label = , label-format = ,) label-width = 12pt MnLargeSymbols’164
MnLargeSymbols’171 section0em1.7em

#  On Harder-Narasimhan slopes of direct images

Siarhei Finski

Abstract. For a polarized family of complex projective manifolds, we study the
asymptotic distribution of Harder-Narasimhan slopes of direct image sheaves
associated with high tensor powers of the polarization. We establish a theorem
of Mehta-Ramanathan type, showing that this asymptotic distribution can be
recovered from the analogous asymptotic distributions associated with base
changes of the family over generic curves.

Table of contents

######

  1. 1 Introduction
  2. 2 Submultiplicative nature of Harder-Narasimhan filtrations
  3. 3 Relative slopes on degenerating families of varieties
  4. 4 Generic base changes and asymptotic distribution of slopes
  5. 5 Numerical expressions for asymptotic slopes

##  1 Introduction

Consider a surjective flat holomorphic map
$\pi\mathrel{\mathop{\ordinarycolon}}X\to B$ between complex projective
manifolds $X$ and $B$ of dimensions $n+m$ and $m$ respectively,
$n,m\in\mathbb{N}$. Let $L$ be a holomorphic line bundle over $X$, which is
relatively ample with respect to $\pi$. We fix a multipolarization
$[\omega_{B}]$, which is a collection of Kähler classes $[\omega_{B,i}]\in
H^{1,1}(B)$, $i=1,\ldots,m-1$. The goal of this paper is to study the Harder-
Narasimhan $[\omega_{B}]$-slopes of direct image sheaves
$E_{k}\mathrel{\mathop{\ordinarycolon}}=R^{0}\pi_{*}L^{k}$ of $L^{k}$ for big
$k\in\mathbb{N}$.

More precisely, recall that a slope (or $[\omega_{B}]$-slope) of a coherent
sheaf $\mathscr{E}$ over $B$ is defined as
$\mu(\mathscr{E})\mathrel{\mathop{\ordinarycolon}}=\deg(\mathscr{E})/{\rm{rk}}(\mathscr{E})$,
where the degree, $\deg(\mathscr{E})$, is given by
$\deg(\mathscr{E})\mathrel{\mathop{\ordinarycolon}}=\int_{B}c_{1}(\det(\mathscr{E}))\cdot[\omega_{B,1}]\cdots[\omega_{B,m-1}]$,
where $\det\mathscr{E}$ is Knudsen-Mumford determinant of $\mathscr{E}$, see
[29]. A torsion-free coherent sheaf $\mathscr{E}$ is called semistable (or
$[\omega_{B}]$-semistable) if for every coherent subsheaf $\mathcal{F}$ of
$\mathscr{E}$, verifying ${\rm{rk}}(\mathcal{F})>0$, we have
$\mu(\mathcal{F})\leq\mu(\mathscr{E})$. When the multipolarization is
symmetric, i.e. $[\omega_{B,1}]=\cdots=[\omega_{B,m-1}]$, we recover the usual
notion of slopes and semistability, as in [30].

Recall that any torsion-free coherent sheaf $\mathscr{E}$ on
$(B,[\omega_{B}])$ admits a unique filtration by subsheaves $\mathscr{E}_{i}$,
$i=1,\ldots,s$, also called Harder-Narasimhan filtration:

$\mathscr{E}=\mathscr{E}_{s}\supset\mathscr{E}_{s-1}\supset\cdots\supset\mathscr{E}_{1}\supset\mathscr{E}_{0}=\\{0\\},$
(1.1)

such that for any $1\leq i\leq s-1$, the quotient sheaf
$\mathscr{E}_{i}/\mathscr{E}_{i-1}$ is the maximal semistable (torsion-free)
subsheaf of $\mathscr{E}/\mathscr{E}_{i-1}$, i.e. for any subsheaf of
$\mathcal{F}$ of a (torsion-free) sheaf $\mathscr{E}/\mathscr{E}_{i-1}$, we
have $\mu(\mathcal{F})\leq\mu(\mathscr{E}_{i}/\mathscr{E}_{i-1})$ and
${\rm{rk}}(\mathcal{F})\leq{\rm{rk}}(\mathscr{E}_{i}/\mathscr{E}_{i-1})$ if
$\mu(\mathcal{F})=\mu(\mathscr{E}_{i}/\mathscr{E}_{i-1})$. For the proof, see
[28, Theorem 1.6.7] for symmetric multipolarization and [21, Corollary 2.27]
for the general case.

We define the Harder-Narasimhan slopes,
$\mu_{1},\ldots,\mu_{{\rm{rk}}(\mathscr{E})}$ of $\mathscr{E}$, so that
$\mu(\mathscr{E}_{i}/\mathscr{E}_{i-1})$ appears among
$\mu_{1},\ldots,\mu_{{\rm{rk}}(\mathscr{E})}$ exactly
${\rm{rk}}(\mathscr{E}_{i}/\mathscr{E}_{i-1})$ times, and the sequence
$\mu_{1},\ldots,\mu_{{\rm{rk}}(\mathscr{E})}$ is non-increasing. We call
$\mu_{\min}\mathrel{\mathop{\ordinarycolon}}=\mu_{{\rm{rk}}(\mathscr{E})}$ and
$\mu_{\max}\mathrel{\mathop{\ordinarycolon}}=\mu_{1}$, the minimal and the
maximal slopes respectively.

By the flatness of $\pi$, for $k\in\mathbb{N}$ big enough (which we assume
implicitly from now on), the coherent sheaves $E_{k}$ are locally free, cf.
[24, Theorem III.9.9]. We denote
$N_{k}\mathrel{\mathop{\ordinarycolon}}={\rm{rk}}(E_{k})$, and let
$\mu_{1}^{k},\ldots,\mu_{N_{k}}^{k}$ be the Harder-Narasimhan slopes of
$E_{k}$. Let $\mu_{\min}^{k},\mu_{\max}^{k}$ be the minimal and the maximal
slopes. We define the probability measure $\eta_{k}^{HN}$ on as follows

$\eta_{k}^{HN}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\delta\Big{[}\frac{\mu_{i}^{k}}{k}\Big{]},$
(1.2)

where $\delta[x]$ is the Dirac mass at $x\in\real$. In Section 2, we establish
the following result, serving as a starting point of the current work.

######  Theorem 1.1.

The sequence of measures $\eta_{k}^{HN}$ converges weakly, as $k\to\infty$, to
a probability measure $\eta^{HN}$ on , which is absolutely continuous with
respect to the Lebesgue measure, except perhaps for a point mass at
$\operatorname*{ess\,sup}\eta^{HN}$. Also, the limits below exist and relate
with $\eta^{HN}$ as follows

$\eta_{\min}^{HN}\mathrel{\mathop{\ordinarycolon}}=\lim_{k\to\infty}\frac{\mu_{\min}^{k}}{k}\leq\operatorname*{ess\,inf}\eta^{HN},\qquad\eta_{\max}^{HN}\mathrel{\mathop{\ordinarycolon}}=\lim_{k\to\infty}\frac{\mu_{\max}^{k}}{k}=\operatorname*{ess\,sup}\eta^{HN}.$
(1.3)

When $\dim B=1$, the convergence part of Theorem 1.1 was established by Chen
[7]. For this, he verified that Harder-Narasimhan filtrations of $E_{k}$ are
submultiplicative, see (2.3), and $\mu_{\max}^{k}$ grows at most linearly in
$k$. Theorem 1.1 then follows from the general theory of bounded
submultiplicative filtrations developed by Chen [7] and Boucksom-Chen [4], cf.
Theorem 2.2. Our contribution in Theorem 1.1 consists in extending the results
of Chen for $\dim B>1$.

When the base of the family is a curve, the measure $\eta^{HN}$ is easier to
study for various reasons: the subsheaves in the Harder-Narasimhan filtration
become locally free, there is a very precise relation between the slope and
the number of holomorphic sections, cf. [8, Lemma 2.1], and the notion of
slope doesn’t depend on the choice of multipolarization.

The main goal of this article is to show that $\eta^{HN}$, $\eta_{\min}^{HN}$,
$\eta_{\max}^{HN}$ can be recovered from the analogous quantities associated
with base changes of our family over generic curves in $B$.

For this, we need to assume that the multipolarization $[\omega_{B}]$ is
integral and very ample, i.e. $[\omega_{B,i}]\in H^{2}(X,\mathbb{Z})$ and very
ample for any $i=1,\ldots,m-1$. Consider a regular curve $C=C(l)\subset B$
given by Bertini theorem as an intersection of $m-1$ generic divisors in the
classes $l_{i}[\omega_{B,i}]$, $i=1,\ldots,m-1$, where
$l\in\mathbb{N}^{*(m-1)}$, $l=(l_{1},\ldots,l_{m-1})$. It is well-known, cf.
Corollary 3.8, that for such generic curves $C$, the Harder-Narasimhan slopes
of $E_{k}|_{C}$ do not depend on the choice of $C$ (for a fixed parameter
$l\in\mathbb{N}^{*(m-1)}$). In particular, the sequence of probability
measures $\eta_{k,l}^{HN|C}$, defined for $E_{k}|_{C}$ analogously to
$\eta_{k}^{HN}$, and their weak limits, as $k\to\infty$, which we denote by
$\eta_{l}^{HN|C}$, as well as the limiting minimal and maximal slopes, cf.
(1.3), which we denote by $\eta_{\min,l}^{HN|C},\eta_{\max,l}^{HN|C}$, are
well-defined and exist by Theorem 1.1.

For $a\in{}^{*}$, we introduce the homothety map
$H^{a}\mathrel{\mathop{\ordinarycolon}}\real\to\real$, $x\mapsto x/a$.

######  Theorem 1.2.

The measures $H^{l_{1}\cdots l_{m-1}}_{*}(\eta_{l}^{HN|C})$ converge weakly to
$\eta^{HN}$, as $l=(l_{1},\ldots,l_{m-1})\to\infty$ (i.e. $l_{i}\to\infty$,
$i=1,\ldots,m-1$). Moreover,

$\lim_{l\to\infty}\frac{\eta_{\min,l}^{HN|C}}{l_{1}\cdots
l_{m-1}}=\eta_{\min}^{HN},\qquad\lim_{l\to\infty}\frac{\eta_{\max,l}^{HN|C}}{l_{1}\cdots
l_{m-1}}=\eta_{\max}^{HN}.$ (1.4)

######  Remark 1.3.

For the rationale behind the normalization by $l_{1}\cdots l_{m-1}$, see
(2.8).

As an application of Theorem 1.2 and a result from Xu-Zhuang [48], we give an
algebraic interpretation of $\eta_{\min}^{HN}$ and $\eta_{\max}^{HN}$. More
precisely, we say that $\alpha\in H^{1,1}(X)$ is $[\omega_{B}]$-generically
fibered nef (resp. psef) with respect to $\pi$ if there is
$l_{0}\in\mathbb{N}^{*}$, such that for any regular curve $C=C(l)\subset B$,
$l=(l_{1},\ldots,l_{m-1})$, $l_{i}\geq l_{0}$, defined as before Theorem 1.2,
the restriction of $\alpha$ to $\pi^{-1}(C)$ is nef (resp. psef), i.e. for any
irreducible curve $C^{\prime}$ in $\pi^{-1}(C)$, the pairing
$\int_{C^{\prime}}\alpha$ is non negative (resp. the restriction of $\alpha$
to $\pi^{-1}(C)$ contains a positive current). For families given by
projectivization of vector bundles, equivalent definition was given by Miyaoka
[37], see also Peternell [38]. In analogy with the definition of the nef
(resp. psef) threshold, for a class $\alpha\in H^{1,1}(X)$, we then define the
$[\omega_{B}]$-generic nef (resp. psef) threshold with respect to $\pi$ as the
supremum over all $T\in\real$, such that
$\alpha-T(\pi^{*}[\omega_{B,0}]/\int_{B}[\omega_{B,0}]\cdots[\omega_{B,m-1}])$
is $[\omega_{B}]$-generically fibered nef (resp. psef) with respect to $\pi$
for some Kähler class $[\omega_{B,0}]\in H^{1,1}(B,\mathbb{C})$ (it is easy to
see that the definition is independent on the choice of $[\omega_{B,0}]$, and
for relatively ample $\alpha$, this supremum is finite).

