < 1 rd such that X (q) t , εD t is klt for general closed points t ∈ T . In particular by the genericity of C the same holds also for general t ∈ C. Then, if we define N := dr(−K X/T +3α f * λ f ), according to Corollary 6.5, the following divisor is nef (Z (q) is normal by Lemma 6.3.(b)).
Set ε := ε qrd . Then we have that N C dr − ε g * A C is nef according to Lemma 8.1.
m is pseudo-effective (as it dots to at least zero with each movable class). Therefore, λ f is the sum of an ample and a pseudoeffective Q-divisor, so λ f is big.
Proof of point (b) of Theorem 1.9 By Nakai-Moishezon it is enough to prove that for all normal varieties V mapping finitely to X , λ f | V dim V > 0. However, using Proposition 3.8, this we may obtain by replacing f : X → T with f V : X × T V → V , and applying point (a) to f V .
Proof of point (c) of Theorem 1.9 Let λ be the CM line bundle on T . According to [80, Thm 6.1], it is enough to prove that for every irreducible closed subspace Z of T we have (λ| Z ) dim Z ≥ 0, with strict inequality if Z intersects U .
The algebraic space Z has a finite cover π : V → Z by a scheme [1, Tag 04V1], and by Nagata's theorem and resolution of singularities we may also assume that V is projective and smooth. To prove (λ| Z ) dim Z ≥ 0, it is enough to show (π * λ| Z ) dim V ≥ 0. As V is smooth and projective, this is equivalent to show that π * λ| Z is nef. Since λ is compatible with base-change (Proposition 3.8 (a)), this follows from our semipositivity result Theorem 1.8 (b).
To prove the strict inequality, as π * λ| Z is nef, we have to show that π * λ| Z is big. This follows from our positivity result Theorem 1.9 (a), remarking that, as the isomorphism class of the family f are finite, the family f Z is of maximal variation, and a finite cover does not affect this maximality.
Proof of Theorem 1.23 Choose q big enough such that −q(K X/T + ) is Cartier and without higher cohomology on the fibers. Let H i ∈ |H | be general for i = 1, . . . , dim T − 1, and set C := dim T −1 i=1 H i . By the above generic choices, Z := X C is normal. Furthermore, C lies in the smooth locus of T , hence for base-change properties along C → T we may assume that T is smooth. In particular, there is an induced boundary Z on Z (Sect. 2.4.2), for which K X/T + | Z = K Z /C + Z (Proposition 2.1), and consequently Therefore, according to Theorem 9.7, ( f C ) * O Z −q K Z /C + Z is a semistable vector bundle of slope 0. However, then the isomorphism (9.8.b) implies that f * O X (−q(K X/T + )) is H -semi-stable of slope 0: if it had a subsheaf F of H -slope bigger than 0, then for the saturation F of F, F | C would be a subbundle of positive degree of ( f C ) * O Z −q K Z /C + Z , which is a contradiction.
Proof of Corollary 1. 24 First, assume that f is analytically locally a fiber bundle. Then, all fibers are isomorphic, and hence uniformly K -stable. In particular, there is an induced moduli map T → M u-K-s n,v . As Second, assume that deg λ f = 0. By Theorem 1.23, the vector bundles E q := f * O X (−q K X/T ) are semistable of slope zero for all q divisible enough. Now, we have to use the language of Higgs bundle to use a result from [107]. Consider the functor from the category of semistable bundles of slopes zero on T to the category of semistable Higgs bundle on T with c 1 = 0 associating to E the pair (E, 0), where 0 be the zero homomorphism E → E ⊗ 1 X . This functor is fully faithfull. By [107,Corollary 3.10], its codomain is equivalent to the category of local systems. Moreover, by the remark at the end of subsection "Examples' 'of [107, Section 3], the local system associated to a semistable Higgs bundle E is isomorphic to E as holomorphic vector bundle.
We conclude that the multiplication map is a actually a morphism of local systems, and its kernel K q,r is a local system too. Taking r = 2, and q big enough such −q K X/T is very ample and the ideals defining the fibers in PE q are defined by quadrics, we conclude that the fibration is locally trivial in the analytic topology.
