Projectivity of moduli and subadditvity of log-Kodaira dimension

# Projectivity of the moduli space of stable log-varieties and subadditvity of log-Kodaira dimension

Sándor J Kovács University of Washington, Department of Mathematics, Box 354350 Seattle, WA 98195-4350, USA  and  Zsolt Patakfalvi Department of Mathematics, Princeton University, Fine Hall, Washington Road, NJ 08544-1000, USA
###### Abstract.

We prove a strengthening of Kollár’s Ampleness Lemma and use it to prove that any proper coarse moduli space of stable log-varieties of general type is projective. We also prove subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.

## 1. Introduction

Since Mumford’s seminal work on the subject, , the moduli space of smooth projective curves of genus , has occupied a central place in algebraic geometry and the study of has yielded numerous applications. An important aspect of the applicability of the theory is that these moduli spaces are naturally contained as open sets in the moduli space of stable curves of genus and the fact that this later space admits a projective coarse moduli scheme.

Even more applications stem from the generalization of this moduli space, , the moduli space of -pointed smooth projective curves of genus and its projective compactification, , the moduli space of -pointed stable curves of genus .

It is no surprise that after the success of the moduli theory of curves huge efforts were devoted to develop a similar theory for higher dimensional varieties. However, the methods used in the curve case, most notably GIT, proved inadequate for the higher dimensional case. Gieseker [Gie77] proved that the moduli space of smooth projective surfaces of general type is quasi-projective, but the proof did not provide a modular projective compactification. In fact, Wang and Xu showed recently that such GIT compactification using asymptotic Chow stability is impossible [WX14]. The right definition of stable surfaces only emerged after the development of the minimal model program allowed bypassing the GIT approach [KSB88] and the existence and projectivity of the moduli space of stable surfaces and higher dimensional varieties have only been proved very recently as the combined result of the effort of several people over several years [KSB88, Kol90, Ale94, Vie95, HK04, AH11, Kol08, Kol13a, Kol13b, Fuj12, HMX14, Kol14].

Naturally, one would also like a higher dimensional analogue of -pointed curves and extend the existing results to that case [Ale96]. The obvious analogue of an -pointed smooth projective curve is a smooth projective log-variety, that is, a pair consisting of a smooth projective variety and a simple normal crossing divisor . For reasons originating in the minimal model theory of higher dimensional varieties, one would also like to allow some mild singularities of and and fractional coefficients in , but we will defer the discussion of the precise definition to a later point in the paper (see section 2). In any case, one should mention that the introduction of fractional coefficients for higher dimensional pairs led Hassett to go back to the case of -pointed curves and study a weighted version in [Has03]. These moduli spaces are more numerous and have greater flexibility than the traditional ones. In fact, they admit natural birational transformations and demonstrate the workings of the minimal model program in concrete highly non-trivial examples. Furthermore, the log canonical models of these moduli spaces of weighted stable curves may be considered to approximate the canonical model of [HH09, HH13].

It turns out that the theory of moduli of stable log-varieties, also known as moduli of semi-log canonical models or KSBA stable pairs, which may be regarded as the higher dimensional analogues of Hassett’s moduli spaces above, is still very much in the making. It is clear what a stable log-variety should be: the correct class (for surfaces) was identified in [KSB88] and further developed in [Ale96]. This notion, is easy to generalize to arbitrary dimension cf. [Kol13a]. On the other hand, at the time of the writing of this article it is not entirely obvious what the right definition of the corresponding moduli functor is over non reduced bases. For a discussion of this issue we refer to [Kol13a, §6]. A major difficulty is that in higher dimensions when the coefficients of are not all greater than a deformation of a log-variety cannot be simplified to studying deformations of the ambient variety and then deformations of the divisor . An example of this phenomenon, due to Hassett, is presented in subsection 1.2, where a family of stable log varieties is given such that does not form a flat family of pure codimension one subvarieties. In fact, the flat limit acquires an embedded point, or equivalently, the scheme theoretic restriction of onto a fiber is not equal to the divisorial restriction. Therefore, in the moduli functor of stable log-varieties one should allow both deformations that acquire and also ones that do not acquire embedded points on the boundary divisors. This is easy to phrase over nice (e.g., normal) bases see section 5 for details. However, at this point it is not completely clarified how it should be presented in more intricate cases, such as for instance over a non-reduced base. Loosely speaking the infinitesimal structure of the moduli space is not determined yet (see subsection 5.1 for a discussion on this), although there are also issues about the implementation of labels or markings on the components of the boundary divisor (cf. section 6).