######  Corollary 1.4.

When the multipolarization $[\omega_{B}]$ is integral, the quantity
$\eta_{\min}^{HN}$ (resp. $\eta_{\max}^{HN}$) equals to $[\omega_{B}]$-generic
nef (resp. psef) threshold of $c_{1}(L)$ with respect to $\pi$.

As we shall explain in Section 5, if $\dim B=1$, from the characterization of
the pseudoeffective cone [5] and [48], we can get the following numerical
formulas

$\eta_{\min}^{HN}=\inf_{C\in\mathcal{C}_{irr}}\frac{\int_{C}c_{1}(L)}{{\deg(\pi|_{C})}},\qquad\eta_{\max}^{HN}=\inf_{C\in\mathcal{C}_{mov}}\frac{\int_{C}c_{1}(L)}{{\deg(\pi|_{C})}},$
(1.5)

where $\mathcal{C}_{irr}$ is the set of irreducible curves $C\subset X$, such
that $\pi|_{C}$ is surjective, $\mathcal{C}_{mov}\subset\mathcal{C}_{irr}$ is
the subset of movable curves, i.e. curves which can be put in a family, which
cover $X$, and $\deg(\pi|_{C})$ is the topological degree of $\pi|_{C}$. By
Corollary 1.4, similar formulas can be established if $\dim B>1$.

To prove Theorem 1.2, we use a theorem of Mehta-Ramanathan [36], showing that
semistability behaves well under restriction over generic curves of
sufficiently big degrees. Using this theorem, in (4.8), we show that the
coupling of $\eta^{HN}$ with an arbitrary test function on can be calculated
as a double limit, where the first limit is taken in $l$ (corresponding to the
degree of the generic curve, $C$) and the second limit is in $k$
(corresponding to the index of the direct image sheaves, $E_{k}$). The main
content of Theorem 1.2 is that we can interchange the order of these limits.
Philosophically, even though we do not literally pursue this direction, this
means that there is a certain weak uniformity in the degree of the generic
curve required in Mehta-Ramanathan theorem for the vector bundles $E_{k}$ for
all parameters $k\in\mathbb{N}$. In order to show that we can change the order
of limits, we rely on the techniques of Shatz [44], to describe how Harder-
Narasimhan slopes behave under the variation of the manifold (in our case, a
specialization of the generic curve) and on the analysis of Chen [7] on the
study of submultiplicative filtrations.

Let us finally mention that recently there’s been a surge of interest in the
study of asymptotics of Harder-Narasimhan slopes, especially when the base of
the family is a curve. Chen studied its applications in Arakelov geometry in
[7] and algebraic geometry in [8]. It was applied in the study of the moduli
space of K-stable Fano varieties by Codogni-Patakfalvi [11], Xu-Zhuang [48]
and more recently by Hattori [25]. Related ideas have been used in diophantine
approximations, see Faltings-Wüstholz [15], cf. Grieve [22]. Sektnan-Tipler
[42] have studied a related question of semistability of pullbacks along
submersions. These developments were part of the motivation for the current
work. Another motivation comes from the sequel of this article, cf. [18],
where we study the relation between $\eta^{HN}$ and differential geometry
using some results from this paper.

This article is organized as follows. In Section 2, we recall some crucial
facts from the theory of submultiplicative filtrations and establish Theorem
1.1. In Section 3, we study how Harder-Narasimhan filtrations behave in
families. The results of that section will be particularily important in
Section 4, where we establish Theorem 1.2. Finally, in Section 5, we establish
Corollary 1.4.

Notation. A word generic in this article means outside a countable union of
proper analytic subspaces in some connected complex manifold. We say that a
curve $C\subset B$ is a complete intersection in an integral multipolarization
$[\omega_{B}]=([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$, if it coincides with
the intersection of $m-1$ divisors in $B$ in the classes $[\omega_{B,i}]$,
$i=1,\ldots,m-1$. For $l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$, we
denote
$l[\omega_{B}]\mathrel{\mathop{\ordinarycolon}}=(l_{1}[\omega_{B,1}],\ldots,l_{m-1}[\omega_{B,m-1}])$.
By an abuse of notations, a $m-1$ tuple of ample line bundles on $B$ will also
be called a multipolarization.

A sequence $a_{n}\in\mathbb{N}$, $n\in\mathbb{N}$, is called multiplicative if
there is $b_{n}\in\mathbb{N}^{*}$, such that $a_{n+1}=a_{n}\cdot b_{n}$. A
sequence $a_{n}\in\real$, $n\in\mathbb{N}$, is called subadditive (resp.
superadditive) if for any $n,m\in\mathbb{N}$, we have $a_{n+m}\leq
a_{n}+a_{m}$ (resp. $a_{n+m}\geq a_{n}+a_{m}$). We say that a property holds
eventually if it holds for sufficiently big parameters.

Acknowledgement. I would like to thank CNRS and École Polytechnique for their
support. A part of this paper was written in the Fall of 2023 during a visit
in Columbia University. I would like to thank the mathematical department of
Columbia University, especially Duong H. Phong, for their hospitality and
Alliance Program for their support.

##  2 Submultiplicative nature of Harder-Narasimhan filtrations

The main goal of this section is to prove Theorem 1.1. For this, we will first
recall some crucial points of the theory of submultiplicative filtrations on
section rings.

Recall that a non-increasing -filtration $\mathcal{F}$ of a vector space $V$
is a map from to vector subspaces of $V$, $t\mapsto\mathcal{F}^{t}V$,
verifying $\mathcal{F}^{t}V\subset\mathcal{F}^{s}V$ for $t>s$, and such that
$\mathcal{F}^{t}V=V$ for sufficiently small $t$ and $\mathcal{F}^{t}V=\\{0\\}$
for sufficiently big $t$. We say that $\mathcal{F}$ is graded if it respects
the grading of $V$. It is left-continuous if for any $t\in\real$, there is
$\epsilon_{0}>0$, such that $\mathcal{F}^{t}V=\mathcal{F}^{t-\epsilon}V$ for
any $0\leq\epsilon\leq\epsilon_{0}$. All filtrations in this article are
assumed to be non-increasing left-continuous and graded if applicable. We
associate with any filtration the probability measure, $\eta(\mathcal{F},V)$,
on , which we also call the jumping measure, defined as follows

$\eta(\mathcal{F},V)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{\dim
V}\sum_{j=1}^{\dim V}\delta\big{[}\eta(\mathcal{F},j)\big{]},$ (2.1)

where $\eta(\mathcal{F},j)$ are the jumping numbers, defined as follows
$\eta(\mathcal{F},j)\mathrel{\mathop{\ordinarycolon}}=\sup\\{t\in\real\mathrel{\mathop{\ordinarycolon}}\dim\mathcal{F}^{t}V\geq
j\\}$. Alternatively, we can define this measure as
$\eta(\mathcal{F},V)=-\frac{1}{\dim
V}\frac{d}{d\lambda}\dim\mathcal{F}^{\lambda}V$.

Now, for an ample line bundle $L$ over a complex projective manifold $Y$,
define the section ring

$R(Y,L)\mathrel{\mathop{\ordinarycolon}}=\oplus_{k=1}^{\infty}H^{0}(Y,L^{k}).$
(2.2)

A graded filtration $\mathcal{F}$ on $R(Y,L)$ is called submultiplicative if
for any $t,s\in\real$, $k,l\in\mathbb{N}$, we have

$\mathcal{F}^{t}H^{0}(Y,L^{k})\cdot\mathcal{F}^{s}H^{0}(Y,L^{l})\subset\mathcal{F}^{t+s}H^{0}(Y,L^{k+l}).$
(2.3)

We say that $\mathcal{F}$ is bounded if there is $C>0$, such that for any
$k\in\mathbb{N}^{*}$, $\mathcal{F}^{Ck}H^{0}(Y,L^{k})=\\{0\\}$. Remark that it
is an immediate consequence of the submultiplicativity and the fact that
$R(Y,L)$ is a finitely generated ring that there is $C>0$, such that
$\mathcal{F}^{-Ck}H^{0}(Y,L^{k})=H^{0}(Y,L^{k})$.

We let $N_{k}=\dim H^{0}(Y,L^{k})$, and denote by
$\eta_{\mathcal{F}}(1,k)\geq\cdots\geq\eta_{\mathcal{F}}(N_{k},k)$ the jumping
numbers of the filtration $\mathcal{F}$ on $H^{0}(X,L^{k})$. We denote by
$\eta_{\mathcal{F},\max}(k)\mathrel{\mathop{\ordinarycolon}}=\eta_{\mathcal{F}}(1,k)$
the maximal and by
$\eta_{\mathcal{F},\min}(k)\mathrel{\mathop{\ordinarycolon}}=\eta_{\mathcal{F}}(N_{k},k)$
the minimal jumping numbers. The following result is well-known, cf. [7,
Propositions 3.2.4, 3.2.6].

######  Proposition 2.1.

For any submultiplicative filtration $\mathcal{F}$ on $R(Y,L)$, the sequence
$\eta_{\mathcal{F},\max}(k)$ (resp. $\eta_{\mathcal{F},\min}(k)$),
$k\in\mathbb{N}$, is (resp. eventually) superadditive. In particular, the
following limits exist $\lim_{k\to\infty}\eta_{\mathcal{F},\max}(k)/k$,
$\lim_{k\to\infty}\eta_{\mathcal{F},\min}(k)/k$, in $\real\cup\\{+\infty\\}$,
and they are finite if $\mathcal{F}$ is bounded.

Define the sequence of jumping measures $\eta_{\mathcal{F},k}$,
$k\in\mathbb{N}^{*}$, of $\mathcal{F}$ on as follows

$\eta_{\mathcal{F},k}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{N_{k}}\sum_{j=1}^{N_{k}}\delta\Big{[}\frac{\eta_{\mathcal{F}}(j,k)}{k}\Big{]}=H^{k}_{*}\Big{(}\eta(\mathcal{F},H^{0}(X,L^{k}))\Big{)},$
(2.4)

where $H^{k}$ is the homothety map, defined before Theorem 1.2.

######  Theorem 2.2 (Chen [7], Boucksom-Chen [4]).

For any bounded submultiplicative filtration $\mathcal{F}$ on $R(Y,L)$, the
jumping measures $\eta_{\mathcal{F},k}$, $k\in\mathbb{N}^{*}$, converge
weakly, as $k\to\infty$, to a probability measure $\eta_{\mathcal{F}}$.
Moreover, the following holds

$\lim_{k\to\infty}\frac{\eta_{\mathcal{F},\min}(k)}{k}\leq\operatorname*{ess\,inf}\eta_{\mathcal{F}},\qquad\lim_{k\to\infty}\frac{\eta_{\mathcal{F},\max}(k)}{k}=\operatorname*{ess\,sup}\eta_{\mathcal{F}}.$
(2.5)

Also, the measure $\eta_{\mathcal{F}}$ is absolutely continuous with respect
to the Lebesgue measure, except perhaps for a point mass at
$\operatorname*{ess\,sup}\eta_{\mathcal{F}}$.

######  Remark 2.3.

Boucksom-Chen in [4] gave an explicit expression of $\eta_{\mathcal{F}}$ in
terms of the Okounkov body associated with $\mathcal{F}$. Witt Nyström [47]
gave an expression of $\eta_{\mathcal{F}}$ in terms the geodesic ray in the
space of positive metrics on $L$ associated with the filtration, when the
filtration is given by the weight of a $\mathbb{C}^{*}$-action; Hisamoto [26]
later generalized it for any finitely generated filtrations; then, the author
in [16] further generalized it to all bounded submultiplicative filtrations by
giving a different proof, based purely on complex pluripotential theory and
semiclassical input from Tian [45] and Phong-Sturm [39]. This was further
generalized by the author in [17] in the relative setting, following Chen-
Maclean [9], cf. also Boucksom-Jonsson [6], and relying on the results of
Zeldtich [49], Chen-Sun [10], Berndtsson [2], Dai-Liu-Ma [12], Ma-Marinescu
[32], [33] and [16].

