# Intersection theory and Chern classes on normal varieties

Adrian Langer

###### Abstract

We study intersection theory and Chern classes of reflexive sheaves on normal
varieties. In particular, we define generalization of Mumford’s intersection
theory on normal surfaces to higher dimensions. We also define and study the
second Chern class for reflexive sheaves on normal varieties. We use these
results to prove some Bogomolov type inequalities on normal varieties in
positive characteristic. We also prove some new boundedness results on normal
varieties in positive characteristic.

Address:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097
Warszawa, Poland
e-mail<EMAIL_ADDRESS>
## Introduction

Let $X$ be a normal projective variety of dimension $n$ defined over an
algebraically closed field $k$. It is well known how to intersect one Weil
divisor $D$ with a collection of $(n-1)$ line bundles $L_{1},...,L_{n-1}$. The
intersection number is simply the degree of the $0$-cycle
$c_{1}(L_{1})\cap...\cap c_{1}(L_{n-1})\cap[D]$. Mumford introduced also a
${\mathbb{Q}}$-valued intersection theory for two Weil divisors in case $X$ is
a surface (see [Fu, Examples 7.1.16 and 8.3.11]). To do so he passes to
resolution of singularities $f:\tilde{X}\to X$, defines the pullback $f^{*}D$
of a Weil divisor $D$ as a ${\mathbb{Q}}$-divisor and then intersects the
pullbacks of Weil divisors on $\tilde{X}$.

M. Enokizono in [En, Appendix] used a similar method for higher dimensional
varieties considering very special pullback taking into account only
exceptional divisors with codimension $2$ centers. His approach uses existence
of resolution of singularities in the characteristic zero case and de Jong’s
alterations [dJ] in positive characteristic.

We give a different approach to this result working directly on the singular
variety. We make some computations in the Grothendieck group of $X$, similar
to Kleiman’s approach to intersection numbers for Cartier divisors (see [Ko,
VI.2]). Using some asymptotic Riemann–Roch formulas, we also reconstruct
Mumford’s intersection theory on surfaces without passing to resolution of
singularities. The final outcome is the following result (see Section 2 for
more precise results).

######  Theorem 0.1.

For any Weil divisors $D_{1}$ and $D_{2}$ on $X$ there exists a unique
${\mathbb{Z}}$-multilinear symmetric form ${\mathop{\rm
Pic\,}}X^{\times(n-2)}\to{\mathbb{Q}}$, $(L_{1},...,L_{n-2})\to
D_{1}.D_{2}.L_{1}...L_{n-2}$ such that:

  1. 1.

If $D_{1}$ and $D_{2}$ are Cartier divisors then

$D_{1}.D_{2}.L_{1}...L_{n-2}=\int_{X}c_{1}({\mathcal{O}}_{X}(D_{1}))\cap
c_{1}({\mathcal{O}}_{X}(D_{2}))\cap c_{1}(L_{1})\cap...\cap
c_{1}(L_{n-2})\cap[X].$

  2. 2.

If $D_{2}$ is a Cartier divisor then

$D_{1}.D_{2}.L_{1}...L_{n-2}=\int_{X}c_{1}({\mathcal{O}}_{X}(D_{2}))\cap
c_{1}(L_{1})\cap...\cap c_{1}(L_{n-2})\cap[D_{1}]\in{\mathbb{Z}}.$

  3. 3.

If $L_{1},...,L_{n-2}$ are very ample then for a general complete intersection
surface $S\in|L_{1}|\cap...\cap|L_{n-2}|$ we have

$D_{1}.D_{2}.L_{1}...L_{n-2}=(D_{1})_{S}.(D_{2})_{S}.$

The main aim of the paper is to see wether one can similarly generalize the
theory of Chern classes of vector bundles to reflexive sheaves on normal
varieties. More precisely, if $X$ is smooth then we have a well defined Chern
character from the Grothendieck group of vector bundles to the Chow ring. This
is a homomorphism of rings that satisfies various remarkable properties like
the Riemann–Roch theorem (see [Fu, Chapter 15]). Using operational Chern
classes, the Riemann–Roch theorem can be also proven for vector bundles on
singular varieties (see [BFM]) but there is no ring structure on the direct
sum of Chow groups. However, if $X$ is a normal projective surface then one
can use Mumford’s intersection theory to define rational Chow ring structure.
In that case one can ask wether there exists a well defined Chern character
from the Grothendieck group of reflexive sheaves on $X$ to this rational (or
real) Chow ring. We provide a definition of such Chern character, that
conjecturally is also a ring homomorphism (see Subsection 5.5). To do so we
revise the theory of Chern classes of reflexive sheaves on normal surfaces
that was studied in [La1] in the complex analytic setup. Here we develop an
algebraic approach that works in an arbitrary characteristic. In particular,
we prove the following theorem (see Definition 3.1 and Proposition 4.2).

######  Theorem 0.2.

For any normal proper algebraic surface $X$ defined over an algebraically
closed field $k$ and any coherent reflexive ${\mathcal{O}}_{X}$-module
${\mathcal{E}}$ on $X$ one can define its second Chern class
$c_{2}({\mathcal{E}})\in A_{0}(X)\otimes{\mathbb{R}}$ so that the following
conditions are satisfied:

  1. 1.

If ${\mathcal{E}}$ has rank $1$ then $c_{2}({\mathcal{E}})=0$.

  2. 2.

If ${\mathcal{E}}$ is a vector bundle on $X$ then this class coincides with
$c_{2}({\mathcal{E}})\cap[X].$

  3. 3.

If $\pi:Y\to X$ is a finite morphism from a normal surface $Y$ then

$\int_{Y}c_{2}(\pi^{[*]}{\mathcal{E}})=\deg\pi\cdot\int_{X}c_{2}({\mathcal{E}}),$

where $\pi^{[*]}{\mathcal{E}}$ is the reflexive hull of
$\pi^{*}{\mathcal{E}}$.

In the above theorem $A_{0}(X)$ stands for the group of $0$-dimensional cycles
modulo rational equivalence on $X$. Our approach to the above theorem is along
similar lines as in [La1] but we give simplified and more detailed versions of
several proofs. We also obtain the following result about additional terms in
the Riemann–Roch theorem (see Theorems 3.12 and 4.4).

######  Theorem 0.3.

Let $X$ be a normal proper algebraic surface defined over an algebraically
closed field. Then there exists a constant $C$ depending only on $X$ such that
for every rank $r$ coherent reflexive ${\mathcal{O}}_{X}$-module
${\mathcal{E}}$ on $X$ we have

$\left|\chi(X,{\mathcal{E}})-\left(\frac{1}{2}c_{1}({\mathcal{E}}).(c_{1}({\mathcal{E}})-K_{X})-\int_{X}c_{2}({\mathcal{E}})+r\chi(X,{\mathcal{O}}_{X})\right)\right|\leq
Cr^{2}.$

We use this result to construct a good theory for the second Chern class (or
character) for reflexive sheaves in higher dimensions. This works well in case
of positive characteristic or for varieties with at most quotient
singularities in codimension $\geq 2$ in characteristic $0$. In particular,
our theory in the characteristic zero case generalizes the one developed in
[GKPT, Section 3].

Here we state one of the main results in positive characteristic (see Theorem
5.2). Let us mention that till now the theory of Chern classes of reflexive
sheaves on singular varieties that are well-behaved under coverings (e.g.,
Mumford’s theory of Chern classes of ${\mathbb{Q}}$-sheaves) was always
restricted to the characteristic zero case.

######  Theorem 0.4.

Assume that $k$ has positive characteristic. For any normal projective variety
$X/k$ and for any coherent reflexive ${\mathcal{O}}_{X}$-module
${\mathcal{E}}$ on $X$ there exists a ${\mathbb{Z}}$-multilinear symmetric
form $\int_{X}c_{2}({\mathcal{E}}):{\mathop{\rm
Pic\,}}X^{\times(n-2)}\to{\mathbb{R}}$ such that:

  1. 1.

If ${\mathcal{E}}$ is a vector bundle on $X$ then

$\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}=\int_{X}c_{2}({\mathcal{E}})\cap
c_{1}(L_{1})\cap...\cap c_{1}(L_{n-2})\cap[X].$

  2. 2.

If $k\subset K$ is an algebraically closed field extension then

$\int_{X_{K}}c_{2}({\mathcal{E}}_{K})(L_{1})_{K}...(L_{n-2})_{K}=\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}.$

  3. 3.

If $n>2$ and $L_{1}$ is very ample then for a very general hypersurface
$H\in|L_{1}|$ we have

$\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}=\int_{H}c_{2}({\mathcal{E}}|_{H})L_{2}|_{H}...L_{n-2}|_{H}.$

It is easy to see that this second Chern class is uniquely determined by
equality

$\int_{X}c_{2}(F_{X}^{[*]}{\mathcal{E}})L_{1}...L_{n-2}=p^{2}\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}$

for the Frobenius morphism $F_{X}$, and by the Riemann–Roch type inequality
analogous to that from Theorem 0.3 (see Remark 5.6). This suggests that one
can define $\int_{X}{\mathop{\rm ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}$ as
the limit

$\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}{p^{2m}}.$

The main problem in the proof is to show that such a limit exists.

We give some applications of the obtained results. For example, we prove a
general Bogomolov type inequality (see Theorem 6.5), Bogomolov’s inequality
for strongly semistable reflexive sheaves on normal varieties (see Corollary
6.6) and various boundedness results (see, e.g., Corollary 6.7). As a final
application we show a quick proof of certain boundedness theorem by Esnault
and Srinivas on F-divided sheaves on normal varieties in positive
characteristic (see [ES, Theorem 2.1]).