By the above reasons, several functors have been suggested, but none of them yet emerged as the obvious “best”. However, our results apply to any moduli functor for which the objects are the stable log-varieties (see section 5 for the precise condition on the functors). In particular, our results apply to any moduli space that is sometimes called a KSBA compactification of the moduli space of log-canonical models.

Our main result is the following. Throughout the article we are working over an algebraically closed base field of characteristic zero. {theorem}[=section 6] Any algebraic space that is the coarse moduli space of a moduli functor of stable log-varieties with fixed volume, dimension and coefficient set (as defined in section 5) is a projective variety over .

For auxiliary use, in subsection 5.1 we also present one particular functor as above, based on a functor suggested by Kollár [Kol13a, §6]. In particular, the above result is not vacuous.

As mentioned Mumford’s GIT method used in the case of moduli of stable curves does not work in higher dimensions and so we study the question of projectivity in a different manner. The properness of any algebraic space as in section 1 is shown in [Kol14]. For the precise statement see section 5. Hence, to prove projectivity over one only has to exhibit an ample line bundle on any such algebraic space. Variants of this approach have been already used in [Knu83, Kol90, Has03]. Generalizing Kollár’s method to our setting [Kol90], we use the polarizing line bundle , where is a stable family and is a divisible enough integer. Following Kollár’s idea and using the Nakai-Moishezon criterion it is enough to prove that this line bundle is big for a maximal variation family over a normal base. However, Kollár’s Ampleness Lemma [Kol90, 3.9,3.13] is unfortunately not strong enough for our purposes and hence we prove a stronger version in section 4. There, we also manage to drop an inconvenient condition on the stabilizers from [Kol90, 3.9,3.13], which is not necessary for the current application, but we hope will be useful in the future. Applying section 4 and some other arguments outlined in subsection 1.1 we prove that the above line bundle is big in section 6.

A side benefit of this approach is that proving a positivity property of opens the doorq to other applications. For example, a related problem in the classification theory of algebraic varieties is the subadditivity of log-Kodaira dimension. We prove this assuming the general fiber is of log general type in section 8. This generalizes to the logarithmic case the celebrated results on the subadditivity of Kodaira dimension [Kaw81, Kaw85, Vie83a, Vie83b, Kol87], also known as Iitaka’s conjecture and its strengthening by Viehweg . For section 8 we refer to section 8, here we only state two corollaries that need less preparation.

{theorem}

(= section 8 and section 8)

1. If is a surjective map of log-smooth projective pairs with coefficients at most , such that and is big, where is the generic point of , then

 κ(KX+D)≥κ(KXη+Dη)+κ(KY+E).
2. Let be a dominant map of (not necessarily proper) algebraic varieties such that the generic fiber has maximal Kodaira dimension. Then

 κ(X)≥κ(Xη)+κ(Y).

In the logarithmic case Fujino obtained results similar to the above in the case of maximal Kodaira dimensional base [Fuj14a, Thm 1.7] and relative one dimensional families [Fuj15]. Another related result of Fujino is subadditivity of the numerical log-Kodaira dimension [Fuj14b]. A version of the latter, under some additional assumptions, was also proved by Nakayama [Nak04, V.4.1]. The numerical log-Kodaira dimension is expected to be equal to the usual log-Kodaira dimension by the Abundance Conjecture. However, the latter is usually considered the most difficult open problem in birational geometry currently. Our proof does not use either the Abundance Conjecture or the notion of numerical log-Kodaira dimension.

Further note that our proof of section 1 is primarily algebraic. That is, we obtain our positivity results, from which section 1 is deduced, algebraically, starting from the semi-positivity results of Fujino [Fuj12, Fuj14a]. Hence, our approach has a good chance to be portable to positive characteristic when the appropriate semi-positivity results (and other ingredients such as the mmp) become available in that setting. See [Pat12b] for the currently available semi-positivity results in positive characteristic, and [CZ13, Pat13] for results on subadditivty of Kodaira-dimension.

section 1 is based on the following theorem stating that the sheaves have more positivity properties than just that their determinants are ample. This is a generalization of [Kol87] and [EV90, Thm 3.1] to the logarithmic case.