Now, for any $\lambda\in\real$, we consider the subalgebra
$R^{(\lambda)}(Y,L)\mathrel{\mathop{\ordinarycolon}}=\oplus_{k=0}^{\infty}\mathcal{F}^{\lambda
k}H^{0}(Y,L^{k})$ of $R(Y,L)$.

######  Theorem 2.4 (Boucksom-Chen [4]).

For any $\lambda\in\real$, the following limit exists

${\rm{vol}}(R^{(\lambda)}(Y,L))\mathrel{\mathop{\ordinarycolon}}=\lim_{k\to\infty}\frac{\dim\mathcal{F}^{\lambda
k}H^{0}(Y,L^{k})}{N_{k}}=\int_{\lambda}^{+\infty}d\eta_{\mathcal{F}}.$ (2.6)

Moreover, the function $\lambda\mapsto{\rm{vol}}(R^{(\lambda)}(Y,L))^{1/\dim
Y}$ is concave on $]-\infty,\operatorname*{ess\,sup}\eta_{\mathcal{F}}[$.

Now, we specialize this general theory to the case of Harder-Narasimhan
filtrations. We use the notations from Introduction. We introduce the (non-
increasing) filtrations $\mathcal{F}_{k}(\lambda)$, $\lambda\in\real$, of
$E_{k}$ by coherent (torsion-free) subsheaves (defined over $B$) so that
$\mathcal{F}_{k}(\lambda)$ is the maximal subsheaf of $E_{k}$ such that all of
its Harder-Narasimhan slopes are bigger than $\lambda$. Clearly, the
filtration $\mathcal{F}_{k}$ is just a “renaming” of the Harder-Narasimhan
filtration of $E_{k}$.

The main idea behind the above “renaming” of Harder-Narasimhan filtration is
that the filtration then becomes submultiplicative. More precisely, for any
$b\in B$, we denote by $\mathcal{F}_{b}$ the filtration induced by
$\mathcal{F}_{k}(\lambda)$ on
$R(X_{b},L_{b})=\oplus_{k=0}^{\infty}H^{0}(X_{b},L_{b}^{k})$, for
$X_{b}=\pi^{-1}(b)$, $L_{b}=L|_{X_{b}}$, $b\in B$.

######  Proposition 2.5.

The filtration $\mathcal{F}_{b}$ is submultiplicative for generic $b\in B$.

When $\dim B=1$, Proposition 2.5 is due to Chen [7, §4.3.2], and the statement
holds non-generically. The proof of the general case remains verbatim modulo
some technical modifications which enter into play due to the fact that
Harder-Narasimhan filtrations are filtrations by subsheaves (and not by vector
bundles) if $\dim B>1$. More precisely, we need the following basic results.

######  Lemma 2.6.

For any $\lambda>0$, we have
$\mu^{HN}_{\max}(E_{k}/\mathcal{F}_{k}(\lambda))<\lambda$.

###### Proof.

It follows directly from the definition of $\mathcal{F}_{k}(\lambda)$. ∎

For the following result, we recall that the determinant line of a coherent
sheaf is canonically trivial if the support of the sheaf has codimension at
least $2$, cf. [30, Proposition 5.6.14]. In particular, Harder-Narasimhan
filtrations can be defined for coherent sheaves, which are torsion-free in
codimension $2$, cf. [28, Theorem 1.6.7]. In this case, the filtration will be
given by coherent subsheaves, which are torsion-free in codimension $2$, and
their subsequent quotients will also be torsion-free in codimension $2$. In
particular, the notions of maximal and minimal slopes are well-defined. We
borrow for those notions the previously-defined notations of the maximal and
the minimal Harder-Narasimhan slopes.

######  Lemma 2.7 ( [28, Proposition 1.2.7 and Theorem 1.6.6] ).

Let $\phi\mathrel{\mathop{\ordinarycolon}}\mathscr{E}\to\mathcal{F}$ be a map
between coherent sheaves $\mathscr{E},\mathcal{F}$, such that both
$\mathscr{E}$ and $\mathcal{F}$ are torsion-free in codimension $2$. Assume
that $\mu^{HN}_{\max}(\mathcal{F})<\mu^{HN}_{\min}(\mathscr{E})$. Then
$\phi=0$ outside an analytic subset of codimension at least $2$.

######  Remark 2.8.

In [28], authors treat symmetric multipolarization, but the generalization is
direct.

The reason why it is necessary to consider sheaves which are not torsion-free
in the context of this article is that in general a tensor product of two
torsion-free coherent sheaves is not torsion-free in general, cf. [41, p.
261]. However, the following result is true.

######  Lemma 2.9.

Let $\mathscr{E}_{1},\mathscr{E}_{2}$ be two torsion-free coherent sheaves,
then their tensor product is torsion-free in codimension $2$. Similarly, for a
torsion-free coherent sheaf $\mathscr{E}$, the sheaves
${\rm{Sym}}^{k}\mathscr{E}$, $k\in\mathbb{N}^{*}$, are torsion-free in
codimension $2$.

###### Proof.

The statement follows directly from the fact that torsion-free coherent
sheaves are locally-free in codimension $2$, cf. [30, Theorem 5.5.8, Corollary
5.5.15] ∎

###### Proof of Proposition 2.5.

It suffices to verify that the natural map
$\mathcal{F}_{k}(\lambda)\otimes\mathcal{F}_{l}(\mu)\to
E_{k}/\mathcal{F}_{k+l}(\lambda+\mu)$ is zero outside a subset of codimension
at least $2$, for any $\lambda,\mu\in\real$, $k,l\in\mathbb{N}$. But this is a
direct consequence of Lemmas 2.6, 2.7, 2.9, the fact that a tensor product of
two semistable sheaves is semistable (in the category of sheaves which are
torsion free in codimension $2$), and the slope behaves additively with
respect to tensor products. The last two results were established by Maruyama
[34] in the case of symmetric mutipolarization and by Greb-Kebekus-Peternell
[21, Theorem 4.2] in general. ∎

The next result proves the boundness of submultiplicative filtrations. In the
context of filtrations associated with test configurations, the importance of
such bounds in analytic approach to the study of submultiplicative filtrations
has been underlined by Phong-Sturm in [40, Lemma 4].

######  Proposition 2.10.

The filtration $\mathcal{F}_{b}$ is bounded for any $b\in B$.

When $m\mathrel{\mathop{\ordinarycolon}}=\dim B=1$, this was established by
Chen [7, Proposition 4.3.5] by relying on the very explicit relation between
the degree and the number of holomorphic sections of a stable vector bundle
(which doesn’t extend to higher dimensions), cf. also [48, Lemma-Definition
2.26]. We will show below that the statement for projective $B$ of any
dimension reduces to the one considered by Chen. For this, the following
result will be crucial.

######  Lemma 2.11.

Assume that the multipolarization $[\omega_{B}]$ is integral and very ample.
Let $\mathscr{E}$ be a torsion-free coherent sheaf over $B$. Then for generic
regular curves $C\subset B$, which are complete intersections in
$l[\omega_{B}]$, $l\in\mathbb{N}^{*(m-1)}$, the restriction,
$\mathscr{E}|_{C}$, of $\mathscr{E}$ to $C$ is torsion-free, and we have

$\displaystyle\mu_{[\omega_{B}],\max}(\mathscr{E})\leq l_{1}\cdots
l_{m-1}\cdot\mu_{\max}(\mathscr{E}|_{C}),$ (2.7)
$\displaystyle\mu_{[\omega_{B}],\min}(\mathscr{E})\geq l_{1}\cdots
l_{m-1}\cdot\mu_{\min}(\mathscr{E}|_{C}).$

###### Proof.

First, the statement about the torsion-free restriction is well-known, cf.
[23, Théorème 12.1.1] or [28, Lemma 3.1.1]. Now, as $C$ is a complete
intersection in $l[\omega_{B}]$, we have

$\mu(\mathscr{E}|_{C})=l_{1}\cdots
l_{m-1}\cdot\mu_{[\omega_{B}]}(\mathscr{E}).$ (2.8)

Remark also that the maximal and the minimal slopes admit the following useful
characterization

$\displaystyle\mu_{[\omega_{B}],\max}(\mathscr{E})=\max\Big{\\{}\mu_{[\omega_{B}]}(\mathcal{F})\mathrel{\mathop{\ordinarycolon}}\mathcal{F}\text{
is a subsheaf of }\mathscr{E}\Big{\\}},$ (2.9)
$\displaystyle\mu_{[\omega_{B}],\min}(\mathscr{E})=\min\Big{\\{}\mu_{[\omega_{B}]}(\mathcal{Q})\mathrel{\mathop{\ordinarycolon}}\mathcal{Q}\text{
is a quotient sheaf of }\mathscr{E}\Big{\\}}.$

The result now follows from (2.8) and (2.9) applied for $\mathscr{E}$ and
$\mathscr{E}|_{C}$. ∎

###### Proof of Proposition 2.10.

First of all, since $B$ is projective, any multipolarization can be bounded
from above by an integral multipolarization. Remark that the slopes will only
increase by this procedure, so it is sufficient to assume that the
multipolarization $[\omega_{B}]$ is integral and very ample.

Now, by Lemma 2.11, the maximal Harder-Narasimhan $[\omega_{B}]$-slope of
$E_{k}$ is bounded from above by the maximal Harder-Narasimhan slope of
$E_{k}|_{C}$, where $C$ is a regular complete intersection curve in
$[\omega_{B}]$. But since the maximal slopes of $E_{k}|_{C}$ grow at most
linearly in $k$ by the already mentioned result of Chen, we deduce the same
for the maximal slopes of $E_{k}$. ∎

###### Proof of Theorem 1.1.

It follows directly from Theorem 2.2 and Propositions 2.5, 2.10. ∎

##  3 Relative slopes on degenerating families of varieties

The main goal of this section is to describe how Harder-Narasimhan filtrations
behave in families. More precisely, we first recall that in any flat family,
Harder-Narasimhan slopes of the generic fiber are well-defined. Following
Shatz [44], we then make a relation between the Harder-Narasimhan slopes of
the generic fiber and the Harder-Narasimhan slopes of the specialization of
the generic fiber. As an application of this, we show that the sequence of
measures composed of Harder-Narasimhan slopes of the restrictions of a given
torsion-free sheaf to generic curves of a given degree satisfy certain
subadditivity properties in terms of degrees of curves.

More precisely, our setting will be as follows. Let
$p\mathrel{\mathop{\ordinarycolon}}B\to S$ be a proper surjective flat
holomorphic map of relative dimension $m$ between complex connected manifolds
$B$ and $S$, and let $F=(F_{1},\ldots,F_{m-1})$ be a relative
multipolarization, which means that $F_{i}$ are relatively ample line bundles
with respect to $p$ on $B$, $i=1,\ldots,m-1$. We assume for simplicity that
$p$ has a smooth fiber, which implies that the set of smooth fibers is an open
dense subset of $S$, see [23, Théorème 12.2.4]. Let $\mathscr{E}$ be a flat
family of torsion-free coherent sheaves on the fibers of $p$, cf. [28,
Definition 2.1.1]. The set of fibers of $p$ such that the restriction of
$\mathscr{E}$ to them is torsion-free is open in $S$ by [23, Théorème 12.1.1].
In particular, the Harder-Narasimhan filtrations (with respect to the
multipolarization
$c_{1}(F)|_{B_{s}}\mathrel{\mathop{\ordinarycolon}}=(c_{1}(F_{1})|_{B_{s}},\ldots,c_{1}(F_{m-1})|_{B_{s}})$)
of the torsion-free coherent sheaf $\mathscr{E}|_{B_{s}}$ are well-defined for
generic $s\in S$.

######  Proposition 3.1.

The $c_{1}(F)|_{B_{s}}$-slopes of the Harder-Narasimhan filtrations of
$\mathscr{E}|_{B_{s}}$ are constant for generic $s\in S$.

The following two results will make the proof of Proposition 3.1.

######  Lemma 3.2 ( [20, Example 20.3.3] ).

The $c_{1}(F)|_{X_{s}}$-slopes of $\mathscr{E}|_{B_{s}}$ are constant for
$s\in S$.