Proof of Corollary 1.12 The proof is very similar to that of point (c) of Theorem 1.9 above. As in the above proof, T has a generically finite cover by a smooth, projective scheme. By base-changing over this cover one may assume that the base is smooth and projective. By Proposition 3.7, we may replace N by the CM-line bundle notion used in the present article, see Definition 3.1, and then point (b) of Theorem 1.8 and point (a) of Theorem 1.9 concludes the proof.

Proof of the main theorem
For the precise definition of the functor of M K-ss n,v we refer to [5,23]. For the present article, the important facts are the following: We start with Lemma 10.1. In the proof of Theorem 1.1, where we apply Lemma 10.1, we want to show the descent of the CM line bundle has positive self-intersection over a proper subspace V of M K-ps n,v . Hence, we want to construct a generically finite cover of V that supports a universal family. Because M K-ss n,v is an Artin stack, this is not possible. However, we can cook up one cover (T → V in the lemma) that supports a universal family over a big open set of V , and this universal family extends on the whole cover to a family f : X → T to which our theorems apply. That is, f is flat, X is normal and klt and −K X/T is Q-Cartier. It is a delicate task to find such an extension, so the second part of the proof of Lemma 10.1 is dedicated to this. The rough idea is to find a flat f : X → T , such that (a) K X + ∼ Q 0, and (b) (X, (1 + ε ) ) is a KSBA stable family.
Indeed, in this situation −K X is up to a scaling Q-linearly equivalent to K X + (1+ε ) , which is ample over T . Although, invoking the KSBA moduli space, guaranteeing condition (b) is quite straightforward, guaranteeing condition (a) is much harder. Hence, in the finishing part of Lemma 10.1, we need to invoke passing to a Q-factorialization, and running an adequate MMP. Additionally, in each step of this process we will need to show that flatness is preserved. As X satisfies the existence part of the valuative criterion for properness, after replacing Z by a finite cover we may assume that Z M K-ss n,v is a morphism in codimension 1. Hence, there is a big regular open setZ ⊆ Z and a family f :X →Z of K -semi-stable Fano varieties. In particular,f is flat,X is normal, −KX is Q-Cartier and ample overZ , andX is klt. We may also assume thatZ maps finitely to an open setṼ of V .
We claim that by possibly shrinkingZ , but still keeping it big in Z , we may find a Q-divisor˜ onX such that KX +˜ ∼ Q,Z 0 and X z ,˜ z is klt for every z ∈Z . For this choose an integer m > 0 and an ample Cartier divisor H onZ such that −m KX /Z +f * H is very ample, and that To show that this is a good choice, it is enough to show that for every codimension 1 point ξ ofZ , X ξ ,˜ ξ is klt. By the genericity assumption on i , using Bertini on the general fiber, there are only finitely many codimension one points ofZ over which X , i is not lc. Using the genericity assumption again, we can also assume that these codimension one points are different for different values of i. Hence, for a fixed codimension 1 point ξ , there is at most one index, say j, such that X ξ , i ξ is not lc. If there is no such index, set j to be a random one. Then, we may writẽ is klt by (10.1.b), and that X ξ , i ξ is lc for all j = i by the choice of j. Then, the fact implied by (10.1.b) that 1 2 + m−1 m 2 < 1, together with equation (10.1.c) yields that X ξ ,˜ ξ is klt. Hence we have showed our claim.
Note that our claim above also implies that for every 0 < ε 1, X , (1 + ε)˜ is a log canonical model overZ . Additionally, by the claim, we In particular, T → M KSBA induces f pre : X pre , (1 + ε) pre → T ) such that X pre , pre | T 0 = X ,˜ ×Z T 0 . Additionally X pre , (1 + ε) pre is klt as it is a family of stable log-varieties with klt general fiber. Note that f pre : X pre → T is flat. We are going to take Q-factorialization of X pre and then we will run a particular MMP and finally we will take a particular canonical model. The point is that all these operations preserve flatness. The reasons is that at each step the statement corresponding to the following one holds, where d := dim T : For each closed t ∈ T and general hyperplanes H 1 , . . . , H d through t : Then, equidimensionality of f pre follows from [36,Prop 34], as X t = d i=1 H i is a union of lc-centers of codimension d, and [36,Prop 34] states that an lccenter is contained in the intersection of d Q-Cartier divisors of coefficient 1 from the boundary (here the H i ), then the codimension of the lc-center has to be at least d. As X pre , pre is klt, X pre is Cohen-Macayulay, an hence by the above shown equidimensionality, f pre is indeed flat. Additionally, [36,Prop 34] Also, (X, ) and (X min , min ) are crepant, by point (h). Hence, by point (g), (10.1.d) is satisfied for f pre , X pre , pre and (1 + ε) replaced by f , X , and 1, respectively. In particular, f is also flat and has reduced fibers.