{theorem}

(=section 7) If is a family of stable log-varieties of maximal variation over a normal, projective variety with klt general fiber, then is big for every divisible enough integer .

Note that section 1 fails without the klt assumption. Also, section 1 allows for numerous applications, such as, the already mentioned section 1, as well as upcoming work in progress on a log-version of [Abr97] in [AT15] and on the ampleness of the CM line bundle on the moduli space of stable varieties in [PX15]. We also state section 1 and our other positivity results over almost projective bases in section 9, that is, over bases that are big open sets in projective varieties. We hope this will be helpful for some applications.

### 1.1. Outline of the proof

As mentioned above, using the Nakai-Moishezon criterion for ampleness, section 1 reduces to the following statement (= section 6): given a family of stable log-varieties with maximal variation over a smooth, projective variety, is big for every divisible enough integer . This follows relatively easily from the bigness of . To be precise it also follows from the bigness of the log canonical divisor of some large enough fiber power for some integer (see section 2 and the proof of section 6). In fact, one cannot expect to do better for higher dimensional bases, see section 6 for details. Here we review the proof of the bigness of these relative canonical divisors, going from the simpler cases to the harder ones.

#### 1.1.1. The case of dimY=1 and dimX=2.

In this situation, roughly speaking, we have a family of weighted stable curves as defined by Hassett [Has03]. The only difference is that in our notion of a family of stable varieties there is no marking (that is, the points are not ordered). This means that the marked points are allowed to form not only sections but multisections as well. However, over a finite cover of these multisections become unions of sections, and hence we may indeed assume that we have a family of weighted stable curves. Denote by the sections given by the marking and let be the images of these sections. Hassett proved projectivity [Has03, Thm 2.1, Prop 3.9] by showing that the following line bundle is ample:

 (.1) detf∗OX(r(KX/Y+D))⊗(m⨂i=1s∗iOX(r(KX/Y+D))).

Unfortunately, this approach does not work for higher dimensional fibers, because according to the example of subsection 1.2, the sheaves corresponding to which is the same as are not functorial in higher dimensions. In fact, the function jumps down in the limit in the case of example of subsection 1.2, which means that there is no possibility to collect the corresponding space of sections on the fibers into a pushforward sheaf. Note that here it is important that means the divisorial restriction of onto . Indeed, with the scheme theoretic restriction there would be no jumping down, since is flat as a scheme over . However, the scheme theoretic restriction of onto contains an embedded point and therefore the space of sections on the divisorial restriction is one less dimensional than on the scheme theoretic restriction.

So, the idea is to try to prove the ampleness of in the setup of the previous paragraph, hoping that that argument would generalize to higher dimensions. Assume that is not ample. Then by the ampleness of Equation .1, for some , must be ample. Therefore, for this value of , . Furthermore, by decreasing the coefficients slightly, the family is still a family of weighted stable curves. Hence is nef for every (see section 6, although this has been known by other methods for curves). Putting these two facts together yields that

 (KX/Y+D)2=(KX/Y+D)⋅(KX/Y+D−εDi)≥0, because KX/Y+D and KX/Y+D−εDi are nef+(KX/Y+D)⋅εDi>0>0.

This proves the bigness of , and the argument indeed generalizes to higher dimensions as explained below.

#### 1.1.2. The case of dimY=1 and arbitrary dimX.

Let be an arbitrary family of non-isotrivial stable log-varieties over a smooth projective curve. Let () be the union of the divisors (with reduced structure) of the same coefficient (cf. section 6). The argument in the previous case suggests that the key is to obtain an inequality of the form

 (.2) ((KX/Y+D)|Di)dimDi>0.

Note that it is considerably harder to reach the same conclusion from this inequality, than in the previous case, because the are not necessarily -factorial and then might not be a stable family. To remedy this issue we pass to a -factorial dlt-blowup. For details see section 6.

Let us now turn to how one might obtain Equation .2. First, we prove using our generalization (see section 4) of the Ampleness Lemma a higher dimensional analogue of Equation .1 in section 6, namely, that the following line bundle is ample:

 (.3) detf∗OX(r(KX/Y+D))⊗(m⨂i=1det(f|Di)∗ODi(r(KX/Y+D)|Di)).