######  Theorem 3.3.

There is a complex manifold $T$ with a birational morphism
$f\mathrel{\mathop{\ordinarycolon}}T\to S$, such that for
$p_{0}\mathrel{\mathop{\ordinarycolon}}B\times_{S}T\to B$, there is a
filtration of $p_{0}^{*}\mathscr{E}$ by subsheaves:

$p_{0}^{*}\mathscr{E}=HN(\mathscr{E})_{s}\supset
HN(\mathscr{E})_{s-1}\supset\cdots\supset HN(\mathscr{E})_{1}\supset
HN(\mathscr{E})_{0}=\\{0\\},$ (3.1)

such that the factors $HN(\mathscr{E})_{i}/HN(\mathscr{E})_{i-1}$ are flat for
$i=1,\ldots,s$, and there is a dense open subset $U\subset T$, such that the
restriction of the above filtration to a fiber at $t\in U$ corresponds to the
Harder-Narasimhan filtration of the fiber.

######  Remark 3.4.

The filtration above is called relative Harder-Narasimhan filtration. The
proof of the above statement for symmetric multipolarizations can be found in
[28, Theorem 2.3.2], and in the general case in [46, Corollary 6.6].

###### Proof of Proposition 3.1.

It follows directly from Theorem 3.3 and Lemma 3.2, applied to all subsheaves
of the relative Harder-Narasimhan filtration. ∎

We will now apply this general theory to study Harder-Narasimhan filtrations
of the restrictions of a given sheaf to generic curves. We fix a complex
projective manifold $B$ of dimension $m$ with very ample line bundles
$F_{1},\ldots,F_{m-1}$. For $l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$,
we consider the space
$S_{l}\mathrel{\mathop{\ordinarycolon}}=\mathbb{P}(H^{0}(B,F_{1}^{l_{1}}))\times\cdots\times\mathbb{P}(H^{0}(B,F_{m-1}^{l_{m-1}}))$,
parametrizing $\dim B-1$ hyperplanes in
$H^{0}(B,F_{1}^{l_{1}})^{*},\ldots,H^{0}(B,F_{m-1}^{l_{m-1}})^{*}$. We define
the incidence variety $\mathcal{C}_{l}$ as follows

$\mathcal{C}_{l}\mathrel{\mathop{\ordinarycolon}}=\Big{\\{}(b\in B,s\in
S_{l})\mathrel{\mathop{\ordinarycolon}}b\text{ lies in the intersection of
hypeplanes parametrized by }s\Big{\\}},$ (3.2)

where we implicitly identified $B$ with its images in
$\mathbb{P}(H^{0}(B,F_{i}^{l_{i}})^{*})$, $i=1,\ldots,m-1$, through the
Kodaira map. We then have the natural maps
$p\mathrel{\mathop{\ordinarycolon}}\mathcal{C}_{l}\to S_{l}$ and
$p_{0}\mathrel{\mathop{\ordinarycolon}}\mathcal{C}_{l}\to B$. It is classical,
cf. [28, §3.1], that $p_{0}$ is a locally trivial bundle with fibers given by
the product of projective spaces. In particular, $\mathcal{C}_{l}$ is a smooth
manifold. The fibers of the projection $p$ are given by the intersection of
the $\dim B-1$ divisors given by the zero-locus of sections parametrized by a
point in $S_{l}$. Hence, these fibers are generically curves. More precisely,
the following result holds true.

######  Proposition 3.5.

The set $U\subset S_{l}$ (resp. $V\subset U$) such that the fibers of $p$ at
$U$ have dimension $1$ (resp. of regular values of $p$) is a non-empty open
subset in Zariski topology, i.e. its complement is an analytic subset.

###### Proof.

The statement for $V$ is the Bertini theorem, cf. [24, Theorem II.8.18]. Let
us now establish the statement about $U$. Define the function
$n\mathrel{\mathop{\ordinarycolon}}S_{l}\to\mathbb{N}$ as

$n(s)=\text{maximal dimension of an irreducible component of }p^{-1}(s).$
(3.3)

Since $p^{-1}(s)$ is given by the intersection of $\dim B-1$ divisors, by
Serre’s inequality on height, cf. [13, Execrice II.11.11], the dimension of
every irreducible component of $p^{-1}(s)$, $s\in S_{l}$, is at least $1$. In
particular, we have $U\mathrel{\mathop{\ordinarycolon}}=\\{s\in
S_{l}\mathrel{\mathop{\ordinarycolon}}n(s)<2\\}$. But $\mathcal{C}_{l}$ is
projective and, hence, $p$ is proper. In particular, the function $n$ is upper
semicontinuous, cf. [23, Théorème 13.1.5]. Hence, the set $\\{s\in
S_{l}\mathrel{\mathop{\ordinarycolon}}n(s)<2\\}$ is open, which finishes the
proof. ∎

######  Proposition 3.6.

The restriction of $p$ to $p^{-1}(U)$ is flat.

######  Remark 3.7.

The same statement appeared in [35, Proposition 1.5] with a different proof.

###### Proof.

Since both $U$ and $p^{-1}(U)$ are smooth manifolds, and the dimension of the
fibers of $p$ over $p^{-1}(U)$ equals to $1$ by the definition of the set $U$,
and $\dim\mathcal{C}_{l}-\dim S_{l}=1$ by a simple dimension count, the result
follows from the miracle flatness theorem, cf. [24, Exercise III.10.9]. ∎

######  Corollary 3.8.

Assume $[\omega_{B}]$ is very ample. For any torsion-free coherent sheaf
$\mathscr{E}$ over $B$, the Harder-Narasimhan slopes of the restriction,
$\mathscr{E}|_{C}$, (locally free by Lemma 2.11) to a generic regular curve
$C\subset B$, which is a complete intersection in $[\omega_{B}]$, do not
depend on the choice of $C$.

###### Proof.

It is a direct consequence of Propositions 3.1, 3.5, 3.6. ∎

In the second part of this section, we discuss the relation between the slopes
of generic and special fibers of the family. For this, we need to introduce a
certain order on filtrations.

Let $\mathscr{E}$ be a torsion-free coherent sheaf of rank $r$ over a complex
Kähler manifold $B$ with a fixed multipolarization $[\omega_{B}]$. We fix an
arbitrary filtration of $\mathscr{E}$ by coherent subsheaves
$\mathscr{E}=\mathcal{F}_{s}\supset\mathcal{F}_{s-1}\supset\cdots\supset\mathcal{F}_{1}\supset\mathcal{F}_{0}=\\{0\\},$
and define the polytope of the filtration as the graph of a partially linear
function, defined on $[0,r]$, as a linear interpolation of nodes
$({\rm{rk}}(\mathcal{F}_{i}),\deg(\mathcal{F}_{i}))$, $i=0,\ldots,s$, where
the degree is calculated with respect to $[\omega_{B}]$. A polytope is then a
certain piecewise linear path, connecting $(0,0)$ with $(r,\deg(E))$. We say
that a polytope dominates another polytope if it lies above it.

The following alternative description of this order will be particularly
useful in what follows. We define the slopes of the filtration as a non-
increasing sequence of numbers
$\mu_{1}^{\mathcal{F}},\ldots,\mu_{r}^{\mathcal{F}}$, such that
$\mu(\mathcal{F}_{r}/\mathcal{F}_{r-1})$ appears exactly
${\rm{rk}}(\mathcal{F}_{r}/\mathcal{F}_{r-1})$ times. We fix two filtrations
of $\mathscr{E}$ by subsheaves, and denote their slopes by
$\mu_{1}^{\mathcal{F},j},\ldots,\mu_{r}^{\mathcal{F},j}$, $j=1,2$. Then by [1,
(12.1)], the first filtration dominates the second one if and only if

$\mu^{\mathcal{F},1}_{1}+\cdots+\mu^{\mathcal{F},1}_{k}\geq\mu^{\mathcal{F},2}_{1}+\cdots+\mu^{\mathcal{F},2}_{k},$
(3.4)

for any $k=1,\ldots,r-1$. Remark that we always have
$\mu^{\mathcal{F},1}_{1}+\cdots+\mu^{\mathcal{F},1}_{r}={\rm{rk}}(\mathscr{E})\cdot\mu(\mathscr{E})=\mu^{\mathcal{F},2}_{1}+\cdots+\mu^{\mathcal{F},2}_{r}$.
Alternatively, we define the probability measures $\mu^{\mathcal{F},j}$,
$j=1,2$, as in (1.2):

$\mu^{\mathcal{F},j}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{r}\sum_{i=1}^{r}\delta\big{[}\mu^{\mathcal{F},j}_{i}\big{]}.$
(3.5)

Then by [27], cf. Atiyah-Bott [1, (12.2)], the above partial order is
equivalent to the following one: for any convex function
$\phi\mathrel{\mathop{\ordinarycolon}}\real\to\real$, we have

$\int\phi\mu^{\mathcal{F},1}\geq\int\phi\mu^{\mathcal{F},2}.$ (3.6)

More generally, for arbitrary probability measures $\mu_{1},\mu_{2}$ on , we
say that $\mu_{1}$ dominates $\mu_{2}$ if the analogue of (3.6) holds.

The relevance of this partial order to Harder-Narasimhan filtrations comes
from the following extremal characterization of them.

######  Theorem 3.9 (Shatz [44, Theorem 2 and Remark on p.174]).

Harder-Narasimhan filtration of a torsion-free coherent sheaf dominates all
the other filtrations by coherent subsheaves.

######  Remark 3.10.

The article [44] is written for symmetric multipolarization, but it readily
extends to the general case, as it only relies on Lemma 2.7 and the existence
of relative Harder-Narasimhan filtrations, which holds for non-symmetric
multipolarization, cf. Theorem 3.3.

As we shall see below, the above partial order is also useful in the study of
how Harder-Narasimhan filtrations behave under specialization. For this, we
assume from now on that our family of manifolds,
$p\mathrel{\mathop{\ordinarycolon}}B\to S$, is defined over a unit disc, i.e.
$S=\mathbb{D}\mathrel{\mathop{\ordinarycolon}}=\\{z\in\mathbb{C}\mathrel{\mathop{\ordinarycolon}}|z|<1\\}$.
We then decompose the central fiber, $B_{0}$, as $\sum n_{j}\cdot B_{0}^{j}$,
where $B_{0}^{j}$, $j=1,\ldots,t$, are irreducible components, which we assume
to be smooth for simplicity, and $n_{j}$ are the multiplicities. We define the
probability measure $\mu^{HN}_{0}$ on as follows

$\mu^{HN}_{0}=\frac{1}{r}\sum_{i=1}^{r}\delta\Big{[}\sum_{j=1}^{t}n_{j}\cdot\mu^{0,j}_{i}\Big{]},$
(3.7)

where $\mu^{0,j}_{i}$, $j=1,\ldots,t$, $i=1,\ldots,r$, are the slopes of the
Harder-Narasimhan filtrations of $\mathscr{E}|_{B_{0}^{j}}$ (which we assume
to be torsion-free) with respect to $c_{1}(F)|_{B_{0}^{j}}$. We denote by
$\mu^{HN}_{*}$ the probability measure composed of the slopes of the Harder-
Narasimhan filtrations of the restriction of $\mathscr{E}$ to generic fibers
(this doesn’t depend on the choice of generic fiber by Proposition 3.1). In
other words, by Theorem 3.3, we have

$\mu^{HN}_{*}=\frac{1}{r}\sum_{i=1}^{r}\delta\big{[}\mu^{*}_{i}\big{]},$ (3.8)

where $\mu^{*}_{i}$ are the slopes of the restriction of the relative Harder-
Narasimhan filtration $HN(\mathscr{E})_{i}$, $i=0,\ldots,s$, from Theorem 3.3
to fibers.

######  Theorem 3.11.

The probability measure $\mu^{HN}_{0}$ dominates $\mu^{HN}_{*}$.

######  Remark 3.12.

When $B=B_{0}\times\mathbb{D}$, and
$p\mathrel{\mathop{\ordinarycolon}}B\to\mathbb{D}$ is the natural projection,
the result is due to Shatz [44, Theorem 3]. When $p$ is of relative dimension
$1$, see [35, Proposition 4.3] for another related result. Our proof of the
general case is very much inspired by [44].

In the proof of Theorem 3.11, we will use the following simple result.