Applications of the above theory to general restriction theorems and
Bogomolov’s inequality (also for Higgs sheaves) can be found in [La4]. Other
applications to non-abelian Hodge theory and Simpson’s correspondence for
singular varieties in positive characteristic are contained in [La5]. After
the first version of the paper was written, there appeared another application
to the abundance conjecture for threefolds in positive characteristic (see
[Xu]).

The structure of the paper is as follows. In the first section we gather some
auxiliary results. In Section 2 we study intersection theory on normal
varieties and in particular we prove Theorem 0.1. In Section 3 we study
relative Chern classes for resolutions of surface singularities. Then in
Section 4 we use them to define Chern classes for reflexive sheaves on normal
proper surfaces. In Section 5 we extend these results to higher dimensions
proving Theorem 0.4. Finally in Section 6 we give some applications of the
obtained results.

### Notation

Let $X$ be an integral normal locally Noetherian scheme. We say that an open
subset $U\subset X$ is _big_ if every irreducible component of $X\backslash U$
has codimension $\geq 2$ in $X$. A vector bundle on $X$ is a finite locally
free ${\mathcal{O}}_{X}$-module. The category of vector bundles on $X$ will be
denoted by ${\mathop{{\rm Vect\,}}}(X)$.

If $X$ is an algebraic variety defined over some algebraically closed field
$k$ then we write $Z^{1}(X)$ for the group of Weil divisors on $X$. We write
$A_{m}(X)$ for the group of $m$-dimensional cycles modulo rational equivalence
on $X$ (see [Fu, 1.3]). By abuse of notation we also write $A^{m}(X)$ for
$A_{\dim X-m}(X)$.

##  1 Preliminaries

###  1.1 Reflexive sheaves

In this subsection $X$ is an integral locally Noetherian scheme. Let
${\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ be the category of
coherent reflexive ${\mathcal{O}}_{X}$-modules. It is a full subcategory of
the category ${\mathop{\operatorname{Coh}\,}}({\mathcal{O}}_{X})$ of coherent
${\mathcal{O}}_{X}$-modules. The inclusion functor
${\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})\to{\mathop{\operatorname{Coh}\,}}({\mathcal{O}}_{X})$
comes with a left adjoint
$(\cdot)^{**}:{\mathop{\operatorname{Coh}\,}}({\mathcal{O}}_{X})\to{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$
given by the reflexive hull. The category
${\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ is an additive category
with kernels and cokernels and it comes with an associative and symmetric
tensor product $\hat{\otimes}$ defined by

${\mathcal{E}}\hat{\otimes}{\mathcal{F}}:=({\mathcal{E}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}})^{**}.$

The following well-known lemma can be found in [SP, Lemma 0EBJ].

######  Lemma 1.1.

Let $j:U\hookrightarrow X$ be an open subscheme with complement $Z$ such that
the depth of ${\mathcal{O}}_{X,z}$ is $\geq 2$ for all $z\in Z$. Then $j_{*}$
and $j^{*}$ define adjoint equivalences of categories
${\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ and
${\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{U})$. In particular, the above
assumptions are satisfied if $U$ is a big open subset of an integral locally
Noetherian normal scheme $X$.

Let us also recall the following standard lemma (see, e.g., [SP, Tag 0EBF]).

######  Lemma 1.2.

Let $f:X\to Y$ be a flat morphism of integral locally Noetherian schemes. If
${\mathcal{F}}$ is a coherent reflexive ${\mathcal{O}}_{Y}$-module then
$f^{*}{\mathcal{F}}$ is reflexive on $X$.

In general, pull back of a reflexive sheaf need not be reflexive. If $f:X\to
Y$ is a morphism between normal schemes and ${\mathcal{E}}$ is a coherent
reflexive ${\mathcal{O}}_{Y}$-module then we set

$f^{[*]}{\mathcal{E}}=(f^{*}{\mathcal{E}})^{**}.$

If ${\mathcal{F}}$ is a coherent reflexive ${\mathcal{O}}_{X}$-module then we
set

$f_{[*]}{\mathcal{F}}=(f_{*}{\mathcal{F}})^{**}.$

Note that in general these operations define functors on reflexive sheaves
that are not functorial with respect to morphisms.

If $X$ is normal and
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ then we
also use the notation

${\mathop{{\rm Sym\,}}}^{[m]}{\mathcal{E}}:=({\mathop{{\rm
Sym\,}}}^{m}{\mathcal{E}})^{**}$

for ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$.

If $X$ is a normal scheme of finite type over an algebraically closed field
$k$ of characteristic $p>0$ then $F_{X}:X\to X$ denotes the absolute Frobenius
morphism. Let us recall that by Kunz’s theorem $F_{X}$ is flat only at regular
points of $X$. If
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ then for
every $m$ we set

$F_{X}^{[m]}{\mathcal{E}}:=(F_{X}^{m})^{[*]}{\mathcal{E}}.$

The following well-known lemma is a corollary of Serre’s criterion on
normality.

######  Lemma 1.3.

Let us assume that $X$ is normal. If ${\mathcal{E}}$ is a coherent torsion
free ${\mathcal{O}}_{X}$-module then the canonical map
${\mathcal{E}}\to{\mathcal{E}}^{**}$ is an isomorphism in codimension $1$,
i.e., it is injective and every irreducible component of the support of its
cokernel has codimension $\geq 2$ in $X$.

Let $X$ be an excellent integral normal scheme (this is the situation in which
we will use the remark below). Then the regular locus $U:=X_{\mathrm{reg}}$ is
open and the above lemma implies that
$j^{*}:{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})\to{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{U})$
and
$j_{*}:{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{U})\to{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$
are adjoint equivalences of categories, where $j:U\hookrightarrow X$ denotes
the open embedding.

###  1.2 Mumford’s intersection theory on normal surfaces

Let $X$ be a normal proper surface defined over an algebraically closed field
and let $f:\tilde{X}\to X$ be any resolution of singularities. For any Weil
divisor $D$ on $X$ one can define its pull back $f^{*}D\in
A^{1}(\tilde{X})\otimes{\mathbb{Q}}$ as the only class of a
${\mathbb{Q}}$-divisor for which $f_{*}f^{*}D=D$ and
$\int_{\tilde{X}}[f^{*}D]\cap[E_{i}]=0$ for all exceptional curves $E_{i}$
(cf. [Fu, Example 8.3.11]). For any two Weil divisors $D_{1}$ and $D_{2}$ we
set

$[D_{1}]\cap[D_{2}]:=f_{*}([f^{*}D_{1}]\cap[f^{*}D_{2}]).$

This defines a ${\mathbb{Z}}$-bilinear symmetric form $A^{1}(X)\times
A^{1}(X)\to A_{0}(X)\otimes{\mathbb{Q}}$,
$(D_{1},D_{2})\to[D_{1}]\cap[D_{2}]$. In the following we write $D_{1}.D_{2}$
for the rational number $\int_{X}[D_{1}]\cap[D_{2}]$.

###  1.3 Bertini’s theorem

Here we recall some Bertini type theorems. In particular, we have Bertini’s
theorem for smoothness, irreducibility and reducedness for unramified
morphisms (see [Jo, Theoreme 6.3 ]) and Seidenberg’s Bertini’s theorem for
normality for embeddings into the projective space (see [Se, Theorem 7’]; see
also [HL, Corollary 1.1.15] for a quick proof). However, we need these
theorems in a slightly more general set-up and it is convenient to state them
uniformly using [CGM, Theorem 1 and Remark below Corollary 2].

Let ${\mathcal{P}}$ be one of the following properties of a locally noetherian
scheme: being regular, normal, geometrically reduced or geometrically
irreducible. Then we have the following result:

######  Theorem 1.4.

Let $X$ be a scheme of finite type over an algebraically closed field and let
$\varphi:X\to{\mathbb{P}}^{n}_{k}$ be a morphism defined by a linear system
$\Lambda$. Let us assume that $\varphi$ has separably generated residue field
extensions. If $X$ has property ${\mathcal{P}}$ then there exists a nonempty
Zariski open subset $U\subset\Lambda$ such that every hypersurface
$H\in\Lambda$ also has property ${\mathcal{P}}$.

As usual we will quote this theorem by saying that a general $H\in\Lambda$ has
property ${\mathcal{P}}$.

###  1.4 Some results on Grothendieck’s group of a normal variety

Let $X$ be a normal projective variety of dimension $n$ defined over an
algebraically closed field $k$. Let $K(X)$ denote the Grothendieck’s group of
$X$. For any $0\leq m\leq n$ we denote by $K_{m}(X)$ the subgroup of $K(X)$
generated by classes of sheaves, whose support has dimension at most $m$. For
a line bundle $L$ on $X$ we have an additive endomorphism of $K(X)$ defined by

${\mathcal{F}}\to
c_{1}(L)\cdot{\mathcal{F}}:={\mathcal{F}}-L^{-1}\otimes{\mathcal{F}}$

(see [Ko, Chapter VI, Definition 2.4]).

Let us recall the following result (see [Ko, Chapter VI, Proposition 2.5] and
its proof):

######  Lemma 1.5.

  1. 1.

$c_{1}(L)\cdot K_{m}(X)\subset K_{m-1}(X)$ for every $m$.

  2. 2.

$c_{1}(L_{1})$ and $c_{1}(L_{2})$ commute for any two line bundles $L_{1}$ and
$L_{2}$.

  3. 3.

For any two line bundles $L_{1}$ and $L_{2}$ we have equality of endomorphisms

$c_{1}(L_{1}\otimes L_{2})=c_{1}(L_{1})+c_{1}(L_{2})-c_{1}(L_{1})\cdot
c_{1}(L_{2}).$

  4. 4.

If $Y\subset X$ is integral and $L|_{Y}\simeq{\mathcal{O}}_{Y}(D)$ for some
effective Cartier divisor $D$ on $Y$ then
$c_{1}(L)\cdot{\mathcal{O}}_{Y}={\mathcal{O}}_{D}$.