The main difference compared to Equation .1 is that is no longer an isomorphism between and as it was in the previous case where the were sections. In fact, has positive dimensional fibers and hence is a vector bundle of higher rank. As before, if is not ample, then for some , has to be. However, since is higher rank now, it is not as easy to obtain intersection theoretic information as earlier.

As a result one has to utilize a classic trick of Viehweg which leads to working with fibered powers. Viehweg’s trick is using the fact that there is an inclusion

 (.4)

where , and where the latter sheaf can be identified with a pushforward from the fiber product space (see section 2). This way one obtains that

 ((KX(d)/Y+DX(d))∣∣D(d)i)dimD(d)i>0,

from which it is an easy computation to prove Equation .2

#### 1.1.3. The case of both dimY and dimX arbitrary.

We only mention briefly what goes wrong here compared to the previous case, and what the solution is. The argument is very similar to the previous case until we show that Equation .3 is big. However, it is no longer true that if is not big, then one of the is big. So, the solution is to treat all the sheaves at once via an embedding as in Equation .4 of the whole sheaf from Equation .3 into a tensor-product sheaf that can be identified with a pushforward from an appropriate fiber product (see Equation .15). The downside of this approach is that one then has to work on for some big , but we still obtain an equation of the type Equation .2, although with replaced with a somewhat cumbersome subvariety of fiber product type.

After that an enhanced version of the previous arguments yields that is big on at least one component, which is enough for our purposes. In fact, in this case we cannot expect that would be big on any particular component, cf. section 6. However, the bigness of on a component already implies the bigness of (see section 6). This argument is worked out in section 6.

#### 1.1.4. Subadditivity of log-Kodaira dimension

First we prove section 1 in section 7 using ideas originating in the works of Viehweg. This implies that although in section 6 we were not able to prove the bigness of (only of ), it actually does hold for stable families of maximal variation with klt general fibers (cf. section 7). Then with a comparison process (see the proof of section 8) of an arbitrary log-fiber space and of the image in moduli of the log-canonical model of its generic fiber, we are able to obtain enough positivity of to deduce subadditivity of log-Kodaira dimension if the log-canonical divisor of the general fiber is big.

### 1.2. An important example

The following example is due to Hassett (cf. [Kol13a, Example 42]), and has been referenced at a couple of places in the introduction.

Let be the cone over with polarization and let be the conic divisor , where is the projection to the second factor, and and are general points. Let be a cone over a hyperplane section of with the given polarization, and a general hyperplane section of (which is isomorphic to ). Note that since , is a cone over a rational normal curve of degree . Let be the pencil of and . It is naturally a subscheme of the blowup of along . Furthermore, the pullback of to induces a divisor on , such that

1. its reduced fiber over is a cone over the intersection of with , that is, over distinct points on with coefficients , and

2. its fiber over is two members of one of the rulings of with coefficients . In the limit both of these lines degenerate to a singular conic, and they are glued together at their singular points.

In case the reader is wondering how this is relevant to stable log-varieties of general type, we note that this is actually a local model of a degeneration of stable log-varieties, but one can globalize it by taking a cyclic cover branched over a large enough degree general hyperplane section of . For us only the local behaviour matters, so we will stick to the above setup. Note that since , the above described reduced structure cannot agree with the scheme theoretic restriction of over , since then would hold. Therefore is non-reduced at the cone point. Furthermore, note that the log canonical divisor of is the cone over a divisor corresponding to . In particular, this log canonical class is -Cartier, and hence does yield a local model of a degeneration of stable log-varieties.

### 1.3. Organization

We introduce the basic notions on general and on almost proper varieties in section 2 and section 3. In section 4 we state our version of the Ampleness Lemma. In section 5 we define moduli functors of stable log-varieties and we also give an example of a concrete moduli functor for auxiliary use. section 6 contains the proof of section 1 as well as of the necessary positivity of . section 7 is devoted to the proof of section 1. section 8 contains the statements and the proofs of the subadditivity statements including section 1. Finally, in section 9 we shortly deduce almost projective base versions of the previously proven positivity statements.