######  Lemma 3.13.

Let $\mu_{i}^{j}=\frac{1}{N}\sum_{r=1}^{N}\delta[a_{i,r}^{j}]$, $i,j=1,2$, be
probability measures on for some $a_{i,r}^{j}\in\real$, ordered in such a way
that $a_{i,r}^{j}$ are non-increasing in $r$, and such that $\sum
a_{i,r}^{1}=\sum a_{i,r}^{2}$. Assume that the measure $\mu_{i}^{1}$ dominates
$\mu_{i}^{2}$ for $i=1,2$. Then for the measures
$\mu^{j}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{N}\sum_{r=1}^{N}\delta[a_{1,r}^{j}+a_{2,r}^{j}]$,
$\mu^{1}$ dominates $\mu^{2}$.

###### Proof.

It follows directly from the characterization (3.4). ∎

###### Proof of Theorem 3.11.

By the flatness of the family and the factors
$HN(\mathscr{E})_{i}/HN(\mathscr{E})_{i-1}$, cf. [20, Example 20.3.3], we
conclude that

$\mu^{*}_{i}=\sum_{j=1}^{t}n_{j}\cdot\mu\Big{(}(HN(\mathscr{E})_{i}/HN(\mathscr{E})_{i-1})|_{B_{0}^{j}}\Big{)}.$
(3.9)

Now, using the notations introduced in (3.7), for any $j=1,\ldots,t$, we
define the measures

$\displaystyle\mu^{HN}_{0,j}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{r}\sum_{i=1}^{r}\delta\big{[}\mu^{0,j}_{i}\big{]},$
(3.10)
$\displaystyle\mu^{HN}_{*,j}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{r}\sum_{i=1}^{r}\delta\Big{[}\mu\Big{(}(HN(\mathscr{E})_{i}/HN(\mathscr{E})_{i-1})|_{B_{0}^{j}}\Big{)}\Big{]}.$

By Theorem 3.9, we conclude that the measures $\mu^{HN}_{0,j}$ dominate
$\mu^{HN}_{*,j}$. We deduce Theorem 3.11 from this, Lemma 3.13 and (3.7),
(3.8), (3.9), (3.10). ∎

We will now apply this general theory to the study of measures associated with
restrictions of torsion-free coherent sheaves. For this, we fix a torsion-free
coherent sheaf $\mathscr{E}$ of generic rank $r$ over $B$. Assume that the
multipolarization $[\omega_{B}]$ is very ample. For any
$l\in\mathbb{N}^{*(m-1)}$, we denote by $\mu^{HN}_{i}(\mathscr{E}|_{C},l)$,
$i=1,\ldots,r$, the Harder-Narasimhan slopes of the restriction of
$\mathscr{E}$ to a generic curve $C$, given by a complete intersection in
$l[\omega_{B}]$, cf. Corollary 3.8. We denote by
$\mu^{HN}_{\min}(\mathscr{E}|_{C},l)$ and
$\mu^{HN}_{\max}(\mathscr{E}|_{C},l)$ the minimal and maximal slopes
respectively. We define the sequence of probability measures, $\eta_{l}$,
$l\in\mathbb{N}^{*(m-1)}$, on as follows

$\eta_{l}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{r}\sum_{i=1}^{r}\delta\Big{[}\frac{\mu^{HN}_{i}(\mathscr{E}|_{C},l)}{l_{1}\cdots
l_{m-1}}\Big{]}.$ (3.11)

We can now state the main result of this section.

######  Theorem 3.14.

For any convex function $\phi\mathrel{\mathop{\ordinarycolon}}\real\to\real$,
the sequence of real numbers
$a_{l}\mathrel{\mathop{\ordinarycolon}}=l_{1}\cdots l_{m-1}\cdot\int\phi
d\eta_{l}$, $l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$, is
coordinatewise subadditive, i.e. for any $l^{\prime}\in\mathbb{N}^{*(m-2)}$,
$l_{1}^{1},l_{1}^{2}\in\mathbb{N}^{*}$, we have $a_{l^{0}}\leq
a_{l^{1}}+a_{l^{2}}$, for
$l^{0}\mathrel{\mathop{\ordinarycolon}}=(l_{1}^{1}+l_{1}^{2},l^{\prime})$,
$l^{1}\mathrel{\mathop{\ordinarycolon}}=(l_{1}^{1},l^{\prime}),l^{2}\mathrel{\mathop{\ordinarycolon}}=(l_{1}^{2},l^{\prime})$,
and similarly for other coordinates.

We will now establish Theorem 3.14. All the curves considered below will
naturally be given as fibers of $p$, which would parametrize them by points in
$S_{l}$, see (3.2). The following result will be crucial for the proof of
Theorem 3.14.

######  Proposition 3.15.

For any $l^{\prime}=(l_{2},\ldots,l_{m-1})\in\mathbb{N}^{*(m-2)}$,
$l_{1}^{1},l_{1}^{2}\in\mathbb{N}^{*}$, we let
$l^{1}\mathrel{\mathop{\ordinarycolon}}=(l_{1}^{1},l^{\prime})$,
$l^{2}\mathrel{\mathop{\ordinarycolon}}=(l_{1}^{2},l^{\prime})$, and fix some
curves $C_{1},C_{2}\subset B$, given by a complete intersection in
$l^{1}[\omega_{B}]$, $l^{2}[\omega_{B}]$, as a zero set of holomorphic
sections $s_{1}^{1}\in H^{0}(B,F_{1}^{l_{1}^{1}})$, $s_{i}\in
H^{0}(B,F_{i}^{l_{i}})$, $i=2,\ldots,m-1$, and $s_{1}^{2}\in
H^{0}(B,F_{1}^{l_{1}^{2}})$, $s_{i}$, $i=2,\ldots,m-1$, respectively. Then
there is a smooth manifold $\mathcal{C}$ and a flat proper map
$p\mathrel{\mathop{\ordinarycolon}}\mathcal{C}\to\mathbb{D}$, such that the
central fiber of $p$ is given by $C_{1}+C_{2}$, and the general fiber is a
complete intersection in $l^{0}[\omega_{B}]$, where
$l^{0}\mathrel{\mathop{\ordinarycolon}}=(l_{1}^{1}+l_{1}^{2},l^{\prime})$.
Moreover, for any countable union of analytic subsets in $S_{l^{0}}$, we can
choose $p$ as a base change along a holomorphic disc in $S_{l^{0}}$, which
doesn’t lie within any of the analytic subsets.

###### Proof.

Remark first that there is a canonical embedding
$\mathbb{P}(H^{0}(B,F^{l_{1}^{1}}))\times\mathbb{P}(H^{0}(B,F^{l_{1}^{2}}))\hookrightarrow\mathbb{P}(H^{0}(B,F^{l_{1}^{1}+l_{1}^{2}}))$,
given by a composition of the Segre map and the multiplication in the section
ring of $B$. The needed family can then be constructed by the base change of
the family $\mathcal{C}_{l^{0}}$ along a holomorphic disc in $S_{l^{0}}$,
which is not contained in the countable union of analytic subsets from the
statement, and which passes through the image of the point representing
$(C_{1},C_{2})$ in $(S_{l^{1}},S_{l^{2}})$, viewed as a point in $S_{l^{0}}$
through the above embedding.

The flatness of the resulting map is a consequence of Proposition 3.6 and the
fact that flatness is preserved by base changes, see [24, Proposition 9.2].
The identification of the central fiber follows from the nature of the Segre
map: if the holomorphic sections $s_{1}^{1},s_{2},\ldots,s_{m-1}$ (resp.
$s_{1}^{2},s_{2},\ldots,s_{m-1}$) vanish along $C_{1}$ (resp. $C_{2}$) with
multiplicity $1$, then the holomorphic sections $s_{1}^{1}\cdot
s_{1}^{2},s_{2},\ldots,s_{m-1}$ vanish along $C_{1}+C_{2}$ with multiplicity
$1$. ∎

We will also need the following technical lemma.

######  Lemma 3.16.

Let $B_{1},B_{2},B$ be compact complex manifolds, and $A_{1}\subset
B_{1}\times B$, $A_{2}\subset B_{2}\times B$, be proper analytic subsets. Then
there are $x_{1}\in B_{1},x_{2}\in B_{2},y\in B$, such that $(x_{1},y)\notin
A_{1}$ and $(x_{2},y)\notin A_{2}$.

###### Proof.

Let $A$ be a proper analytic subspace in $C\times B$, where $C$ is a compact
complex manifold. We denote by $D\subset B$, the locus of $x\in B$, such that
$A$ doesn’t contain $C\times\\{x\\}$. By using the properness of the
projection map $p\mathrel{\mathop{\ordinarycolon}}A\to B$ and properness of
$A$, we argue similarly to (3.3), that $D$ is open in $B$ (in Zariski
topology). It is also non-empty since $A$ is a proper subset of $C\times D$.
By applying this for $A_{1}$ and $A_{2}$, we see that the locus of $y\in B$,
such that $A_{i}$ doesn’t contain $B_{i}\times\\{x\\}$, $i=1,2$, is non-empty.
It suffices then to pick any $y\in B$ from this locus, and choose
$x_{1},x_{2}$ arbitrarily away from $A_{1},A_{2}$. ∎

###### Proof of Theorem 3.14..

From Corollary 3.8, the slopes of the Harder-Narasimhan filtrations of
$\mathscr{E}|_{C}$ are constant for curves $C\subset B$ given by a complete
intersection in $l^{i}[\omega_{B}]$, $i=0,1,2$, located away from a proper
analytic subset, $A(l^{i})$, in $S_{l^{i}}$. Using Lemma 3.16, we fix curves
$C_{i}\in S_{l^{i}}\setminus A(l^{i})$, given by complete intersection in
$l^{i}[\omega_{B}]$, $i=1,2$, as required in Proposition 3.15. By Proposition
3.15, $C_{1}+C_{2}$ can be put as a central fiber in a flat family of curves,
obtained as a base change along a holomorphic disc from $S_{l^{0}}$, not
contained in $A(l^{0})$. Since the generic curve from this family doesn’t lie
in $A(l^{0})$, the Harder-Narasimhan slopes of the restriction of
$\mathscr{E}$ to a generic curve in this family will then coincide with the
Harder-Narasimhan slopes of the restriction of $\mathscr{E}$ to generic curves
given by complete intersection in $l^{0}[\omega_{B}]$. Directly from this,
Theorem 3.11 and (3.6),

$\sum_{i=1}^{r}\phi\big{(}\mu^{HN}_{i}(\mathscr{E}|_{C},l^{1})+\mu^{HN}_{i}(\mathscr{E}|_{C},l^{2})\big{)}\geq\sum_{i=1}^{r}\phi\big{(}\mu^{HN}_{i}(\mathscr{E}|_{C},l^{0})\big{)}.$
(3.12)

By making a homothety and using the convexity of $\phi$, we finish the proof.
∎

As a byproduct of Theorem 3.14 and (3.4), we deduce the following.

######  Proposition 3.17.

For any $l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$, we have

$\displaystyle\mu^{HN}_{\min}(E_{k}|_{C},l)\geq l_{1}\cdots
l_{m-1}\cdot\mu^{HN}_{\min}(\mathscr{E}|_{C},(1,\ldots,1)),$ (3.13)
$\displaystyle\mu^{HN}_{\max}(E_{k}|_{C},l)\leq l_{1}\cdots
l_{m-1}\cdot\mu^{HN}_{\max}(\mathscr{E}|_{C},(1,\ldots,1)).$

##  4 Generic base changes and asymptotic distribution of slopes

The main goal of this section is to show that the asymptotic distribution of
Harder-Narasimhan slopes of direct images can be calculated through generic
base changes, i.e. to establish Theorem 1.2. The following famous result will
be at the heart of our approach.

######  Theorem 4.1.

For any semistable torsion-free coherent sheaf $\mathscr{E}$ over a complex
projective manifold $B$ with an integral multipolarization $[\omega_{B}]$,
there is $l_{0}\in\mathbb{N}$, such that for any generic curve $C\subset B$,
which is a complete intersection in $l[\omega_{B}]$,
$l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$, $l_{i}\geq l_{0}$,
$i=1,\ldots,m-1$, the restriction of $\mathscr{E}$ to $C$ is semistable.