We will also need the following lemmas:

######  Lemma 1.6.

If $D$ is a Weil divisor on $X$ then $[{\mathcal{O}}_{D}]\in K_{n-1}(X)$.
Moreover, if we write $D=\sum a_{i}D_{i}$ for some Weil divisors $D_{i}$ and
some integers $a_{i}$ then the class $[{\mathcal{O}}_{D}]-\sum
a_{i}[{\mathcal{O}}_{D_{i}}]$ lies in $K_{n-2}(X)$.

###### Proof.

Since $X$ is normal, Weil divisors are Cartier outside of a closed subset of
codimension $\geq 2$. But for any closed subscheme $Y\subset X$ we have a
short exact sequence

$K(Y)\to K(X)\to K(X\backslash Y)\to 0.$

This together with Lemma 1.5, (3) shows that $[{\mathcal{O}}_{D}]-\sum
a_{i}[{\mathcal{O}}_{D_{i}}]$ lies in $K_{n-2}(X)$.

To prove the first part of the lemma let us write $D$ as $D=D_{1}-D_{2}$ for
some effective Weil divisors $D_{1}$ and $D_{2}$. Since $D_{i}$, $i=1,2$ are
effective, we have short exact sequences

$0\to{\mathcal{O}}_{X}(-D_{i})\to{\mathcal{O}}_{X}\to{\mathcal{O}}_{D_{i}}\to
0$

showing that $[{\mathcal{O}}_{D_{i}}]\in K_{n-1}(X)$. Now the first part of
the proof implies that
$[{\mathcal{O}}_{D}]-[{\mathcal{O}}_{D_{1}}]+[{\mathcal{O}}_{D_{2}}]\in
K_{n-2}(X)$, so $[{\mathcal{O}}_{D}]\in K_{n-1}(X)$. ∎

######  Lemma 1.7.

Let ${\mathcal{F}}$ be a coherent torsion free ${\mathcal{O}}_{X}$-module of
rank $r$. Then the class $\alpha({\mathcal{F}})$ of
${\mathcal{F}}-{\mathcal{O}}_{X}^{\oplus
r}+[{\mathcal{O}}_{-c_{1}({\mathcal{F}})}]$ lies in $K_{n-2}(X)$.

###### Proof.

If $r=1$ and ${\mathcal{F}}$ is reflexive then by definition of
$c_{1}({\mathcal{F}})$ we have $\alpha({\mathcal{F}})=0$ in $K(X)$. In
general, if $r=1$ then

$\alpha({\mathcal{F}})=\alpha({\mathcal{F}})-\alpha({\mathcal{F}}^{**})={\mathcal{F}}-{\mathcal{F}}^{**},$
${\mathcal{F}}-{\mathcal{O}}_{X}^{\oplus
r}+[{\mathcal{O}}_{-c_{1}({\mathcal{F}})}]={\mathcal{F}}-{\mathcal{O}}_{X}^{\oplus
r}+[{\mathcal{O}}_{-c_{1}({\mathcal{F}})}]-({\mathcal{F}}^{**}-{\mathcal{O}}_{X}^{\oplus
r}+[{\mathcal{O}}_{-c_{1}({\mathcal{F}})}])={\mathcal{F}}-{\mathcal{F}}^{**},$

so the assertion follows from the fact that the cokernel of
${\mathcal{F}}\to{\mathcal{F}}^{**}$ is supported in codimension $\geq 2$.

If $r>1$ then ${\mathcal{F}}$ has a filtration
${\mathcal{F}}={\mathcal{F}}^{0}\supset{\mathcal{F}}^{1}\supset...\supset{\mathcal{F}}^{r}=0$,
whose quotients ${\mathcal{L}}_{i}={\mathcal{F}}^{i}/{\mathcal{F}}^{i+1}$ are
rank $1$ torsion free sheaves. Then we have

$\alpha({\mathcal{F}})=[{\mathcal{O}}_{-c_{1}({\mathcal{F}})}]-\sum[{\mathcal{O}}_{-c_{1}({\mathcal{L}}_{i})}]+\sum\alpha({\mathcal{L}}_{i}).$

Since $c_{1}({\mathcal{F}})=\sum c_{1}({\mathcal{L}}_{i})$ Lemma 1.6 shows
that
$[{\mathcal{O}}_{-c_{1}({\mathcal{F}})}]-\sum[{\mathcal{O}}_{-c_{1}({\mathcal{L}}_{i})}]$
lies in $K_{n-2}(X)$. Therefore the assertion follows from the rank $1$ case.
∎

##  2 Intersection theory on normal varieties

Let $X$ be a normal projective variety of dimension $n$ defined over an
algebraically closed field $k$. If $D$ is a Weil divisor on $X$ then we write
$[{\mathcal{O}}_{D}]$ for the class of
${\mathcal{O}}_{X}-{\mathcal{O}}_{X}(-D)$ in $K(X)$. We also write $N^{1}(X)$
for the group of line bundles on $X$ modulo numerical equivalence. By the
Néron–Severi theorem of the base, $N^{1}(X)$ is finitely generated free
${\mathbb{Z}}$-module.

###  2.1 Intersection of a Weil divisor with Cartier divisors

In this subsection we define a K-theoretic intersection of a Weil divisor with
Cartier divisors and compare it with standard definition using intersections
with Chern classes of line bundles.

######  Lemma 2.1.

The image of the map $Z^{1}(X)\times{\mathop{\rm Pic\,}}X^{\times(n-1)}\to
K(X)$ defined by

$(D,L_{1},...,L_{n-1})\to c_{1}(L_{1})\cdot...\cdot
c_{1}(L_{n-1})\cdot[{\mathcal{O}}_{D}]$

is contained in $K_{0}(X)$. Moreover, this map is ${\mathbb{Z}}$-linear with
respect to each variable and symmetric with respect $(L_{1},...,L_{n-1})$ to
for fixed Weil divisor $D$.

###### Proof.

Let us write $D=\sum a_{i}D_{i}$ for some prime Weil divisors $D_{i}$ and some
integers $a_{i}$. By Lemma 1.5, (1) we have
$c_{1}(L_{1})...c_{1}(L_{n-1})K_{n-1}(X)\subset K_{0}(X)$ and
$c_{1}(L_{1})...c_{1}(L_{n-1})K_{n-2}(X)=0$. Therefore Lemma 1.6 implies the
first assertion. It also shows that

$c_{1}(L_{1})...c_{1}(L_{n-1})[{\mathcal{O}}_{D}]=\sum
a_{i}c_{1}(L_{1})...c_{1}(L_{n-1})[{\mathcal{O}}_{D_{i}}].$

This proves that the map is ${\mathbb{Z}}$-linear with respect to $D$. By
Lemma 1.5, (2) the map is symmetric for fixed $D$. Moreover, we have

$\displaystyle c_{1}(L_{1}\otimes
M_{1})c_{1}(L_{2})...c_{1}(L_{n-1})\cdot[{\mathcal{O}}_{D}]-c_{1}(L_{1})...c_{1}(L_{n-1})\cdot[{\mathcal{O}}_{D}]-c_{1}(M_{1})c_{1}(L_{2})...c_{1}(L_{n-1})\cdot[{\mathcal{O}}_{D}]$
$\displaystyle=-c_{1}(L_{1})c_{1}(M_{1})c_{1}(L_{2})...c_{1}(L_{n-1})\cdot[{\mathcal{O}}_{D}]=0,$

which finishes the proof that the map is ${\mathbb{Z}}$-linear with respect to
all variables. ∎

We also have a map ${\mathop{\rm Pic\,}}X^{\times(n-1)}\to A_{0}(X)$ defined
by

$(L_{1},...,L_{n-1})\to c_{1}(L_{1})\cap...\cap c_{1}(L_{n-1})\cap[D]$

(see [Fu, Section 2.5]). This map is also symmetric and multilinear (see [Fu,
Proposition 2.5]).

To compare the above maps we can use the map $\psi:K_{0}(X)\to A_{0}(X)$ given
by sending ${\mathcal{F}}$ to $\sum_{x\in X(k)}l_{x}({\mathcal{F}})\,[x]$,
where $l_{x}({\mathcal{F}})$ is the length of ${\mathcal{F}}_{x}$ as an
${\mathcal{O}}_{X.x}$-module (cf. [Fu, Example 18.3.11], where an analogous
map is defined also for cycles of higher dimension but it goes into the Chow
group with rational coefficients). This map is an isomorphism with the inverse
$\varphi:A_{0}(X)\to K_{0}(X)$ given by sending $[x]$ to ${\mathcal{O}}_{x}$.

The following lemma, generalizing the Riemann–Roch theorem on curves, compares
the above maps.

######  Lemma 2.2.

For any Weil divisor $D$ and any line bundles $L_{1}$, …, $L_{n-1}$ we have

$c_{1}(L_{1})\cap...\cap
c_{1}(L_{n-1})\cap[D]=\psi(c_{1}(L_{1})...c_{1}(L_{n-1})\cdot[{\mathcal{O}}_{D}]).$

###### Proof.

By the above we know that both sides of our equality are ${\mathbb{Z}}$-linear
in $L_{i}$. But any line bundle $L$ can be written as $A\otimes B^{-1}$ for
some very ample line bundles $A$ and $B$. So it is sufficient to prove the
above equality assuming that all $L_{i}$ are very ample.