We also note that as (X, (1 + ε ) ) is klt and K X is Q-Cartier, X is also klt. Take now the Stein factorization of T → V . As T 0 is finite over an open set of V , we obtain diagram (10.1.a) from our statement. The k-points of M K-ss n,v correspond to K-semistable Fano varieties over k, and their stabilizers correspond to the automorphism group of the variety.
Fix a K-semistable Fano variety F, its automorphism group G = Aut(F) is a linear algebraic group, and the fiber λ [F] of the CM line bundle over [F] ∈ M K-ss n,v is a one dimensional representation of G. Let G 0 be the connected component of the identity of G, and G 0 = R U be its Levi decomposition.
We first show that G 0 acts trivially on λ [F] . The unipotent part U acts trivially because all one dimensional representations of unipotent groups are trivial. To prove that also the reductive part R acts trivially, we have to show that for every one parameter subgroup γ : G m → R, the weight of the action of γ on λ [F] is zero. This weight equals the Donaldson-Futaki invariant of the product test configuration of X induced by γ . This invariant vanishes because F is K-semistable.
The quotient G/G 0 is a finite group, it does not necessarily act trivially on λ [F] , however it acts trivially on λ ⊗M [F] for every integer M divisible by the cardinality of G/G 0 . To conclude, we have to show that the cardinality of G/G 0 is bounded as we vary [F] in M K-ss n,v . By the boundness of K-semistable Fano varieties of dimension n and volume v proved in [64], there exists a projective family f : Y → T of Fano varieties over a smooth base with the following property: for every As, T 0 is a big open set in S, (10.2.b) implies that ξ * L = φ * λ f . As λ f is big, so is φ * λ f , and then by the finiteness of ξ so is L| V . This concludes the proof of our second claim.

Boundedness of the volume
Proof of Corollary 1. 17 We have For the second inequality, if F is smooth and = 0, we apply the bound on the volume of K-semi-stable Fano varieties obtained in [

Examples
In this section, we give examples showing the sharpness of our theorems.
Example 12.1 Here, we give an example of a family of Fano varieties which are not K-semistable and such that the degree of the Chow-Mumford line bundle is strictly negative. The members of this family are smooth del Pezzo surfaces of degree 8, and the family is isotrivial but not trivial. The relevance of this example for the study of the Chow-Mumford line bundle was already pointed out by J. Fine and J. Ross in [43,Example 5.2]. Let us warn the reader that, in contrast with [43], our projective bundles parametrizes rank one quotients rather than sub-bundles. Let Then, f is a family of smooth degree 8 del Pezzo surfaces. We want to compute We compute the four monomials appearing in the above expression separately.
Before describing the other two terms we need to have a better understanding of E. The ideal J C of C corresponds to the graded ideal I of Sym We have: Wrapping up, we obtain Example 12.2 In this example we exhibit a family f : X → T of smooth degree 8 del Pezzo surfaces over a curve such that deg λ f > 0, or equivalently (−K X/T ) 3 < 0, but −K X/T is big. So, the statement (−K X/T ) 3 < 0 is a negativity condition independent of −K X/T being big or not. However, let us also note that there is one missing piece of our example: it is a family of non-K -semi-stable Fano varieties, although we suspect that a K -semi-stable one exists also.
Modify Example 12.1 replacing V with its dual; so we take V = O T (2) ⊕ O T (−1) ⊕ O T (−1) and we blow-up the curve defined by the quotient V → O P 1 (2). In this case, deg λ f = 12 > 0. However, −K X/T is still big. Indeed, write V = L ⊕ K ⊕ M, where L = O T (2) and K = M = O T (−1). Then for every integer m > 0: As I C is generated in Sym V by K ⊕ M, we obtain that To show that −K X/T is big, it is enough to prove that lim Equation (12.2.a) yields: So, we showed indeed that −K X/T is big, and we even computed that vol(−K X/T ) = 12 (a coincidence with the previous number 12 above).