######  Remark 4.2.

When the multipolarization is symmetric, the theorem is due to Mehta-
Ramanathan [35, Theorem 6.1], cf. also [28, Theorem 7.11] and [36]. The proof
for the general multipolarization remains verbatim.

In particular, from Theorem 4.1 and (2.8), we conclude that for any torsion-
free coherent sheaf $\mathscr{E}$ over $B$, the restriction of Harder-
Narasimhan filtration of $\mathscr{E}$ over generic curves $C\subset B$ as in
Theorem 4.1 coincides with the Harder-Narasimhan filtration of
$\mathscr{E}|_{C}$.

The main difficulty in the proof of Theorem 1.2 is due to the fact that the
value $l_{0}$ from Theorem 4.1 depends heavily on $\mathscr{E}$. Despite many
works on the effective version of Mehta-Ramanathan theorem, see for example
Flenner [19] or Bogomolov [3], cf. [28, §7], it seems that all known effective
bounds on $l_{0}$ depend on the rank and the discriminant of $\mathscr{E}$.
However, in the notations from Introduction, when $k\to\infty$, both rank and
discriminant of $E_{k}$ tend to infinity, and so it seems implausible that the
value $l_{0}$ can be chosen uniformly for all $k\in\mathbb{N}$. Remark also
that there are some results that relate the Harder-Narasimhan slopes of a
sheaf with the Harder-Narasimhan slopes of its restriction to a curve
associated with any fixed value $l_{0}$, see for example Grauert-Mülich
theorem, cf. [28, §3], or Langer [31, Theorem 3.1]. To the best of authors
knowledge, none of these implies Theorem 1.2 directly. Our approach for
Theorem 1.2 is then, by necessity, a different one, and it relies on
subadditive properties enjoyed by the measures $\eta_{l}^{HN|C}$ and
$\eta^{HN}$, coming from the submultiplicative nature of Harder-Narasimhan
filtrations and the results of Section 3.

We will first establish the easier part of Theorem 1.2 about the minimal
slopes. For this, we conserve the notations from Introduction and (3.11).
Directly from Theorem 4.1, the remark below it and Lemma 2.11, we deduce the
following result.

######  Lemma 4.3.

The following convergence takes place

$\lim_{l\to\infty}\frac{\mu^{HN}_{\min}(E_{k}|_{C},l)}{l_{1}\cdots
l_{m-1}}=\mu^{HN}_{\min}(E_{k}),$ (4.1)

where $l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$. Moreover, the above
limit becomes stationary for $l$ big enough (i.e. for big enough
$l_{0}\in\mathbb{N}$, and $l_{i}\geq l_{0}$, $i=1,\ldots,m-1$), and we always
have

$\frac{\mu^{HN}_{\min}(E_{k}|_{C},l)}{l_{1}\cdots
l_{m-1}}\leq\mu^{HN}_{\min}(E_{k}).$ (4.2)

######  Remark 4.4.

The same proof shows that the analogous statements hold (with inverse sign in
(4.2)) for the maximal slope.

We will also use the following statement, the proof of which is left to the
reader.

######  Lemma 4.5.

Let $c_{k,l}\in\real$, $k\in\mathbb{N}$, $l\in\mathbb{N}^{*(m-1)}$, be such
that $c_{k,l}$ increases in $k\in\mathbb{N}$, and the limit
$\lim_{l\to\infty}c_{k,l}$ exists and satisfies
$c_{k,l}\leq\lim_{l\to\infty}c_{k,l}$. Then
$\lim_{l\to\infty}\lim_{k\to\infty}c_{k,l}=\lim_{k\to\infty}\lim_{l\to\infty}c_{k,l}$.

###### Proof of the first convergence of (1.4).

By (4.1) and the definitions of $\eta_{\min,l}^{HN|C}$ and $\eta_{\min}^{HN}$,
we see that the first convergence of (1.4) is equivalent to the following
statement

$\lim_{l\to\infty}\lim_{k\to\infty}\frac{\mu^{HN}_{\min}(E_{k}|_{C},l)}{kl_{1}\cdots
l_{m-1}}=\lim_{k\to\infty}\lim_{l\to\infty}\frac{\mu^{HN}_{\min}(E_{k}|_{C},l)}{kl_{1}\cdots
l_{m-1}},$ (4.3)

where $k$ goes to infinity along any subsequence which we assume from now on
to be multiplicative (as $k=2^{r}$, $r\in\mathbb{N}$). Remark, however, that
by Proposition 2.1, the sequence $\mu^{HN}_{\min}(E_{k}|_{C},l)/k$ increases
in $k$, when $k$ runs over a multiplicative subsequence. By this and Lemma
4.3, the assumptions of Lemma 4.5 are satisfied for the restriction of the
sequence
$c_{k,l}\mathrel{\mathop{\ordinarycolon}}=\mu^{HN}_{\min}(E_{k}|_{C},l)/(kl_{1}\cdots
l_{m-1})$ to multiplicative subsequences in $k$, which finishes the proof. ∎

######  Remark 4.6.

Curiously, the analogous proof for the maximal slope doesn’t work due to
Remark 4.4 and the fact that the analogue of Lemma 4.5 with inverse inequality
sign doesn’t hold.

We will now concentrate on the proof of the rest of Theorem 1.2. For this, we
define the sequence of probability measures, $\eta_{k,l}$,
$k\in\mathbb{N}^{*}$, $l\in\mathbb{N}^{*(m-1)}$, on as follows

$\eta_{k,l}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\delta\Big{[}\frac{\mu^{HN}_{i}(E_{k}|_{C},l)}{kl_{1}\cdots
l_{m-1}}\Big{]},$ (4.4)

where $N_{k}={\rm{rk}}(E_{k})$, and we used the notations from (3.11).

The key in the proof of Theorem 1.2 lies in the following result, the proof of
which we defer to the end of this section.

######  Theorem 4.7.

For any convex non-increasing Lipschitz function
$\phi\mathrel{\mathop{\ordinarycolon}}\real\to\real$, and any
$k\in\mathbb{N}^{*}$, $l\in\mathbb{N}^{*(m-1)}$, we have the following
decomposition

$k\int\phi d\eta_{k,l}=a_{k,l}+kb_{k,l},$ (4.5)

where $b_{k,l}\to 0$, as $k\to\infty$, uniformly on $l\in\mathbb{N}^{*(m-1)}$,
and $a_{k,l}$ is subadditive in $k\in\mathbb{N}$ over multiplicative
subsequences.

We will also use the following statement, the proof of which repeats the proof
of classical Fekete’s Subadditive Lemma, and is left to the interested reader.

######  Lemma 4.8.

Let $c_{k,l}\in\real$, $k\in\mathbb{N}^{*}$,
$l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$, be such that $c_{k,l}$ is
coordinatewise subadditive in $l\in\mathbb{N}$ in the sense of Theorem 3.14,
and it admits the following decomposition $c_{k,l}=a_{k,l}+kb_{k,l}$, where
$b_{k,l}\to 0$, as $k\to\infty$, uniformly on $l\in\mathbb{N}^{*(m-1)}$, and
$a_{k,l}$ is subadditive in $k\in\mathbb{N}$. Then
$\lim_{l\to\infty}\lim_{k\to\infty}\frac{c_{k,l}}{kl_{1}\cdots
l_{m-1}}=\lim_{k\to\infty}\lim_{l\to\infty}\frac{c_{k,l}}{kl_{1}\cdots
l_{m-1}}$.

###### Proof of Theorem 1.2.

We will first establish the weak convergence of measures. The statement about
the convergence of maximal slopes will follow from the convergence of
measures.

Clearly, to verify weak convergence, it suffices to verify that for any smooth
test function $\phi\mathrel{\mathop{\ordinarycolon}}\real\to\real$ of compact
support, as $l\to\infty$ over multiplicative subsequences, we have

$\lim_{l\to\infty}\int\phi dH^{l^{m-1}}_{*}(\eta_{l}^{HN|C})=\int\phi
d\eta^{HN}.$ (4.6)

Remark now that any smooth function of compact support can be represented as a
difference of two convex non-increasing Lipschitz functions (to see this, it
suffices to decompose $\phi=(\phi+A\min(x-a,0)^{2})-A\min(x-a,0)^{2}$ for
carefully chosen $a,A$), and, hence, it suffices to establish (4.6) for such
functions. We, hence, assume from now on that $\phi$ is convex, non-increasing
and Lipschitz. By the definition of $\eta_{l}^{HN|C}$, we have

$\lim_{l\to\infty}\int\phi
dH^{l^{m-1}}_{*}(\eta_{l}^{HN|C})=\lim_{l\to\infty}\lim_{k\to\infty}\int\phi
d\eta_{k,l}.$ (4.7)

From Theorem 4.1 and the remark after it, we also have

$\int\phi d\eta^{HN}=\lim_{k\to\infty}\lim_{l\to\infty}\int\phi d\eta_{k,l}.$
(4.8)

B Theorem 3.14, the sequence $(k,l)\mapsto kl_{1}\cdots l_{m-1}\int\phi
d\eta_{k,l}$, $k\in\mathbb{N}^{*}$,
$l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$, is coordinatewise
subadditive in $l$. We establish (4.6) by Theorem 4.7, Lemma 4.8 and (4.7),
(4.8).

We will now establish the second convergence of (1.4). First of all, from
Lemma 2.11, we see

$\liminf_{l\to\infty}\frac{\eta_{\max,l}^{HN|C}}{l_{1}\cdots
l_{m-1}}\geq\eta_{\max}^{HN}.$ (4.9)

Now, by Theorem 2.4, the cumulative function
$F_{HN}(\lambda)\mathrel{\mathop{\ordinarycolon}}=\int_{\lambda}^{\infty}d\eta^{HN}$
(resp.
$F_{HN,l}(\lambda)\mathrel{\mathop{\ordinarycolon}}=\int_{\lambda}^{\infty}dH^{l^{m-1}}_{*}(\eta_{l}^{HN|C})$)
is such that $F_{HN}^{1/n}$ (resp. $F_{HN,l}^{1/n}$) is concave on
$]-\infty,\operatorname*{ess\,sup}\eta^{HN}[$. In particular, the function
$F_{HN}$ is continuous on $\real\setminus\operatorname*{ess\,sup}\eta^{HN}$.
Recall that weak convergence of measures implies pointwise convergence of the
cumulative functions over the continuity set. In particular,
$F_{HN,l}(\lambda)$ converge pointwise, as $l\to\infty$, to $F_{HN}(\lambda)$
away from $\operatorname*{ess\,sup}\eta^{HN}$.

Assume now, for the sake of contradiction, that we have a strict inequality

$\limsup_{l\to\infty}\frac{\eta_{\max,l}^{HN|C}}{l_{1}\cdots
l_{m-1}}>\eta_{\max}^{HN}.$ (4.10)

Upon restricting to a subsequence in $l$, we may assume that we have $\liminf$
instead of $\limsup$ in (4.10). Let $\epsilon>0$ be such that the inequality
(4.10) is still satisfied if we replace $\eta_{\max}^{HN}$ by
$\eta_{\max}^{HN}+4\epsilon$. We assume that $l$ is big enough so that
$\eta_{\max}^{HN}+2\epsilon<\eta_{\max,l}^{HN|C}/(l_{1}\cdots l_{m-1})$.
Remark that by Theorem 2.2, $\eta_{\max}^{HN}$ coincides with the essential
supremum of $\eta^{HN}$, hence $F_{HN}(\eta_{\max}^{HN}-\epsilon)$ is strictly
positive. By the concavity of $F_{HN,l}^{1/n}$ on the interval
$]-\infty,\eta_{\max}^{HN}+2\epsilon]\subset]-\infty,\eta_{\max,l}^{HN|C}/(l_{1}\cdots
l_{m-1})[$, the fact, following from the first part of the proof, that
$F_{HN,l}(\eta_{\max}^{HN}-\epsilon)$ converges, as $l\to\infty$, to
$F_{HN}(\eta_{\max}^{HN}-\epsilon)$, which is strictly positive, and the fact
that $F_{HN,l}(\eta_{\max}^{HN}+\epsilon)$ is non-negative, we conclude that
there is a constant $\delta>0$, such that $F_{HN,l}(\lambda)>\delta$ for any
$\lambda\in]-\infty,\eta_{\max}^{HN}+\epsilon]$. This clearly contradicts the
fact that $F_{HN,l}(\eta_{\max}^{HN}+\epsilon)$ converges to
$F_{HN}(\eta_{\max}^{HN}+\epsilon)=0$. Hence, the initial assumption (4.10)
was false, which finishes the proof. ∎

We will now prove Theorem 4.7, and for this, the following result will be
crucial.