For general divisors $H_{i}\in|L_{i}|$ the complete intersection
$C=H_{1}\cap...\cap H_{n-1}$ is a smooth curve and $D\cap C$ is a $0$-cycle
representing $c_{1}(L_{1})\cap...\cap c_{1}(L_{n-1})\cap[D]$. But by Lemma
1.5, (4) we have equality
$c_{1}(L_{1})...c_{1}(L_{n-1})[{\mathcal{O}}_{D}]={\mathcal{O}}_{D\cap C}$ in
$K(X)$, so the required assertion is clear. ∎

Note that the above defined maps descend to the intersection product
$A^{1}(X)\times({\mathop{\rm Pic\,}}X)^{\times(n-1)}\to A_{0}(X)$ given by
$(D,L_{1},...,L_{n-1})\to c_{1}(L_{1})\cap...\cap c_{1}(L_{n-1})\cap[D]$.
Since $\chi(X,\cdot)=\int_{X}\circ\psi$, the above lemma shows that this
descends to an intersection product $A^{1}(X)\times
N^{1}(X)^{\times(n-1)}\to{\mathbb{Z}}$ given by

$D.L_{1}...L_{n-1}:=\chi(X,c_{1}(L_{1})...c_{1}(L_{n-1})\cdot[{\mathcal{O}}_{D}])=\int_{X}c_{1}(L_{1})\cap...\cap
c_{1}(L_{n-1})\cap[D].$

###  2.2 Intersection of two Weil divisors with Cartier divisors

In this subsection we define intersection number for two Weil divisors and a
collection of Cartier divisors. We assume that $n\geq 2$.

######  Proposition 2.3.

Let $D$ be a Weil divisor and let $L_{1},...,L_{n-2}$ be line bundles on $X$.
Then the sequence

$\left(\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}{m^{2}}\right)_{m\in{\mathbb{N}}}$

is convergent and its limit is a rational number.

###### Proof.

By Lemma 1.5 (3) we have

$\displaystyle{\chi(X,c_{1}(L_{1}\otimes
M_{1})c_{1}(L_{2})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}={\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}$
$\displaystyle+{\chi(X,c_{1}(M_{1})c_{1}(L_{2})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}-{\chi(X,c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}.$

Since

$\displaystyle\lim_{m\to\infty}\frac{\chi(X,c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}{m^{2}}=\lim_{m\to\infty}\frac{mD.c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})}{m^{2}}=0,$

we see that if
$\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}{m^{2}}$
and
$\lim_{m\to\infty}\frac{\chi(X,c_{1}(M_{1})c_{1}(L_{2})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}{m^{2}}$
exist then $\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1}\otimes
M_{1})c_{1}(L_{2})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}{m^{2}}$ exists
and it is equal to the sum of these limits.

Since any line bundle $L$ can be written as $A\otimes B^{-1}$ for some very
ample line bundles $A$ and $B$ and the formula in the sequence is symmetric in
$(L_{1},...,L_{n-2})$, it is sufficient to prove existence of (rational) limit
assuming that all line bundles $L_{i}$ are very ample.

Let us write $D=\sum d_{i}D_{i}$ as an integral combination of prime Weil
divisors $D_{i}$. By Theorem 1.4 for a general sequence $H_{i}\in|L_{i}|$,
$i=1,...,n-2,$ each intersection $X_{j}:=\bigcap_{i\leq j}H_{i}$ is normal and
all $D_{i}\cap X_{j}$ have codimension $1$ in $X_{j}$. So $D_{X_{j}}=\sum
d_{i}(D_{i}\cap X_{j})$ is a well defined Weil divisor on $X_{j}$ and for all
$m\in{\mathbb{Z}}$ and for all $1\leq j\leq n-2$ we have short exact sequences

$0\to{\mathcal{O}}_{X_{j-1}}(mD)\otimes
L_{j}^{-1}\to{\mathcal{O}}_{X_{j-1}}(mD_{X_{j-1}})\to{\mathcal{O}}_{X_{j}}(mD_{X_{j}})\to
0.$

These sequences show that
${\mathcal{O}}_{X_{j}}(mD_{X_{j}})=c_{1}(L_{j})\cdot{\mathcal{O}}_{X_{j-1}}(mD_{X_{j-1}})$
in $K(X)$. So if we set $S:=X_{n-2}$ then
${\mathcal{O}}_{S}(mD_{S})=c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{\mathcal{O}}_{X}({mD})$
in $K(X)$. Now by the Riemann–Roch theorem on $S$ (see Theorems 4.4 and 3.12)
we get

$\displaystyle\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}{m^{2}}$
$\displaystyle=\lim_{m\to\infty}\frac{-\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{\mathcal{O}}_{X}(-mD))}{m^{2}}$
$\displaystyle=\lim_{m\to\infty}\frac{-\chi({S},{\mathcal{O}}_{S}(-mD_{S}))}{m^{2}}=\frac{1}{2}D_{S}^{2},$

where $D_{S}^{2}$ is the self intersection of $D_{S}$ in the sense of
Mumford’s intersection pairing on $S$. ∎

For any Weil divisor $D$ and any line bundles $L_{1},...,L_{n-2}$ we set

$D^{2}.L_{1}...L_{n-2}:=2\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{mD}])}{m^{2}}.$

######  Theorem 2.4.

Let us consider the map $Z^{1}(X)\times Z^{1}(X)\times{\mathop{\rm
Pic\,}}X^{\times(n-2)}\to{\mathbb{Q}}$ defined by sending
$(D_{1},D_{2},L_{1},...,L_{n-2})$ to

$D_{1}.D_{2}.L_{1}...L_{n-2}:=\frac{1}{2}\left((D_{1}+D_{2})^{2}.L_{1}...L_{n-2}-D_{1}^{2}.L_{1}...L_{n-2}-D_{2}^{2}.L_{1}...L_{n-2}\right).$

This map satisfies the following conditions:

  1. 1.

It is ${\mathbb{Z}}$-linear in each variable.

  2. 2.

It is symmetric in $D_{1}$ and $D_{2}$.

  3. 3.

It is symmetric in $L_{1},...,L_{n-2}$.

  4. 4.

If $D_{2}$ is a Cartier divisor then
$D_{1}.D_{2}.L_{1}...L_{n-2}=D_{1}.{\mathcal{O}}_{X}(D_{2})L_{1}...L_{n-2}$.

  5. 5.

If we fix $D_{1},D_{2}\in Z^{1}(X)$ and assume that $n>2$ and $L_{1}$ is very
ample then for a general hypersurface $H_{1}\in|L_{1}|$ we have

$D_{1}.D_{2}.L_{1}...L_{n-2}=(D_{1})_{H_{1}}.(D_{2})_{H_{1}}.L_{2}|_{H_{1}}...L_{n-2}|_{H_{1}}.$

###### Proof.

The fact that the map is ${\mathbb{Z}}$-linear in $L_{i}$ follows from the
first part of the proof of Proposition 2.3. If we fix $D_{1},D_{2}\in
Z^{1}(X)$ and assume that all $L_{1},...,L_{n-2}$ are very ample then for a
general complete intersection surface $S\in|L_{1}|\cap...\cap|L_{n-2}|$ we
have

$D_{1}.D_{2}.L_{1}...L_{n-2}=(D_{1})_{S}.(D_{2})_{S}.$

This follows from the proof of Proposition 2.3 and the fact that on a general
complete intersection surface $S$ we have equality

$(D_{1})_{S}.(D_{2})_{S}=\frac{1}{2}((D_{1}+D_{2})_{S}^{2}-(D_{1})_{S}^{2}-(D_{2}^{2})_{S}.$

Now (5) follows from the above mentioned linearity, which allows us to assume
that also $L_{2},...,L_{n-2}$ are very ample.

(2) is clear and (3) follows from Lemma 1.5, (2). To prove that the map is
${\mathbb{Z}}$-linear in $D_{1}$ and $D_{2}$ we can assume that
$L_{1},...,L_{n-2}$ are very ample. In this case the assertion follows from
(5). In the same way we can reduce (4) to the normal surface case, where the
assertion is clear. ∎

The above theorem implies that we have a well defined induced intersection
form

$A^{1}(X)\times A^{1}(X)\times N^{1}(X)^{\times(n-2)}\to{\mathbb{Q}},$

generalizing Mumford’s intersection pairing on surfaces. Note that our
approach reconstructs Mumford’s intersection pairing on normal surfaces
without using any resolution of singularities.

######  Corollary 2.5.

Let $D_{1},D_{2}$ be Weil divisors and $L_{1},...,L_{n-2}$ line bundles on
$X$. Let $\pi:Y\to X$ is a finite morphism from a normal projective variety
$Y$. Let us assume that $\pi$ has separably generated residue field
extensions. Then we have

$\pi^{*}D_{1}.\pi^{*}D_{2}.\pi^{*}L_{1}...\pi^{*}L_{n-2}=\deg\pi\cdot
D_{1}.D_{2}.L_{1}...L_{n-2}.$

###### Proof.

We can use similar arguments to that from the proof of Proposition 2.3 to
reduce to the case, when all $L_{i}$ are very ample. Then Theorem 1.4 allows
us to reduce to the surface case. In this case the result follows from Remark
4.3. ∎

Unfortunately, it is not clear how to prove the above corollary for arbitrary
finite morphisms in positive characteristic. Clearly, it holds also for the
Frobenius morphism as $F_{X}^{*}D=pD$ for any Weil divisor $D$.

Now let fix very ample line bundles $L_{1},...,L_{n-2}$ and consider a
${\mathbb{Q}}$-valued intersection pairing
$\langle\cdot,\cdot\rangle:A^{1}(X)\times A^{1}(X)\to{\mathbb{Q}}$ by setting

$\langle D_{1},D_{2}\rangle:=D_{1}.D_{2}.L_{1}...L_{n-2}.$

Let us write $N_{L}(X)$ for the quotient of $A^{1}(X)$ by the radical of this
intersection pairing. Then we have an induced non-degenerate intersection
pairing

$\langle\cdot,\cdot\rangle:N_{L}(X)\times N_{L}(X)\to{\mathbb{Q}}.$

If $n=2$ then we write $N(X)$ instead of $N_{L}(X)$.