Example 12.3
Here we give an example of a family f : X → T of smooth Del-Pezzo surfaces of degree 6 such that δ X t = 1 for all closed point t ∈ T , deg λ f = 0 but −K X/T not nef. This shows that the hypothesis δ > 1 in Theorem 1.20 is necessary. For this, we modify Example 12.1 in two respects: (a) We take V to be the dual vector bundle, that is, Instead of one curve, we blow up 3 curves. That is, we set X := Let E i , F i and W i (and for i = 1 also D 1 ) to be defined for each C i as E, F and W (and for i = 1 also D) was defined for C in Example 12.1. We do not define D i also as in Example 12.1 because for i = 2, 3, Note that the E i are disjoint, and hence any intersection product involving different E i is automatically 0. We write out below the computations where the result is different than in Example 12.1, where i = 2 or 3: Set E := E 1 + E 2 + E 3 . Then, we conclude that The fibres of f are smooth del Pezzo surfaces of degree 6, they are wellknown to be K-poly-stable (so, in particular, K-semi-stable), but they are not uniformly K-stable because they have a positive dimensional automorphism group. The delta invariant is thus equal to 1 (e.g., Corollary 4.9). Furthermore, for i = 2, 3: Hence, −K X/T | E i is not nef, and then also −K X/T is not nef. Example 12.4 In this example, for each choice of an integer d > 0 we exhibit families f : X → T of uniformly K-stable del Pezzo surfaces of degree 4 over a smooth projective curve. In this situation, Theorem 1.9.(a) tells us that deg λ f > 0, or equivalently (−K X/T ) 3 < 0. So, one would expect −K X/T to have only a few sections. Here, we show that both the expected and the unexpected behavior can happen. More precisely, | − K X/T | Q = ∅ for d > 3, and κ(−K X/T ) ≥ 1 for d = 1.
Let p 1 , . . . , p 4 be four points in P 2 in general position, and denote by L i j the line trough p i and p j . Let ι : T → P 2 be a degree d smooth curve in P 2 which avoids the four points. Let ∼ = T be the graph of ι in P 2 × T , and let T i be the curve { p i } × T in P 2 × T . We want to look at the blow-up π : Y → P 2 × T of and T i , for i = 1, . . . , 4. Denote by g : Y → T the natural projection.
The family g : Y → T is generically a family of degree 4 smooth del Pezzo surfaces of maximal variation. The only exception is at the points t ∈ T where ι(T ) intersects one of the lines L i j . In these cases, Y t = Bl p 1 , p 2 , p 3 , p 4 , p P 2 , where p lies on L i j . In particular, −K Y t is big and semi-ample, and there is a unique curve C for which C · −K Y t = 0: the proper transform of L i j . The anti-canonical model is the contraction of L i j to an A 1 singularity, so in particular it has canonical singularities.
Let f : X → T be the relative anti-canonical model of g (remark that R i g * (−m K Y /T ) = 0 for i > 0 and m big enough, by Kawamata-Viehweg vanishing theorem, so we do have base change). The family f satisfies the hypotheses of Theorem 1.9(a), so deg λ f > 0.
We show that, if d > 3, then | − K X/T | Q = ∅ and if d = 1, then κ(−K X/T ) ≥ 1. In either case the crucial remark is that H 0 (X, −m K X/T ) can be identified with the subspace ofH 0 (P 2 ×T, O P 2 ×T (3m)) ∼ = H 0 (P 2 , O P 2 (3m)) which vanish along ι(T ) and p i with multiplicity at least m. Hence: • If d > 3, then there are no such sections, as d is exactly the degree of ι(T ).
• If d = 1, then ι(T ) is a line L. So, | − K X/T | is the set of cubics C on P 2 such that C goes through p i and Supp C contains L. Hence, C = L + C , where C is a conic through p i . There is a one parameter family of such conics.