######  Proposition 4.9 ( Chen [7, Proposition 3.3.3] ).

For any convex non-increasing function
$\phi\mathrel{\mathop{\ordinarycolon}}\real\to\real$ and any submultiplicative
filtration $\mathcal{F}$ on the ring
${\rm{Sym}}V\mathrel{\mathop{\ordinarycolon}}=\oplus_{k=0}^{\infty}{\rm{Sym}}^{k}V$
for some vector space $V$, the sequence $k\mapsto k\int\phi\cdot
dH^{k}_{*}(\eta(\mathcal{F},{\rm{Sym}}^{k}V))$ is subadditive.

Remark that if the fibers of $p$ were projective spaces, then Theorem 4.7
would already follow from Proposition 4.9, as the section ring of the
projective space polarized by a hyperplane bundle is isomorphic to the
symmetric tensor algebra. Proposition 4.9, however, seemingly doesn’t hold if
the ring ${\rm{Sym}}V$ is replaced by a section ring of an arbitrary complex
projective manifold. To circumvent this problem, we need a more refined
argument, and our solution, which is inspired by [7, §3.4.2], goes through the
use of Noether normalization lemma.

More precisely, Noether normalization lemma, cf. [14, Theorem 13.3], says that
for any projective manifold $Y$ of dimension $n$, and an ample line bundle $L$
over $Y$, there is $d\in\mathbb{N}^{*}$, and $s_{1},\ldots,s_{n+1}\in
H^{0}(Y,L^{d})$, which are algebraically independent in the section ring,
$R(Y,L)$, and such that the subring, $R_{(d)}(Y,L)$, of $R(Y,L)$, defined as
$R_{(d)}(Y,L)=\oplus_{k=0}^{\infty}H^{0}(Y,L^{kd})$, is a finitely generated
module over the (polynomial) subring $A\subset R_{(d)}(Y,L)$ induced by
$s_{1},\ldots,s_{n+1}$, i.e. $A={\rm{Sym}}\langle
s_{1},\ldots,s_{n+1}\rangle$.

Now, let $\mathcal{F}$ be an arbitrary bounded submultiplicative filtration on
$R(Y,L)$. The following proposition says that the spectral measure of the
filtration $\mathcal{F}$ on $R(Y,L)$ can be reconstructed from the spectral
measure of the restriction of the filtration $\mathcal{F}$ to $A\subset
R(Y,L)$.

######  Proposition 4.10 ( Chen [7, proof of Theorem 3.4.3] ).

For any convex non-increasing Lipshitz function $\phi$ on , there is $C>0$,
such that for any $k\in\mathbb{N}^{*}$, we have

$\Big{|}\int\phi\cdot dH^{kd}_{*}(\eta(\mathcal{F},A_{kd}))-\int\phi\cdot
dH^{kd}_{*}(\eta(\mathcal{F},H^{0}(Y,L^{kd})))\Big{|}\leq\frac{C}{k}.$ (4.11)

######  Remark 4.11.

A combination of Propositions 4.9 and 4.10, and a decomposition argument from
the proof of Theorem 1.2 was used in [7] to prove the weak convergence in
Theorem 2.2.

Our proof for Theorem 4.7 will then be a combination of Proposition 4.9 and a
uniform version of Proposition 4.10. In order to explain the latter statement
and its proof, we first recall the main steps from [7] of the proof of
Proposition 4.10.

######  Lemma 4.12 ( Chen [7, Proposition 1.2.5] ).

Let $0\rightarrow V_{0}\rightarrow V_{1}\rightarrow V_{2}\rightarrow 0$ be a
short exact sequence of vector spaces. We fix a filtration $\mathcal{F}$ on
$V_{1}$, and induce the filtrations on $V_{0}$, $V_{2}$, which we denote by an
abuse of notation by the same letter. Then the following identity holds

$\dim V_{2}\cdot\eta(\mathcal{F},V_{1})=\dim
V_{2}\cdot\eta(\mathcal{F},V_{0})+\dim V_{2}\cdot\eta(\mathcal{F},V_{2}),$
(4.12)

where the jumping measures, $\eta(\mathcal{F},V_{i})$, $i=0,1,2$, are defined
in (2.1).

We denote by ${\rm{Frac}}(A)$ the field of fractions of $A$. As $R_{(d)}(Y,L)$
is a finitely generated module over $A$, the ${\rm{Frac}}(A)$-vector space
${\rm{Frac}}(A)\otimes_{A}R_{(d)}(Y,L)$ is finitely generated. We denote by
$x_{1},\ldots,x_{r}$, $r\in\mathbb{N}$, some homogeneous elements of
$R_{(d)}(Y,L)$, which form a basis of the above vector space. Then the space
$B\mathrel{\mathop{\ordinarycolon}}=x_{1}A+\cdots+x_{r}A$,
$B=\oplus_{k=0}^{\infty}B_{kd}$, is a free $A$-submodule of $R_{(d)}(Y,L)$,
and the quotient $R_{(d)}(Y,L)/B$ is a $A$-torsion module, since
${\rm{Frac}}(A)\otimes_{A}R_{(d)}(Y,L)/B=\\{0\\}$. Then, as $R_{(d)}(Y,L)$ is
a finitely generated $A$-module, the annihilator,
${\rm{Ann}}(R_{(d)}(Y,L)/B)\subset A$, of $R_{(d)}(Y,L)/B$ is non-empty.
Hence, the Krull dimension of the $A$-module $R_{(d)}(Y,L)/B$, which is
defined as the Krull dimension of $A/{\rm{Ann}}(R_{(d)}(Y,L)/B)$, is strictly
smaller that the Krull dimension of $A$. Since Krull dimension of a module
coincides with the degree of its Hilbert polynomial, see Serre [43,
Proposition 6, p. 276], we conclude that there is a constant such that the
dimension of the $k$-th graded part, $H^{0}(Y,L^{kd})/B_{kd}$, of
$R_{(d)}(Y,L)/B$ is smaller than $Ck^{\dim Y-1}$. We conclude by this and
Lemma 4.12 that for any bounded function $\phi$, there is $C>0$, such that for
any $k\in\mathbb{N}^{*}$, we have

$\Big{|}\int\phi\cdot dH^{kd}_{*}(\eta(\mathcal{F},B_{kd}))-\int\phi\cdot
dH^{kd}_{*}(\eta(\mathcal{F},H^{0}(Y,L^{kd})))\Big{|}\leq\frac{C}{k}.$ (4.13)

By (4.13), we see that Proposition 4.10 will follow if we prove the following
result.

######  Lemma 4.13.

For any homogeneous element $x\in H^{0}(Y,L^{d_{0}})$,
$d_{0}\in\mathbb{N}^{*}$, $d|d_{0}$, the spectral measures of the restriction
of the filtration $\mathcal{F}$ to the submodules $xA$ and $A$ compare as
follows. For any convex non-increasing function $\phi$ on , there is $C>0$,
such that for any $k\in\mathbb{N}^{*}$, we have

$\Big{|}\int\phi\cdot dH^{kd}_{*}(\eta(\mathcal{F},(xA)_{kd}))-\int\phi\cdot
dH^{kd}_{*}(\eta(\mathcal{F},A_{kd}))\Big{|}\leq\frac{C}{k}.$ (4.14)

###### Proof.

In order to prove this, remark that since the ${\rm{Frac}}(A)$-vector space
${\rm{Frac}}(A)\otimes_{A}R_{(d)}(Y,L)$ is finitely generated, there is
$N\in\mathbb{N}^{*}$, $N\leq r+1$, and homogeneous elements $a_{0}\in
A_{d_{1}},a_{1}\in A_{d_{1}+d_{0}},\ldots,a_{N}\in A_{d_{1}+Nd_{0}}$,
$a_{N}\neq 0$, for some $d_{1}\in\mathbb{N}^{*}$, such that $d|d_{1}$ and
$a_{0}x^{N}+a_{1}x^{N-1}+\cdots+a_{N}=0.$ From [7, (61)], for any non-
increasing function $\psi$ on , we have

$\int\psi\cdot d\eta(\mathcal{F},A_{kd})\geq\int\psi\cdot
d\eta(\mathcal{F},(xA)_{kd+d_{0}})\geq\int\psi\cdot
d\eta(\mathcal{F},(a_{N}A)_{kd+d_{1}+Nd_{0}}).$ (4.15)

Now, remark that the Krull dimension of the $A$-module $A/a_{N}A$, is strictly
smaller that the Krull dimension of $A$, as $a_{N}\neq 0$. Hence, as in
(4.13), we deduce that for any bounded function $\psi$ on , there is $C>0$,
such that we have

$\Big{|}\int\psi\cdot d\eta(\mathcal{F},(a_{N}A)_{kd})-\int\psi\cdot
d\eta(\mathcal{F},A_{kd})\Big{|}\leq\frac{C}{k}.$ (4.16)

Also, for any Lipschitz function $\psi$ on , and any $d_{3}\in\real$, by the
boundness of the filtration $\mathcal{F}$, there is $C>0$, such that for any
$k\in\mathbb{N}^{*}$, we have

$\Big{|}\int\psi\cdot dH^{kd}_{*}\eta(\mathcal{F},A_{kd})-\int\psi\cdot
dH^{kd+d_{3}}_{*}\eta(\mathcal{F},A_{kd})\Big{|}\leq\frac{C}{k}.$ (4.17)

From (4.15), (4.16), (4.17) and the fact that the limit, as $k\to\infty$, of
$\int\psi\cdot dH^{kd}_{*}\eta(\mathcal{F},A_{kd})$ exists, which follows from
Proposition 4.9, we deduce (4.14). ∎

We will now explain that a part of the above reasoning can be done in a family
setting. More precisely, consider a surjective flat holomorphic map
$\pi\mathrel{\mathop{\ordinarycolon}}X\to B$ between complex projective
manifolds $X$ and $B$ of dimensions $n+m$ and $m$. Let $L$ be a holomorphic
line bundle over $X$, which is relatively ample with respect to $\pi$. We
argue that there is a subset $S\subset B$, which is a countable union of
analytic subsets, and $d^{*}\in\mathbb{N}$, such that the following holds. For
any $b\in B\setminus S$, there is $d(b)\in\mathbb{N}^{*}$, verifying $d(b)\leq
d^{*}$, and $s_{1}(b),\ldots,s_{n+1}(b)\in H^{0}(X_{b},L|^{d(b)}_{X_{b}})$,
which are algebraically independent in the section ring,
$R(X_{b},L|_{X_{b}})$, and such that $R_{(d(b))}(X_{b},L|_{X_{b}})$ is a
finitely generated module over the (polynomial) subring $A(b)\subset
R_{(d(b))}(X_{b},L|_{X_{b}})$ induced by $s_{1}(b),\ldots,s_{n+1}(b)$.
Moreover, there are $r^{*},d_{0}^{*}\in\mathbb{N}$ and $C>0$, such that for
any $b\in B\setminus S$, there are $r(b),d_{0}(b)\in\mathbb{N}^{*}$, verifying
$r(b)\leq r^{*},d_{0}(b)\leq d_{0}^{*}$, and $x_{1}(b),\ldots,x_{r(b)}(b)\in
H^{0}(X_{b},L|^{d_{0}(b)}_{X_{b}})$, such that the $A(b)$-module
$B(b)\mathrel{\mathop{\ordinarycolon}}=x_{1}(b)A(b)+\cdots+x_{r(b)}(b)A(b)$ is
a free $A(b)$-submodule of $R_{(d(b))}(X_{b},L|_{X_{b}})$, such that

$\dim H^{0}(X_{b},L|^{d(b)k}_{X_{b}})-\dim B(b)_{{d(b)k}}\leq Ck^{n-1}.$
(4.18)

To see this, we first recall that in an arbitrary vector bundle, the rank of a
set of holomorphic sections at a given point is a lower semicontinuous
function (in Zariski topology). We pick an arbitrary $b_{0}\in S$, and
construct the sections $s_{1}(b_{0}),\ldots,s_{n+1}(b_{0})\in
H^{0}(X_{b_{0}},L|^{d(b_{0})}_{X_{b_{0}}})$ by Noether normalization lemma as
we did above. Extend them holomorphically (as sections of
$R^{0}\pi_{*}L^{d(b_{0})}$) to $s_{1}(b),\ldots,s_{N(b_{0})}(b)\in
H^{0}(X_{b},L|^{d(b_{0})}_{X_{b}})$ for $b$ in a small neighborhood $U$ of
$b_{0}$.