#### Acknowledgement

The authors are thankful to János Kollár for many insightful conversations on the topic; to Maksym Fedorchuk for the detailed answers on their questions about the curve case; to James McKernan and Chenyang Xu for the information on the results in the article [HMX14].

## 2. Basic tools and definitions

We will be working over an algebraically closed base field characteristic zero in the entire article. In this section we give those definitions and auxiliary statements that are used in multiple sections of the article. Most importantly we define stable log-varieties and their families here.

{definition}

A variety will mean a reduced but possibly reducible separated scheme of finite type over . A vector bundle on a variety in this article will mean a locally free sheaf. Its dual is denoted by .

{remark}

It will always be assumed that the support of a divisor does not contain any irreducible component of the conductor subscheme. Obviously this is only relevant on non-normal schemes. The theory of Weil, Cartier, and -Cartier divisors work essentially the same on demi-normal schemes, i.e., on schemes that satisfy Serre’s condition and are semi-normal and Gorenstein in codimension . For more details on demi-normal schemes and their properties, including the definition and basic properties of divisors on demi-normal schemes see [Kol13b, §5.1].

{definition}

Let be a scheme. A big open subset of is an open subset such that . If is , e.g., if it is normal, then this is equivalent to the condition that .

{definition}

The dual of a coherent sheaf on a scheme will be denoted by and the sheaf is called the reflexive hull of . If the natural map is an isomorphism, then is called reflexive. For the basic properties of reflexive sheaves see [Har80, §1].

Let be an scheme and a coherent sheaf on . Then the reflexive powers of are the reflexive hulls of tensor powers of and are denoted the following way:

 F[m]:=(F⊗m)∗∗

Obviously, is reflexive if and only if . Let be coherent sheaf on . Then the reflexive product of and (resp. reflexive symmetric power of ) is the reflexive hull of their tensor product (resp. of the symmetric power of ) and is denoted the following way:

 F[⊗]G:=(F⊗G)∗∗Sym[a](F):=(Syma(F))∗∗
{notation}

Let and be morphisms of schemes. Then the base change to will be denoted by

 fZ:XZ→Z,

where and . If for a point , then we will use and to denote and .

{lemma}

Let and be surjective morphisms such that is normal and let and be line bundles on and respectively. Assume that there is a big open set of over which and are flat and and are locally free. Then

Furthermore, if and are flat and and are locally free over the entire , then the above isomorphism is true without taking reflexive hulls.

###### Proof.

Since the statement is about reflexive sheaves, we may freely pass to big open sets. In particular, we may assume that and are flat and and are locally free. Then

{notation}

Let be a flat equidimensional morphism of demi-normal schemes, and a morphism between normal varieties. Then for a -divisor on that avoids the generic and codimension singular points of the fibers of , we will denote by the divisorial pull-back of to , which is defined as follows: As avoids the singular codimension points of the fibers, there is a big open set such that is -Cartier. Clearly, is also a big open set in and we define to be the unique divisor on whose restriciton to is .

{remark}

Note that this construction agrees with the usual pullback if itself is -Cartier, because the two divisors agree on .

Also note that is not necessarily the (scheme theoretic) base change of as a subscheme of . In particular, for a point , is not necessarily equal to the scheme theoretic fiber of over . The latter may contain smaller dimensional embedded components, but we restrict our attention to the divisorial part of this scheme theoretic fiber. This issue has already come up multiple times in section 1, in particular in the example of subsection 1.2.

Finally, note that if is Cartier, then using this definition the line bundle is compatible with base-change, that is, for a morphism ,

 (OX(q(KX/Y+D)))Z≃OZ(q(KXZ/Z+DZ)).

To see this, recall that this holds over by definition and both sheaves are reflexive on . (See section 2 for the precise definition of .)

{definition}

A pair consist of an equidimensional demi-normal variety and an effective -divisor . A stable log-variety is a pair such that

1. is proper,

2. has slc singularities, and

3. the -Cartier -divisor is ample.