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Appendix A Computations concerning the definition of the CM line bundle
The following work is needed to prove the statements of Sect. 3. These are verifications of technical issues concerning the singular situation. We need the following lemmas as we work with singular varieties, and hence Riemann-Roch computations do not work directly. It turns out that if the spaces are normal then singularities do not mess up any of the terms involving any of the definitions of the CM line bundle. However, in the non-normal situation, which we do not deal with in the present article, Lemma A.2 seems to suggest that one has to face extra difficulties.
Lemma A.1 Consider the following situation: • f : X → T is a projective morphism to a normal quasi-projective variety (allowing T = Spec k), • M is an f -ample Q-divisor on X , • E is a coherent sheaf on X , and • r ≥ 0 is an integer such that dim Supp E t ≤ r for t ∈ T the generic point and dim Supp E t ≤ r + 1 for t ∈ T a codimension 1 point.
Then there are Q-divisors D i (resp. d i ∈ Q), determined uniquely up to Qlinear equivalence (resp. determined uniquely), such that for all q divisible enough, if dim T > 0, then Proof In the case of T = Spec k, h 0 (X, O X (q M) ⊗ E) equals the Hilbert polynomial for q divisible enough, and hence the statement follows. So, from now we assume that dim T > 0. Let s > 0 be an integer such that s M is relatively very ample. As the statement is for all q divisible enough, by replacing M with s M we may assume that M is relatively very ample and f * O X (M) is locally free, in which case we will exhibit Z-divisors D i . Furthermore, as the statement is about codimension 1 behavior over T , and T reg is a big open set of T , by replacing T with T reg we may assume that T is regular.
As M is relatively very ample, it induces an embedding ι : X → P := Proj T f * O X (M). Let π : P → T be the natural morphism. As P is regular, ι * E has a locally free resolution P • , which in particular is a perfect complex on P. Hence, for q divisible enough, the following holds (where, following [69], det is the alternating tensor product of the determinants of the elements of a locally free resolution, which exists as T is regular): Note that in the following lemma we do not assume any Q-Cartier hypothesis on K X/T . Still, our intersection in (13.2.a) is well defined as M is Q-Cartier.
Lemma A.2 Let f : X → T be a surjective morphism from a normal projective variety of dimension n + d to a smooth variety of dimension d ≥ 0 with n ≥ 1, and let M be a Q-Cartier f -ample divisor on X .
(a) If dim T > 0, then for all divisible enough integers q > 0, Proof As Grothendieck-Riemann-Roch works directly only for smooth X (or also on locally complete intersection singularities, which does not include klt singularities with Cartier index greater than 1), we need to compare X with a resolution. Let σ : Z → X be a resolution of singularities and set g := f • σ .
First, we claim that for all integers i > 0 and 1||q, in the respective cases: Indeed, fix an integer i > 0. There is a spectral sequence with E 2 -terms R p f * (O X (q M) ⊗ R r σ * O Z ) abutting to R i g * O Z (qσ * M) for i = p + r . As M is f -ample and q is divisible enough, this spectral sequence degenerates. Therefore, Then Lemma A.1 applied to E := R i σ * O Z concludes our claim, using that Supp R i σ * O Z is contained in the non-normal locus, which is at most n − 2 dimensional in the generic fiber and at most n − 1 dimensional over the fibers over codimension 1 points. Having shown our claim, in the dim T > 0 case the statement of the proposition is shown by the following computation, which holds for every q divisible enough (so q M is f -very ample, Cartier and without higher cohomologies on the fibers): The case of (b) is verbatim the same with s n , λ f,L and L CM, f,s L replaced by −s n+1 , λ f, and M n+1 , respectively.
Step 2: The proof assuming that T is smooth. Set M i := c 1 (M i ).

Lemma A.4
Let h : V → S be a flat n-relative dimensional morphism from a reduced projective scheme to a smooth projective curve, and let L be an h-very ample line bundle on V . Let π : Z → V be the normalization of V with g : Z → S being the induced morphism, and assume also that π * L is g-very ample. Then the n + 1-th (so highest) Knudsen-Mumford coefficients of L with respect to g (as in Notation 3.6) agrees with that of π * L.
Proof Consider the exact sequence on V given by the normalization: This yields a natural inclusion h * (L q ) → g * (π * L q ). Hence, it is enough to prove that for q divisible enough, deg h * (L q ⊗ E) = O(q n ), which is given by Lemma A.1 as dim Supp E ≤ n.