We argue that away from a countable union of analytic subsets in $U$, the
sections $s_{1}(b),\ldots,s_{n+1}(b)\in H^{0}(X_{b},L|^{d(b_{0})}_{X_{b}})$
are algebraically independent. Indeed, this follows from the above lower
semicontinuity property, as for any $k\in\mathbb{N}^{*}$, the set of
$k$-homogeneous monomials in $s_{1}(b),\ldots,s_{n+1}(b)$, $b\in U$, is free
away from an analytic subset (which is a proper subset of $U$, as it doesn’t
contain $b_{0}$).

Now, we fix homogeneous elements $x_{1}(b_{0}),\ldots,x_{r(b_{0})}(b_{0})$, in
$R_{(d(b_{0}))}(X_{b_{0}},L|_{X_{b_{0}}})$, such that the associated
$A(b_{0})$-module
$B(b_{0})\mathrel{\mathop{\ordinarycolon}}=x_{1}(b_{0})A(b_{0})+\cdots+x_{r(b_{0})}(b_{0})A(b_{0})$
is free, and there is $C>0$, such that for any $k\in\mathbb{N}^{*}$, we have
(4.18) for $b\mathrel{\mathop{\ordinarycolon}}=b_{0}$. We extend
$x_{1}(b_{0}),\ldots,x_{r(b_{0})}(b_{0})$ holomorphically in a neighborhood of
$b_{0}$. A similar argument to the one before shows that for any $b$ in this
neighborhood away from a countable union of analytic subsets, the
$A(b)$-module $B(b)$, defined analogously to $B(b_{0})$, is free. Then, since
by the flatness, $\dim H^{0}(X_{b},L|^{d(b)k}_{X_{b}})$ is a locally constant
function of $b\in S$, cf. [24, Theorem III.9.9], we conclude that inequality
(4.18) holds for $b$ in this neighborhood away from a countable union of
analytic subsets.

We then repeat this procedure for every point $b_{0}\in B$, and use the
compactness of $B$, to extract a finite cover of $B$ by open subsets. In
conclusion, a direct repetition of the argument in the proof of Proposition
4.10 yields the following result.

######  Proposition 4.14.

There is a countable union of proper analytic subsets $S$ in $B$, such that
the following holds. For any $D>0$, any sequence $b_{l}\in B\setminus S$,
$l\in\mathbb{N}$, any family of submultiplicative filtrations $\mathcal{F}(l)$
on $R(X_{b_{l}},L|_{X_{b_{l}}})$, which is uniformly bounded in the sense that
the absolute value of the maximal and the minimal jumping numbers on degree
$k$-part are bounded by $Dk$ (uniformly on $l$), and any convex Lipshitz
function $\phi$ on , there is $C>0$, such that for any sufficiently divisible
$k\in\mathbb{N}^{*}$, and any $l\in\mathbb{N}^{*}$, we have

$\Big{|}\int\phi\cdot
dH^{k}_{*}(\eta(\mathcal{F}(l),A_{k}(b_{l})))-\int\phi\cdot
dH^{k}_{*}(\eta(\mathcal{F}(l),H^{0}(X_{b_{l}},L|^{k}_{X_{b_{l}}})))\Big{|}\leq\frac{C}{k}.$
(4.19)

Finally, to finish the argument, we need the following statement.

######  Proposition 4.15.

There is $C>0$, such that for any $k\in\mathbb{N}^{*}$,
$l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$, for any generic regular
curve $C$, which is a complete intersection in $l[\omega_{B}]$, we have

$\mu^{HN}_{\min}(E_{k}|_{C},l)\geq-Ckl_{1}\cdots
l_{m-1},\qquad\mu^{HN}_{\max}(E_{k}|_{C},l)\leq Ckl_{1}\cdots l_{m-1}.$ (4.20)

###### Proof.

The result now follows from the linear (in $k$) bounds on
$\mu^{HN}_{\min}(E_{k}|_{C},(1,\ldots,1))$,
$\mu^{HN}_{\max}(E_{k}|_{C},(1,\ldots,1))$, cf. Proposition 2.10, and
Proposition 3.17. ∎

###### Proof of Theorem 4.7.

We denote by $S$ the subset from Proposition 4.14. For any
$l=(l_{1},\ldots,l_{m-1})\in\mathbb{N}^{*(m-1)}$, consider a generic regular
curve $C=C(l)\subset B$, which is a complete intersection in $l[\omega_{B}]$,
and which is not contained in $S$. We consider an arbitrary point $b_{l}\in
C(l)\setminus S$, and denote by $\mathcal{F}(l)$ the filtration on
$R(X_{b_{l}},L|_{X_{b_{l}}})$ induced by the Harder-Narasimhan filtration on
$E_{k}|C$ with slopes, divided by $l_{1}\cdots l_{m-1}$. Then by Proposition
4.15, the family of filtrations $\mathcal{F}(l)$ is uniformly bounded in the
sense of Proposition 4.14. By the results of Section 2, the filtrations
$\mathcal{F}(l)$ are submultiplicative. Then by Propositions 4.10 and 4.14,
the decomposition (4.5) holds for the sequence $a_{k,l}=k\int\phi
dH^{k}_{*}(\eta(\mathcal{F}(l),A_{k}(b_{l})))$. ∎

##  5 Numerical expressions for asymptotic slopes

The main goal of this section is to prove Corollary 1.4. For this, let us
first recall the results of Xu-Zhuang from [48]. We conserve the notations
from the introduction, and assume for the moment that $\dim B=1$. For a class
$\alpha\in H^{1,1}(X)$, we then define the nef (resp. psef) threshold of
$\alpha$ with respect to $\pi$ as the supremum over all $T\in\real$, such that
$\alpha-T(\pi^{*}[\omega_{B,0}]/\int_{B}[\omega_{B,0}])$ is nef (resp. psef)
for some Kähler class $[\omega_{B,0}]$ on $B$.

######  Proposition 5.1.

If $\dim B=1$, then $\eta_{\min}^{HN}$ (resp. $\eta_{\max}^{HN}$) equals to
nef (resp. psef) threshold of $c_{1}(L)$ with respect to $\pi$. Moreover, we
have $\eta_{\min}^{HN}=\operatorname*{ess\,inf}\eta^{HN}$.

###### Proof.

In [48, Proposition 2.28], authors proved that
$\operatorname*{ess\,sup}\eta^{HN}$ equals to psef threshold of $c_{1}(L)$
with respect to $\pi$. From this and Theorem 2.2, we deduce the statement of
Proposition 5.1 for $\eta_{\max}^{HN}$. In the proof of [48, Lemma 2.26],
authors established that $\eta_{\min}^{HN}$ is not smaller than nef threshold
of $c_{1}(L)$ with respect to $\pi$. This finishes the proof of Proposition
5.1, as by Theorem 2.2, we have
$\eta_{\min}^{HN}\leq\operatorname*{ess\,inf}\eta^{HN}$, and by [48,
Proposition 2.28], $\operatorname*{ess\,inf}\eta^{HN}$ equals to nef threshold
of $c_{1}(L)$ with respect to $\pi$. ∎

Let us now deduce formulas (1.5) from Proposition 5.1. By the definition of
nefness, we see that the nef threshold of $c_{1}(L)$ with respect to $\pi$
equals

$\inf_{C\in\mathcal{C}_{irr}}\frac{\int_{C}c_{1}(L)\cdot\int_{C}\pi|_{C}^{*}[\omega_{B,0}]}{\int_{B}[\omega_{B,0}]},$
(5.1)

where $\mathcal{C}_{irr}$ was defined before (1.5). This gives us the first
formula from (1.5) by the definition of the topological degree. The proof of
the second formula is identical, except that it is necessary to use the
characterization of the dual to the pseudoeffective cone in terms of movable
curves due to Boucksom-Demailly-Paun-Peternell [5, Theorem 2.2].

###### Proof of Corollary 1.4.

Directly from Proposition 5.1, we have

$\eta_{\min,l}^{HN|C}=\sup\bigg{\\{}T\in\real\mathrel{\mathop{\ordinarycolon}}\text{
the class
}c_{1}(L)|_{C}-T\cdot\frac{\pi^{*}[\omega_{B,0}]|_{C}}{\int_{C}[\omega_{B,0}]}\text{
is nef}\bigg{\\}},$ (5.2)

where the curves $C=C(l)$ are as in Theorem 1.2. Similar formulas hold for the
maximal asymptotic slope. We deduce Corollary 1.4 by (5.2), the identity
$\int_{C}[\omega_{B,0}]=l^{m-1}\cdot\int_{B}[\omega_{B,0}]\cdots[\omega_{B,m-1}]$
and Theorem 1.2. ∎

Let us now give another interpretation of Corollary 1.4. We say that a
relatively ample $\mathbb{Q}$-line bundle $L$ on $X$ is stably
$([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically fibered nef with
respect to $\pi$ if for some (or any) ample line bundle $L_{0}$ on $X$, for
any $\epsilon>0$, $\epsilon\in\mathbb{Q}$, the $\mathbb{Q}$-line bundle
$L\otimes L_{0}^{\epsilon}$ is
$([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically fibered nef with
respect to $\pi$.

######  Proposition 5.2.

A relatively ample line bundle $L$ is stably
$([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically fibered nef with
respect to $\pi$ if and only if $\eta_{\min}^{HN}\geq 0$.

###### Proof.

We assume first that $\eta_{\min}^{HN}\geq 0$. By Corollary 1.4, for any
$\epsilon>0$, $\epsilon\in\mathbb{Q}$, the $\mathbb{Q}$-line bundle
$L\otimes\pi^{*}L_{1}^{\epsilon}$ is
$([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically fibered nef with
respect to $\pi$. But then for any ample line bundle $L_{0}$ on $X$, any
$\epsilon>0$, $\epsilon\in\mathbb{Q}$, the $\mathbb{Q}$-line bundle $L\otimes
L_{0}^{\epsilon}$ is $([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically
fibered nef with respect to $\pi$, which means that $L$ is stably
$([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically fibered nef with
respect to $\pi$.

Inversely, assume that $L$ is stably
$([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically fibered nef with
respect to $\pi$. Let $N\in\mathbb{N}^{*}$ be such that the line bundle
$L_{0}\mathrel{\mathop{\ordinarycolon}}=L\otimes\pi^{*}L_{1}^{N}$ is ample
(such $N$ exists since $L$ is relatively ample). Then for any $\epsilon>0$,
$\epsilon\in\mathbb{Q}$, the $\mathbb{Q}$-line bundle $L\otimes
L_{0}^{\epsilon}$ is $([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically
fibered nef with respect to $\pi$. However, since $L\otimes
L_{0}^{\epsilon}=L^{1+\epsilon}\otimes\pi^{*}L_{1}^{N\epsilon}$, we see that
the $([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generic fibered nef threshold
of $c_{1}(L)$ is non-negative, which by Corollary 1.4 means that
$\eta_{\min}^{HN}\geq 0$, and this finishes the proof. ∎

######  Remark 5.3.

From the proof, we see that in the definition of stably
$([\omega_{B,1}],\ldots,[\omega_{B,m-1}])$-generically fibered nefness with
respect to $\pi$, instead of ample line bundle $L_{0}$ over $X$, we can
consider the pull-back of an ample line bundle over the base.

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Siarhei Finski, CNRS-CMLS, École Polytechnique F-91128 Palaiseau Cedex,
France.

E-mails<EMAIL_ADDRESS>or<EMAIL_ADDRESS>