######  Lemma 2.6.

$N_{L}(X)$ is a free ${\mathbb{Z}}$-module of finite rank. If ${\mathop{\rm
rk\,}}N_{L}(X)=s$ then the intersection pairing $\langle\cdot,\cdot\rangle$
has signature $(1,s-1)$. Moreover, there exists a positive integer $N$ such
that the intersection pairing $\langle\cdot,\cdot\rangle$ takes values in
$\frac{1}{N}{\mathbb{Z}}\subset{\mathbb{Q}}$.

###### Proof.

The lemma is well known in case of normal surfaces. Now let $Y$ be the product
of $(n-2)$ projective spaces $|L_{i}|$ and let us consider the incidence
scheme ${\mathcal{S}}\subset X\times_{k}Y$, whose points are of the form
$(x;H_{1},...,H_{n-2})\in X\times_{k}Y$ with $x\in\bigcap_{j\leq n-2}H_{j}$.
Let $\bar{\eta}:{\mathop{{\rm Spec\,}}}K\to Y$ be a geometric generic point of
$Y$. Then the fiber ${\mathcal{S}}_{K}$ of the projection ${\mathcal{S}}\to Y$
over $\bar{\eta}$ is a normal surface contained in $X\otimes_{k}K$. Note that
if $D$ is a prime Weil divisor on $X$ then ${\mathcal{S}}_{K}$ is not
contained in $D_{K}$. Otherwise, ${\mathcal{S}}$, which is the closure of the
image of ${\mathcal{S}}_{K}$ in $X\times_{k}Y$, would be contained in
$D\times_{k}Y$, a contradiction. So
$D_{{\mathcal{S}}_{K}}=D_{K}\cap{\mathcal{S}}_{K}$ is a prime Weil divisor on
${\mathcal{S}}_{K}$. Now Theorem 2.4 implies that for any two divisors $D_{1}$
and $D_{2}$ on $X$ we have

$\langle
D_{1},D_{2}\rangle=(D_{1})_{{\mathcal{S}}_{K}}.(D_{2})_{{\mathcal{S}}_{K}}.$

Note that if the class of a Weil divisor $D_{1}$ on $X$ is $0$ in $N_{L}(X)$
then the class of $(D_{1})_{{\mathcal{S}}_{K}}$ is $0$ in
$N({\mathcal{S}}_{K})$. This follows from the fact that the class of
$(D_{1})_{{\mathcal{S}}_{K}}$ is $0$ in $N({\mathcal{S}}_{K})$ if and only if
$((D_{1})_{{\mathcal{S}}_{K}})^{2}=0$ and
$(D_{1})_{{\mathcal{S}}_{K}}.H_{{\mathcal{S}}_{K}}=0$ for some ample divisor
$H$ on $X$ (cf. [La4, Lemma 1.15]). It follows that $N_{L}(X)$ is a
${\mathbb{Z}}$-submodule of a free ${\mathbb{Z}}$-module
$N({\mathcal{S}}_{K})$ of finite rank $t$. So $N_{L}(X)$ is also a free
${\mathbb{Z}}$-module of finite rank. This implies that there exists $N$ such
that our intersection pairing takes values in
$\frac{1}{N}{\mathbb{Z}}\subset{\mathbb{Q}}$. To see the signature of
$\langle\cdot,\cdot\rangle$, we note that for any ample divisor $H$ we have
$\langle H,H\rangle>0$ and Mumford’s intersection pairing on
$N({\mathcal{S}}_{K})$ has signature $(1,t-1)$. ∎

###  2.3 Comparison with the de Fernex–Hacon pullback and Enokizono’s
construction

In this subsection we compare the obtained intersection theory with the one
obtained by intersecting de Fernex–Hacon pullbacks of Weil divisors. This
subsection is not needed in the following but we will need to use some results
and notation introduced in further part of the paper.

Let us recall that if $X$ is a normal variety and $f:Y\to X$ is a birational
morphism then for any Weil divisor $D$ on $X$ in [dFH, Definition 2.5] the
authors define $f^{\sharp}D$ so that
${\mathcal{O}}_{X}(-f^{\sharp}D)=f^{[*]}{\mathcal{O}}_{X}(-D)$. If
$f:\tilde{X}\to X$ is a resolution of singularities and $X$ is a proper normal
surface then in the notation of Subsection 4.1 we have

$f^{*}(-mD)=-f^{\sharp}(mD)-\sum_{x\in
f(E)}c_{1}(f_{x},f^{[*]}{\mathcal{O}}_{X}(-mD)),$

where on the left hand side we have Mumford’s pullback. But by Theorem 3.9
there are only finitely many possibilities for
$c_{1}(f_{x},f^{[*]}{\mathcal{O}}_{X}(-mD))$, so by ${\mathbb{Z}}$-linearity
of Mumford’s pullback we have

$f^{*}(D)=\lim_{m\to\infty}\frac{f^{\sharp}(mD)}{m}.$

This shows that in the surface case de Fernex–Hacon pullback of Weil divisors
defined in [dFH, Definition 2.9] coincides with Mumford’s pullback.

In higher dimensions (in characteristic $0$) the de Fernex–Hacon pullback
satisfies $f^{*}(-D)\neq-f^{*}D$, so it is not useful for defining
intersection form on $X$. If one wants to get pullback of Weil divisors
similar to Mumford’s rational pullback, one needs to consider only very
special morphisms (see [Sch]). One can also consider a partial (Mumford’s)
pullback

$f^{*}_{M}D:=\sum_{{\rm codim}\,f(E)\leq 2}a(E)\cdot E,$

where $a(E)$ is the coefficient of $E$ in the de Fernex–Hacon pullback
$f^{*}D$, and the sum is taken over all prime divisors $E$ on $\tilde{X}$ with
centers of codimension $1$ or $2$. Cutting to the surface case and using the
above surface case, we see that this coincides with Enokizono’s construction
of Mumford’s pullback (see [En, Appendix]). Note that even if $X$ is smooth,
the above pullback does not coincide with the usual pullback of Cartier
divisors. However, this partial pullback can be used to define intersection of
two Weil divisors with Cartier divisors.

##  3 Local relative Chern classes for resolutions of normal surfaces

In this section we revise the theory of local relative Chern classes for
resolutions of normal surfaces. This theory was studied in [Wa] in the rank
$2$ case and in [La1] in general, but only in the complex analytic setup. Here
we develop an algebraic approach in an arbitrary characteristic. Our approach
is along similar lines as in [La1] but we give simplified and more detailed
versions of several proofs.

Let $k$ be an algebraically closed field and let $A$ be an excellent normal
$2$-dimensional Henselian local $k$-algebra. The developed theory seems to
work in more general situations as in [Li] but we will need it only for
henselizations of local rings of algebraic surfaces defined over an
algebraically closed field.

Let $X={\mathop{{\rm Spec\,}}}A$ and let $x\in X$ be the closed point of $X$.
Let $f:\tilde{X}\to X$ be a desingularization of $X$, i.e., a proper
birational morphism from a regular surface $\tilde{X}$. Here a surface is a
reduced noetherian separated $k$-scheme of dimension $2$ but we do not assume
that it is of finite type over $k$. Let $E$ be the exceptional locus of $f$
considered with the reduced scheme structure. One can also assume that $f$ is
_good_ , i.e., $E$ is a simple normal crossing divisor, but this will not be
used in the following.

###  3.1 Local relative Chern classes of vector bundles for resolutions of
surfaces

Let ${\mathcal{F}}$ be a vector bundle on $\tilde{X}$. If $\\{E_{i}\\}$ denote
the irreducible components of $E$ then the intersection matrix $[E_{i}.E_{j}]$
of the exceptional divisor is negative definite (see, e.g., [Li, Lemma 14.1]).
So there exists a unique ${\mathbb{Q}}$-divisor $c_{1}(f,{\mathcal{F}})$
supported on $E$ such that for every irreducible component $E_{i}$ of $E$ we
have

$c_{1}(f,{\mathcal{F}}).E_{i}=\deg{\mathcal{F}}|_{E_{i}}.$

We call $c_{1}(f,{\mathcal{F}})$ the _first relative Chern class_ of
${\mathcal{F}}$ with respect to $f$.

Let $\mathop{\rm Div\,}_{E}(\tilde{X})$ denote the group of divisors on
$\tilde{X}$ that are supported on $E$. Then $c_{1}(f,{\mathcal{F}})$ is an
element of $\mathop{\rm Div\,}_{E}(\tilde{X})\otimes{\mathbb{Q}}$. Since the
canonical map $\mathop{\rm Div\,}_{E}(\tilde{X})\to{\mathop{\rm
Pic\,}}\tilde{X}$ is injective, we can, without any loss of information,
consider $c_{1}(f,{\mathcal{F}})$ as an element of ${\mathop{\rm
Pic\,}}\tilde{X}\otimes{\mathbb{Q}}$.

Let $\tilde{\pi}:\tilde{Y}\to\tilde{X}$ be a generically finite proper
morphism from a regular surface $\tilde{Y}$. Let us consider the Stein
factorization of $f\circ\tilde{\pi}$ into a proper birational morphism
$g:\tilde{Y}\to Y$ and a finite morphism $\pi:Y\to X$. Clearly, $Y\to X$
corresponds to a finite extension $A\to B$ with $B$ a normal domain. Since $A$
is Henselian, $B$ is local and also Henselian. Since $A$ is excellent, $B$ is
stil excellent (of dimension $2$). Possibly further blowing up $\tilde{Y}$ we
can also assume that $g$ is good.