For the definition of slc singularities the reader is referred to [Kol13b, 5.10]

{definition}

If is either

1. a flat projective family of equidimensional demi-normal varieties, or

2. a surjective morphism between normal projective varieties,

then is defined to be . In particular, if is Gorenstein (e.g., is smooth), then . In any case, is a reflexive sheaf (c.f., [PS14, Lemma 4.9]) of rank . Furthermore, if either in the first case is also normal or in the second case is smooth, then is trivial at the codimension one points, and hence it corresponds to a Weil divisor that avoids the singular codimension one points [Kol13b, 5.6]. This divisor can be obtained by fixing a big open set over which is a line bundle, and hence over which it corresponds to a Cartier divisor, and then extending this Cartier divisor to the unique Weil-divisor extension on . Note that in the first case can be chosen to be the relative Gorenstein locus of , and in the second case the regular locus of . Furthermore, in the first case, we have for any base-change from a normal variety (here restriction is taken in the sense of section 2).

{definition}

A family of stable log-varieties, over a normal variety consists of a pair and a flat proper surjective morphism such that

1. avoids the generic and codimension singular points of every fiber,

2. is -Cartier, and

3. is a connected stable log-variety for all .

{notation}

For a morphism of schemes and , define

 X(m)Y:= m1\scriptsize YX =X×YX×Y⋯×YXm times,

and let be the induced natural map. For a sheaf of -modules define

 F(m)Y:=m⨂i=1p∗iF,

where is the -th projection . Similarly, if is flat, equidimensional with demi-normal fibers, then for a divisor on define

 ΓX(m)Y:=m∑i=1p∗iΓ,

a divisor on .

Finally, for a subscheme , is naturally a subscheme of . Notice however that if and has positive codimension in , then is never a divisor in . In particular, if is normal, is flat, equidimensional and has demi-normal fibers, and is an effective divisor that does not contain any generic or singular codimension points of the fibers of , then

 (.5) (Γ(m)Y)red=(m⋂i=1p∗iΓ)red.

Notice the difference between and . The former corresponds to taking the box-power of a divisor as a sheaf, while the latter to taking fiber power as a subscheme. In particular,

 OX(m)Y(ΓX(m)Y)≃(OX(Γ))(m)Y,

while is not even a divisor if .

In most cases, we omit from the notation. I.e., we use , , , and instead of , , , and , respectively.

## 3. Almost proper varieties and big line bundles

{definition}

An almost proper variety is a variety that admits an embedding as a big open set into a proper variety . If is almost proper, then a proper closure will mean a proper variety with such an embedding. The proper closure is not unique, but also, obviously, an almost proper variety is not necessarily a big open set for an arbitrary embedding into a proper (or other) variety. An almost proper variety is called almost projective when it has a proper closure which is projective. Such a proper closure will be called a projective closure.

{lemma}

Let be an almost projective variety of dimension and a Cartier divisor on . Then there exists a constant such that for all

 h0(Y,OY(mB))≤c⋅mn
###### Proof.

Let be a projective closure of and set . Let be a very ample invertible sheaf on such that where is the dual of . It follows that there exists an embedding and hence for all another embedding . Pushing this forward to one obtains that . Note that the last isomorphism follows by the condition of being almost projective/proper, that is, because . Finally this implies that

 h0(Y,OY(mB))=h0(¯¯¯¯Y,Bm)≤h0(¯¯¯¯Y,Hm)∼c⋅mn,

where the last inequality follows from [Har77, I.7.5]. ∎

{definition}

Let be an almost proper variety of dimension . A Cartier divisor on is called big if for some constant and integer. A line bundle is called big if the associated Cartier divisor is big.

{lemma}

Let be an almost proper variety of dimension and a projective closure of . Let be a Cartier divisor on and denote its restriction to by . Then is big if and only if is big.

###### Proof.

Clear from the definition and the fact that for every . ∎

{remark}

Note that it is generally not assumed that extends to as a Cartier divisor.

{lemma}

Let be an almost projective variety of dimension and a Cartier divisor on . Then the following are equivalent:

1. where is ample and is effective for some ,

2. the rational map associated to the linear system is birational for some ,

3. the projective closure of the image has dimension for some , and

4. is big.

###### Proof.

The proof included in [KM98, 2.60] works almost verbatim. We include it for the benefit of the reader since we are applying it in a somewhat unusual setup.

Clearly, the implications are obvious. To prove , let . By assumption , so by [Har77, I.7.5] the Hilbert polynomial of is By definition of the associated rational map induces an injection , which proves .