Let us fix a rank $r$ vector bundle ${\mathcal{F}}$ on $\tilde{X}$ and let us
consider a filtration
$\tilde{\pi}^{*}{\mathcal{F}}={\mathcal{F}}^{0}\supset{\mathcal{F}}^{1}\supset...\supset{\mathcal{F}}^{r}=0$
such that all quotients
${\mathcal{L}}_{i}={\mathcal{F}}^{i}/{{\mathcal{F}}^{i+1}}$ are line bundles.
Note that such a filtration always exists, possibly after further blowing up
$\tilde{Y}$.

Let us note that

$(r-1)(\sum L_{i})^{2}-2r\sum_{i<j}L_{i}L_{j}=\sum_{i<j}(L_{i}-L_{j})^{2}\leq
0,$

where $L_{i}=c_{1}(g,{\mathcal{L}}_{i})$. But $(\sum
L_{i})^{2}=c_{1}(g,\tilde{\pi}^{*}{\mathcal{F}})^{2}=\deg\tilde{\pi}\cdot
c_{1}(f,{\mathcal{F}})^{2}$, so we have

$\frac{\sum_{i<j}L_{i}L_{j}}{\deg\tilde{\pi}}\geq\frac{r-1}{2r}c_{1}(f,{\mathcal{F}})^{2}.$

Therefore the following definition makes sense:

######  Definition 3.1.

The _second relative Chern class_ $c_{2}(f,{\mathcal{F}})$ of ${\mathcal{F}}$
with respect to $f$ is defined as the real number

$c_{2}(f,{\mathcal{F}})=\inf\left(\frac{\sum_{i<j}L_{i}L_{j}}{\deg\tilde{\pi}}\right),$

where the infimum is taken over all generically finite proper morphisms
$\tilde{\pi}:\tilde{Y}\to\tilde{X}$ from a regular surface and all filtrations
$\tilde{\pi}^{*}{\mathcal{F}}={\mathcal{F}}^{0}\supset{\mathcal{F}}^{1}\supset...\supset{\mathcal{F}}^{r}=0$,
whose quotients ${\mathcal{L}}_{i}={\mathcal{F}}^{i}/{{\mathcal{F}}^{i+1}}$
are line bundles.

In the rank $2$ case in the complex analytic setting this definition was
introduced by Wahl in [Wa]. His definition was generalized to arbitrary rank
in [La1] and studied there also in the complex analytic setting.

The following proposition summarizes some basic properties of the relative
second Chern class.

######  Proposition 3.2.

  1. 1.

For any line bundle ${\mathcal{L}}$ on $\tilde{X}$ we have
$c_{2}(f,{\mathcal{F}})=0$.

  2. 2.

(relative Bogomolov’s inequality) For any rank $r$ vector bundle
${\mathcal{F}}$ on $\tilde{X}$ we have

$\Delta(f,{\mathcal{F}}):=2rc_{2}(f,{\mathcal{F}})-(r-1)c_{1}(f,{\mathcal{F}})^{2}\geq
0.$

  3. 3.

For any rank $r$ vector bundle ${\mathcal{F}}$ and any line bundle
${\mathcal{L}}$ on $\tilde{X}$ we have

$\Delta(f,{\mathcal{F}}\otimes{\mathcal{L}})=\Delta(f,{\mathcal{F}}).$

  4. 4.

If $\tilde{\pi}:\tilde{Y}\to\tilde{X}$ is a generically finite proper morphism
from a regular surface $\tilde{Y}$ and
$\tilde{Y}\xrightarrow{g}Y\xrightarrow{\pi}X$ is the Stein factorization of
$f\circ\tilde{\pi}$ then

$c_{2}(g,\tilde{\pi}^{*}{\mathcal{F}})=\deg\pi\cdot c_{2}(f,{\mathcal{F}}).$

###### Proof.

Properties (1)–(3) are obvious from the definition. To show (4) let us first
remark that the category of finite extensions $A\to B$ with $B$ a normal
domain, is filtered. Given two extensions $A\to B$ and $A\to C$ we can embed
the quotient fields $K(B)$ and $K(C)$ into some fixed algebraic closure of
$K(A)$. Then we can find a finite field extension $K(A)\subset L$ that
contains both $K(B)$ and $K(C)$. Then the normalization $D$ of $A$ in $L$
gives a finite normal extension $A\to D$ dominating both $A\to B$ and $A\to
C$.

This implies that the category of generically finite proper morphisms from a
regular surface to $\tilde{X}$ is cofiltered. Indeed, if
$\tilde{\pi}:\tilde{Y}\to\tilde{X}$ and $\tilde{\tau}:\tilde{Z}\to\tilde{X}$
are generically finite proper morphisms from regular surfaces $\tilde{Y}$ and
$\tilde{Z}$ then we consider the corresponding Stein factorizations
$\tilde{Y}\xrightarrow{g}Y\xrightarrow{\pi}X$ and
$\tilde{Z}\xrightarrow{h}Z\xrightarrow{\tau}X$. By the above we can find a
finite morphism $T\to X$ from a normal surface $T$, that dominates both $\pi$
and $\tau$. Then we can find a good resolution of singularities $\tilde{T}\to
T$ with generically finite proper morphisms to both $\tilde{Y}$ and
$\tilde{Z}$ (it is sufficient to find a resolution dominating main irreducible
components of both $\tilde{Y}\times_{Y}T$ and $\tilde{Z}\times_{Z}T$).

The above fact allows us to pull-back to $\tilde{T}$ any filtration of the
pullback of ${\mathcal{F}}$ to $\tilde{Z}$. This gives a filtration of the
pull back of $\tilde{\pi}^{*}{\mathcal{F}}$ implying (4). ∎

######  Remark 3.3.

In the following we write ${\mathop{\rm ch\,}}_{2}(f,{\mathcal{F}})$ for
$\frac{1}{2}c_{1}(f,{\mathcal{F}})^{2}-c_{2}(f,{\mathcal{F}})$. Note that
definition of $c_{2}(f,\cdot)$ implies that for any two vector bundles
${\mathcal{F}}_{1},{\mathcal{F}}_{2}$ on $\tilde{X}$ we have

${\mathop{\rm
ch\,}}_{2}(f,{\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2})\geq{\mathop{\rm
ch\,}}_{2}(f,{\mathcal{F}}_{1})+{\mathop{\rm ch\,}}_{2}(f,{\mathcal{F}}_{2}).$

Later we prove that if the base field $k$ has positive characteristic then we
have equality (see Corollary 3.17).

######  Remark 3.4.

One of the main open problems related to local relative Chern classes is their
behaviour under tensor operations (see [La1, Conjecture 8.1]). For example, if
we knew that one can compute $c_{2}(f,{\mathop{{\rm
Sym\,}}}^{m}{\mathcal{F}})$ using $c_{1}(f,{\mathcal{F}})$ and
$c_{2}(f,{\mathcal{F}})$ using the same formulas as follow from the splitting
principle for usual Chern classes, then Conjecture 3.18 holds.

###  3.2 Local relative Riemann–Roch theorem

######  Definition 3.5.

For a vector bundle ${\mathcal{F}}$ on $\tilde{X}$ we define a _relative Euler
characteristic_ $\chi(f,{\mathcal{F}})$ by

$\chi(f,{\mathcal{F}}):=\dim
H^{0}(X,f_{[*]}{\mathcal{F}}/f_{*}{\mathcal{F}})+\dim
H^{0}(X,R^{1}f_{*}{\mathcal{F}}).$

######  Definition 3.6.

For any rank $r$ vector bundle ${\mathcal{F}}$ on $\tilde{X}$ we set

$a(f,{\mathcal{F}}):=\chi(f,{\mathcal{F}})-r\,\chi(f,{\mathcal{O}}_{\tilde{X}})+\frac{1}{2}c_{1}(f,{\mathcal{F}})(c_{1}(f,{\mathcal{F}})-K_{\tilde{X}})-c_{2}(f,{\mathcal{F}}).$

The proof of the following proposition is essentially the same as that of
[La1, Proposition 2.9] so, since we cannot improve upon it, we skip it.

######  Proposition 3.7.

Let $f^{\prime}:\tilde{X}^{\prime}\to X$ be a desingularization of $X$. Let
${\mathcal{F}}\in{\mathop{{\rm Vect\,}}}(\tilde{X})$ and
${\mathcal{F}}^{\prime}\in{\mathop{{\rm Vect\,}}}(\tilde{X}^{\prime})$ be
vector bundles such that $f_{[*]}{\mathcal{F}}$ and
$f^{\prime}_{[*]}{\mathcal{F}}^{\prime}$ are isomorphic. Then we have
$a(f,{\mathcal{F}})=a(f^{\prime},{\mathcal{F}}^{\prime})$.

If ${\mathcal{E}}$ be a reflexive coherent ${\mathcal{O}}_{X}$-module then we
set

$a(x,{\mathcal{E}}):=a(f,{\mathcal{F}}),$

where $f:\tilde{X}\to X$ is any desingularization of $X$ and ${\mathcal{F}}$
is any vector bundle on $\tilde{X}$ such that
$f_{[*]}{\mathcal{F}}\simeq{\mathcal{E}}$. The above proposition implies that
$a(x,{\mathcal{E}})$ is well defined. Therefore we get a function
$a(x,\cdot):{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})\to{\mathbb{R}}$.
If $j:U:=X\backslash\\{x\\}\hookrightarrow X$ denotes the open embedding then
by Lemma 1.1 the functor $j_{*}:{\mathop{{\rm
Vect\,}}}(U)\to{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ is an
equivalence of categories, so we can also treat $a(x,\cdot)$ as a function
${\mathop{{\rm Vect\,}}}(U)\to{\mathbb{R}}$.

In the remaining part of the section we reprove the results of [La1, Section
4] giving more details and providing simpler proofs that avoid the use of
reduction cycles. First we note the following lemma.