To prove , let be a Cartier divisor on and let be a projective closure of . Further let be a general member of a very ample linear system on . Then is an almost projective variety by [Fle77, 5.2]. It follows by Lemma 3 that , which, combined with the exact sequence

 0→H0(Y,OY(mB−A))→H0(Y,OY(mB))→H0(A,OA(mB|A)),

shows that if is big, then for which implies as desired. ∎

The notion of weak-positivity used in this article is somewhat weaker than that of [Vie95]. The main difference is that we do not require being global generated on a fixed open set for every in the next definition. This is a minor technical issue and proofs of the basic properties works just as for the definitions of [Vie95], after disregarding the fixed open set. The reason why this weaker form is enough for us is that we use it only as a tool to prove bigness, where there is no difference between our definition and that of [Vie95].

{definition}

Let be a normal, almost projective variety and an ample line bundle on .

1. A coherent sheaf on is weakly-positive, if for every integer there is an integer , such that is generically globally generated. Note that this does not depend on the choice of [Vie95, Lem 2.14.a].

2. A coherent sheaf on is big if there is an integer such that is generically globally generated. This definition also does not depend on the choice of by a similar argument as for the previous point. Further, this definition is compatible with the above definition of bigness for divisors and the correspondence between divisors and rank one reflexive sheaves.

{lemma}

Let be a normal, almost projective variety, a weakly-positive and a big coherent sheaf. Then

1. , , , are weakly-positive,

2. generically surjective images of are weakly-positive, and those of are big,

3. if is an ample line bundle, then is big, and

4. if is of rank , then is big.

###### Proof.

Let us fix an ample line bundle . item 1 follows verbatim from [Vie95, 2.16(b) and 2.20], and item 2 follows immediately from the definition. Indeed, given generically surjective morphisms and , there are generically surjective morphisms and proving the required generic global generation.

To prove item 3, take an , such that is effective and is very ample for . Then for a such that is globally generated, the embedding

 Sym[3b](F)⊗Ab↪Sym[3b](F)⊗A3b−a≃Sym[3b](F⊗A)⊗A−a↪Sym[3b](F⊗A)⊗H−1

is generically surjective which implies the statement.

To prove item 4 take an , such that is generically globally generated. This corresponds to a generically surjective embedding . According to item 1 and item 3, is big. Hence, by item 2, is also big. Therefore, for some , is generically globally generated and then the surjection concludes the proof. ∎

## 4. Ampleness Lemma

{theorem}

Let be a weakly-positive vector bundle of rank on a normal almost projective variety with a reductive structure group the closure of the image of which in the projectivization of the space of matrices is normal and let be vector bundles of rank on admitting generically surjective homomorphisms for . Let be the induced classifying map of sets. Assume that this map has finite fibers on a dense open set of . Then is big.

{remark}

One way to define the above classifying map is to choose a basis on every fiber of over every closed point up to the action of . For this it is enough to fix a basis on one fiber of over a closed point, and transport it around using the -structure. In fact, a little less is enough. Given a basis, multiplying every basis vector by an element of does not change the corresponding rank quotient space, and hence the classifying map, so we only need to fix a basis up to scaling by an element of . To make it easier to talk about these in the sequel we will call a basis which is determined up to scaling by an element of a homogenous basis.

{remark}

The normality assumption in section 4 is satisfied if with and acting via the representation . Indeed, in this case the closure of the image of in agrees with the image of the embedding . In particular, it is isomorphic to , which is smooth.

For more results regarding when this normality assumption is satisfied in more general situations see [Tim03, DC04, BGMR11] and other references in those papers.

{remark}

section 4 is a direct generalization of the core statement [Kol90, 3.13] of Kollár’s Ampleness Lemma [Kol90, 3.9]. This statement is more general in several ways:

• The finiteness assumption on the classifying map is weaker (no assumption on the stabilizers).

• The ambient variety is only assumed to be almost projective instead of projective.

Our proof is based on Kollár’s original idea with some modifications to allow for weakening the finiteness assumptions.

Note that if is projective and is nef on , then it is also weakly positive [Vie95, Prop. 2.9.e].

We will start by making a number of reduction steps to simplify the statement. The goal of this reduction is to show that it is enough to prove the following theorem which contains the essential statement.