######  Lemma 3.8.

Let ${\mathcal{G}}$ be a vector bundle of rank $r$ on $\tilde{X}$. If
${\mathcal{G}}$ is globally generated outside of a finite number of $k$-points
$T$ of $E$ then for general $(r+2)$ sections of ${\mathcal{G}}$ the cokernel
of the induced map ${\mathcal{O}}^{\oplus(r+2)}_{X}\to{\mathcal{G}}$ is
supported on $T$.

###### Proof.

Let us choose a finite dimensional $k$-vector subspace $V\subset
H^{0}(\tilde{X},{\mathcal{G}})$ such that the evaluation map
$V\otimes_{k}{\mathcal{O}}_{\tilde{X}}\to{\mathcal{G}}$ is surjective on
$U:=\tilde{X}\backslash T$. We can assume that $V$ has large dimension so that
$s:=\dim V-(r+2)>0$.

Let $G$ be the Grassmannian of $s$-dimensional quotient spaces of $V$. Then
$U\times_{k}G\to U$ is the Grassmann scheme representing rank $s$ quotient
vector bundles of the trivial bundle $V\otimes_{k}{\mathcal{O}}_{U}$. Let
${\mathcal{K}}$ be the kernel of the evaluation map
$V\otimes_{k}{\mathcal{O}}_{\tilde{X}}\to{\mathcal{G}}$ and let us consider
the subfunctor of the appropriate Grassmann functor, such that $S$-points
consist of those quotients $V\otimes_{k}{\mathcal{O}}_{S}\to{\mathcal{F}}$ for
which the induced map
$\bigwedge^{s}{\mathcal{K}}\to\bigwedge^{s}{\mathcal{F}}$ vanishes. By [Kl,
Proposition 2.2] this functor is represented by a closed subscheme $Z$ of
$U\times_{k}G$. For $x\in U(k)$, $k$-points of $Z_{x}\subset G$ correspond to
$(r+2)$-dimensional vector subspaces $W\subset V$ such that $\dim(W\otimes
k(x)\cap{\mathcal{K}}\otimes k(x))\geq 3$ (which is why $Z$ is also called the
$3$-rd special Schubert cycle defined by ${\mathcal{K}}$). Naively speaking,
$Z$ parametrizes pairs $(x,[W\subset V])$ such that the induced map $W\otimes
k(x)\to{\mathcal{G}}\otimes k(x)$ is not surjective.

By [Kl, Corollary 2.9] the scheme $Z$ has relative dimension $\dim G-3$ over
$U$ (this also follows from the standard dimension computation of the Schubert
cell defined by condition $\dim(W\cap{\mathcal{K}}\otimes k(x))\geq 3$ in
$G$). It follows that $\dim Z=\dim U+\dim G-3<\dim G$. So there exists
$W\subset V$ of dimension $(r+2)$ (corresponding to some point of $G\backslash
p_{2}(Z)$, where $p_{2}:Z\to G$ comes from the projection $U\times_{k}G\to G$)
such that the evaluation map
$W\otimes_{k}{\mathcal{O}}_{\tilde{X}}\to{\mathcal{G}}$ is surjective over
$U$. ∎

######  Proposition 3.9.

Let ${\mathcal{E}}$ be a reflexive coherent ${\mathcal{O}}_{X}$-module of rank
$r$ and let ${\mathcal{F}}=f^{[*]}{\mathcal{E}}$. Then the following
conditions are satisfied:

  1. 1.

$f_{*}{\mathcal{F}}={\mathcal{E}}$,

  2. 2.

$\dim H^{0}(X,R^{1}f_{*}{\mathcal{F}})\leq(r+2)\dim
H^{0}(X,R^{1}f_{*}{\mathcal{O}}_{\tilde{X}})$,

  3. 3.

There exists some $C>0$ depending only on $f:\tilde{X}\to X$ such that or
every irreducible component $E_{j}$ of $E$ we have

$0\leq c_{1}(f,{\mathcal{F}})\cdot E_{j}\leq C\cdot r.$

###### Proof.

To see (1) note that we have the canonical map

${\mathcal{E}}\to f_{*}f^{*}{\mathcal{E}}\to f_{*}f^{[*]}{\mathcal{E}},$

which is an isomorphism on $X\backslash f(E)$. Since ${\mathcal{E}}$ is
reflexive and $f_{*}f^{[*]}{\mathcal{E}}$ is torsion free, it is an
isomorphism on the whole $X$.

To see (2) let us note that ${\mathcal{E}}$ is globally generated as $X$ is
affine. So ${\mathcal{F}}$ is globally generated outside a finite number of
$k$-points (lying on $E$) and by Lemma 3.8 for general $(r+2)$ sections of
${\mathcal{F}}$ the cokernel of the corresponding map
${\mathcal{O}}^{\oplus(r+2)}_{\tilde{X}}\to{\mathcal{F}}$ is supported on a
finite set of points. This shows that we have a surjective map
$(R^{1}f_{*}{\mathcal{O}}_{\tilde{X}})^{\oplus(r+2)}\to
R^{1}f_{*}{\mathcal{F}}$, which gives (2). Note also that for every $j$ the
restriction of the above map to $E_{j}$ gives
${\mathcal{O}}^{\oplus(r+2)}_{E_{j}}\to{\mathcal{F}}|_{E_{j}}$ with cokernel
supported on a finite set of points. Therefore $c_{1}(f,{\mathcal{F}})\cdot
E_{j}=\deg{\mathcal{F}}|_{E_{j}}\geq 0$.

Now let us consider a short exact sequence

$0\to{\mathcal{F}}\to{\mathcal{F}}(E_{j})\to{\mathcal{F}}(E_{j})|_{E_{j}}\to
0.$

(1) implies that
$f_{*}{\mathcal{F}}={\mathcal{E}}=f_{*}({\mathcal{F}}(E_{j}))$, so the map
$f_{*}({\mathcal{F}}(E_{j})|_{E_{j}})\to R^{1}f_{*}{\mathcal{F}}$ is
injective. Therefore we have

$\chi({\mathcal{F}}(E_{j})|_{E_{j}})\leq\dim
H^{0}(\tilde{X},{\mathcal{F}}(E_{j})|_{E_{j}})\leq\dim
H^{0}(X,R^{1}f_{*}{\mathcal{F}})\leq(r+2)\dim
H^{0}(X,R^{1}f_{*}{\mathcal{O}}_{\tilde{X}}).$

But by the Riemann–Roch theorem on $E_{j}$ we have
$\chi({\mathcal{F}}(E_{j})|_{E_{j}})=r\chi({\mathcal{O}}_{E_{j}})+rE_{j}^{2}+\deg{\mathcal{F}}|_{E_{j}}$,
so we get

$c_{1}(f,{\mathcal{F}})\cdot E_{j}\leq(r+2)\dim
H^{0}(X,R^{1}f_{*}{\mathcal{O}}_{\tilde{X}})-r\chi({\mathcal{O}}_{E_{j}})-rE_{j}^{2},$

which gives the second inequality in (3). ∎

######  Corollary 3.10.

In the notation of Proposition 3.9 the ${\mathbb{Q}}$-divisor
$-c_{1}(f,{\mathcal{F}})$ is effective and there exists a constant
$\tilde{C}>0$ depending only on $f:\tilde{X}\to X$ such that for every
${\mathcal{E}}$ we have

$-c_{1}(f,{\mathcal{F}})^{2}\leq\tilde{C}\cdot r^{2}.$

###### Proof.

The first assertion follows from the inequalities $0\leq
c_{1}(f,{\mathcal{F}})\cdot E_{j}$ (see [Gi, (7)]). To see the second
assertion let us write $c_{1}(f,{\mathcal{F}})=-\sum\alpha_{i}E_{i}$ for some
non-negative rational numbers $\alpha_{i}$. Then by Cramer’s rule the
$\alpha_{i}$ depend linearly on the numbers $c_{1}(f,{\mathcal{F}})\cdot
E_{j}$. So the assertion follows from the last part of Proposition 3.9. ∎

######  Lemma 3.11.

If ${\mathcal{G}}$ is a globally generated rank $r$ vector bundle on
$\tilde{X}$ then $c_{2}(f,{\mathcal{G}})\leq 0$.

###### Proof.

Let us choose a finite dimensional $k$-vector subspace $V\subset
H^{0}(\tilde{X},{\mathcal{G}})$ such that the evaluation map
$V\otimes_{k}{\mathcal{O}}_{\tilde{X}}\to{\mathcal{G}}$ is surjective. Let $G$
be the Grassmannian of $(r-1)$-dimensional $k$-subspaces of $V$ and let
$Z\subset\tilde{X}\times G$ be the subscheme parametrizing pairs $(x,[W\subset
V])$ such that the induced map $W\otimes k(x)\to{\mathcal{G}}\otimes k(x)$ is
not injective. Strictly speaking, we should again use the functorial approach
of [Kl] (as in the proof of Lemma 3.8) but we sketch a naive approach leaving
formalization of proof to the reader.

Let us consider projections $p_{1}:Z\to\tilde{X}$ and $p_{2}:Z\to G$. For any
$k$-point of $\tilde{X}$ the map $V\otimes k(x)\to{\mathcal{G}}\otimes k(x)$
is surjective. So if $K$ denotes its kernel (which is of dimension $\dim V-r$)
then we have

$p_{1}^{-1}(x)\simeq\\{[W\subset V]\in G:\dim((W\otimes k(x))\cap K)\geq
1\\}.$

A standard computation shows that this Schubert cell has codimension $2$ in
$G$. It follows that $\dim p_{1}^{-1}(E)=\dim E+\dim G-2<\dim G$. So there
exists $W\subset V$ of dimension $(r-1)$ (corresponding to some point of
$G\backslash p_{2}(p_{1}^{-1}(E))$) such that the evaluation map
$W\otimes_{k}{\mathcal{O}}_{\tilde{X}}\to{\mathcal{G}}$ is injective and its
cokernel is locally free along $E$. This implies that we have a short exact
sequence

$0\to{\mathcal{O}}_{\tilde{X}}^{\oplus(r-1)}\to{\mathcal{G}}\to{\mathcal{L}}\to
0,$

where ${\mathcal{L}}$ is a line bundle. This immediately implies the required
inequality. ∎

The following theorem is an important improvement upon [La1, Theorem 4.12 and
Corollary 4.13]. In particular, this theorem immediately implies [La1,
Conjecture 8.2]. Since this conjecture is equivalent to the more natural
formula in [La1, Corollary 8.3] that is the content of theorem below, we do
not repeat the original formulation.