{theorem}

Let be a weakly-positive vector bundle of rank on a normal almost projective variety with a reductive structure group the closure of the image of which in the projectivization of the space of matrices is normal and let be a surjective morphism onto a vector bundle of rank . Let be the induced classifying map. If this map has finite fibers on a dense open set of , then the line bundle is big.

{lemma}
###### Proof.

Step 1. We may assume that the are surjective. Let . Then there exists a big open subset such that is locally free of rank . If is big, then so is and hence so is . Therefore we may replace with and with .

Step 2. It is enough to prove the statement for one quotient bundle. Indeed, let with the diagonal -action, , and the induced morphism. If all the are surjective, then so is .

Furthermore, there is a natural injective -invariant morphism

Since the -action on is the restiction of the -action on via this embedding it follows that the induced map on the quotients remain injective:

It follows that the classifying map of also has finite fibers and then the statement follows because . ∎

{lemma}

If is a -invariant sub-vector bundle of the -vector bundle on a normal almost projective variety , and is weakly positive, then so is .

###### Proof.

corresponds to a subrepresentation of , and by the characteristic zero and reductivity assumptions it follows that is a direct summand of , so is also weakly positive. ∎

{remark}

The above lemma, which is used in the last paragraph of the proof, is the only place where the characteristic zero assumption is used in the proof of section 4. In particular, the statement holds in positive characteristic for a given if the -subbundles of are weakly-positive whenever is. According to [Kol90, Prop 3.5] this holds for example if is projective and is nef satisfying the assumption of [Kol90, Prop 3.6]. The latter is satisfied for example if for a nef vector bundle of rank and .

###### Proof of section 4.

We start with the same setup as in [Kol90, 3.13]. Let , which can be viewed as the space of matrices with columns in , and consider the universal basis map

 ς:w⨁j=1OP(−1)→π∗W,

formally given via the identification by the identity sections of the different summands of the form . Informally, the closed points of over can be thought of as -tuples and hence a dense open subset of corresponds to the choice of a basis of up to scaling by an element of , i.e., to a homogenous basis. Similarly, the map gives local sections of which over take the values , up to scaling by an element of where this scaling corresponds to the transition functions of .

As explained in section 4, to define the classifying map we need to fix a homogenous basis of a fiber over a fixed closed point. Let us fix such a point and a homogenous basis on and keep these fixed throughout the proof. This choice yields an identification of with . Notice that the dense open set of corresponding to the different choices of a homogenous basis of is identified with the image of in and the point in representing the fixed homogenous basis above is identified with the image of the identity matrix in . Now we want to restrict to a orbit inside all the choices of homogenous bases. Let denote the closure of the image of in . Via the identification of and , corresponds to a -invariant closed subscheme of , which carried around by the -action defines a -invariant closed subscheme . Note that since is assumed to be normal, so is by [EGA-IV, II 6.5.4]. To simplify notation let us denote the restriction also by . Restricting the universal basis map to and twisting by gives

Let be the divisor where this map is not surjective, i.e., those points that correspond to non-invertible matrices via the above identification of and . By construction, gives a trivialization of over . It is important to note the following fact about this trivialization: let be the closed point that via the above identification of and corresponds to the image of the identity matrix in . Then the trivialization of given by gives a basis on which is compatible with our fixed homogenous basis on . Furthermore, for any the basis on given by corresponds to the fixed homogenous basis of twisted by the matrix (which is only given up to scaling by an element of ) corresponding to the point via the identification of and . Note that as is reductive, it is closed in and hence is transitive on . It follows that then the choices of homogenous bases of given by on for form a -orbit, and this orbit may be identified with . Transporting this identification around using the -action we obtain: For every ,

 (.6) (P∖Υ)y may be identified with the G(k)-orbit of homogenous bases of Wy containing the homogenous basis obtained from the fixed homogenous basis of Wy0 via the G-structure.

Next consider the composition of and :

which is surjective on . Taking wedge products yields

which is still surjective outside and hence gives a morphism

 ν:P∖Υ→Gr(w,q)⊆P(q⋀(k⊕w))=:PGrPl\"{u}cker embedding,

such that

• according to Equation .6, on the -points is a lift of the classifying map , where is the Grassmannian of rank quotients of a rank vectorspace, and

• , where is the restriction of