######  Theorem 3.12.

There exists some constants $A$ and $B$ depending only on $X$ such that for
every reflexive coherent ${\mathcal{O}}_{X}$-module ${\mathcal{E}}$ of rank
$r$ we have

$Ar^{2}\leq a(x,{\mathcal{E}})\leq Br.$

###### Proof.

Let ${\mathcal{E}}$ be a reflexive coherent ${\mathcal{O}}_{X}$-module of rank
$r$ and let ${\mathcal{F}}=f^{[*]}{\mathcal{E}}$. Let us fix an $f$-very ample
line bundle ${\mathcal{O}}_{\tilde{X}}(1)$. By Serre’s theorem there exists
some $n_{0}$ such that for all $n\geq n_{0}$ such that for all irreducible
components $E_{j}$ of $E$ we have
$R^{1}f_{*}({\mathcal{O}}_{\tilde{X}}(-E_{j})\otimes{\mathcal{O}}_{\tilde{X}}(n))=0$.

Let us recall that by Lemma 3.8 the cokernel of the map
${\mathcal{O}}^{\oplus(r+2)}_{\tilde{X}}\to{\mathcal{F}}$ determined by
$(r+2)$ general sections of ${\mathcal{F}}$ is supported on a finite set of
points. Twisting it by
${\mathcal{O}}_{\tilde{X}}(-E_{j})\otimes{\mathcal{O}}_{\tilde{X}}(n)$ we see
that

$R^{1}f_{*}({\mathcal{F}}(n)\otimes{\mathcal{O}}_{\tilde{X}}(-E_{j}))=0$

for all $n\geq n_{0}$. So we have surjective maps

$H^{0}(\tilde{X},{\mathcal{F}}(n))\to H^{0}(E_{j},{\mathcal{F}}(n)|_{E_{j}}).$

This implies that for all $n\geq n_{0}$ the bundle ${\mathcal{F}}(n)$ is
globally generated. So by Lemma 3.11 we have
$c_{2}(f,{\mathcal{F}}(n_{0}))\leq 0$ and hence

$\displaystyle 0\leq\Delta(f,{\mathcal{F}})$
$\displaystyle=\Delta(f,{\mathcal{F}}(n_{0}))\leq-(r-1)c_{1}(f,{\mathcal{F}}(n_{0}))^{2}$
$\displaystyle=-(r-1)\left(c_{1}(f,{\mathcal{F}})^{2}+n_{0}c_{1}(f,{\mathcal{F}}).c_{1}(f,{\mathcal{O}}_{X}(1))+n_{0}^{2}c_{1}(f,{\mathcal{O}}_{X}(1))^{2}\right).$

By Proposition 3.9 we also know that
$0\leq\chi(f,{\mathcal{F}})\leq(r+2)\chi(f,{\mathcal{O}}_{\tilde{X}})$. Since

$a(x,{\mathcal{E}})=\chi(f,{\mathcal{F}})-r\,\chi(f,{\mathcal{O}}_{\tilde{X}})+\frac{1}{2r}c_{1}(f,{\mathcal{F}})(c_{1}(f,{\mathcal{F}})-rK_{\tilde{X}})-\frac{\Delta(f,{\mathcal{F}})}{2r},$

the required inequalities follow from Proposition 3.9 and Corollary 3.10 . ∎

######  Remark 3.13.

The proof shows that in the above theorem one can find $A$ and $B$ that depend
only on numerical invariants of $X$ and its fixed resolution $f:\tilde{X}\to
X$. More precisely, these constants can be determined by $\dim
R^{1}f_{*}{\mathcal{O}}_{X}$ (called the geometric genus of the singularity
$X$), discrepancies of the exceptional divisor and the intersection matrix of
exceptional curves of $f$. This fact is useful when studying how Chern classes
change in families of reflexive sheaves on normal surfaces.

######  Remark 3.14.

It is natural to expect that in notation of the proof of Theorem 3.12, there
exists a constant $\tilde{A}$ such that
$\Delta(f,{\mathcal{F}})\leq\tilde{A}r^{2}$. This would imply that one can
find $A$ such that $Ar^{2}\leq a(x,{\mathcal{E}})$. This conjecture is
equivalent to [La1, Conjecture 8.2].

###  3.3 Characterization of the relative second Chern class in positive
characteristic

In this subsection we assume that the base field $k$ has characteristic $p>0$.

######  Theorem 3.15.

There exists a uniquely determined function $c_{2}(f,\cdot):{\mathop{{\rm
Vect\,}}}(\tilde{X})\to{\mathbb{R}}$, ${\mathcal{F}}\to
c_{2}(f,{\mathcal{F}})$ such that the following conditions are satisfied:

  1. 1.

For every ${\mathcal{F}}\in{\mathop{{\rm Vect\,}}}(\tilde{X})$ we have
$c_{2}(f,F_{\tilde{X}}^{*}{\mathcal{F}})=p^{2}c_{2}(f,{\mathcal{F}}).$

  2. 2.

There exists a function $\varphi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$ such
that for every rank $r$ vector bundle ${\mathcal{F}}\in{\mathop{{\rm
Vect\,}}}(\tilde{X})$ we have

$\left|\chi(f,{\mathcal{F}})+\frac{1}{2}c_{1}(f,{\mathcal{F}})(c_{1}(f,{\mathcal{F}})-K_{\tilde{X}})-c_{2}(f,{\mathcal{F}})-r\,\chi(f,{\mathcal{O}}_{\tilde{X}})\right|\leq\varphi(r).$

###### Proof.

Existence of the function $c_{2}(f,\cdot)$ was proven in previous sections.
More precisely, (1) follows from the last part of Proposition 3.2 applied to
the Frobenius morphism. Condition (2) follows from Theorem 3.12.

To prove uniqueness note that
$c_{1}(f,(F_{X}^{m})^{*}{\mathcal{F}})=p^{m}c_{1}(f,{\mathcal{F}})$. So (1)
and (2) imply that

$\displaystyle\left|\chi(f,(F_{X}^{m})^{*}{\mathcal{F}})+\frac{1}{2}p^{m}c_{1}(f,{\mathcal{F}}).(p^{m}c_{1}(f,{\mathcal{F}})-K_{\tilde{X}})-p^{2m}c_{2}(f,{\mathcal{F}})-r\,\chi(f,{\mathcal{O}}_{\tilde{X}})\right|$
$\displaystyle\leq\varphi(r)$

Dividing the above inequality by $p^{2m}$ and passing to the limit we get

$c_{2}(f,{\mathcal{F}})=\frac{1}{2}c_{1}(f,{\mathcal{F}})^{2}+\lim_{m\to\infty}\frac{\chi(f,(F_{X}^{m})^{*}{\mathcal{F}})}{p^{2m}}.$

∎

######  Remark 3.16.

Theorem 3.12 shows that in fact we can take $\varphi$ to be quadratic and
independent of $f$. This is not needed for the proof of uniqueness of
$c_{2}(f,\cdot)$.

The last formula in the above proof and additivity of the relative Euler
characteristic imply the following corollary.

######  Corollary 3.17.

For any ${\mathcal{F}}_{1},{\mathcal{F}}_{2}\in{\mathop{{\rm
Vect\,}}}(\tilde{X})$ we have

${\mathop{\rm
ch\,}}_{2}(f,{\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2})={\mathop{\rm
ch\,}}_{2}(f,{\mathcal{F}}_{1})+{\mathop{\rm ch\,}}_{2}(f,{\mathcal{F}}_{2}).$

Moreover, for any coherent reflexive ${\mathcal{O}}_{X}$-modules
${\mathcal{E}}_{1}$ and ${\mathcal{E}}_{2}$ we have

$a(x,{\mathcal{E}}_{1}\oplus{\mathcal{E}}_{2})=a(x,{\mathcal{E}}_{1})+a(x,{\mathcal{E}}_{2}).$

###  3.4 Conjectural characterization of the relative second Chern class in
characteristic zero

Assume that the base field $k$ has characteristic zero. In this case one
expects that the following asymptotic Riemann–Roch formula works (see the
conjecture in [Wa, Introduction] for the rank $2$ case; we need this
conjecture in an arbitrary rank as hinted in [La1, Section 8]).

######  Conjecture 3.18.

If ${\mathcal{F}}$ is a rank $r$ vector bundle on $\tilde{X}$ then

$\chi(f,{\mathop{{\rm
Sym\,}}}^{m}{\mathcal{F}})=-\frac{m^{r+1}}{r!}\left(c_{1}(f,{\mathcal{F}})^{2}-c_{2}(f,{\mathcal{F}})\right)+O(m^{r}).$

This conjecture would allow us to characterize $c_{2}(f,{\mathcal{F}})$ as

$c_{2}(f,{\mathcal{F}})=c_{1}(f,{\mathcal{F}})^{2}+r!\lim_{m\to\infty}\frac{\chi(f,{\mathop{{\rm
Sym\,}}}^{m}{\mathcal{F}})}{m^{r+1}}.$

Using Theorem 3.12 it is easy to prove inequality $\leq$ in the above
conjecture (cf. [La1, Theorem 4.15] for the rank $2$ case). This allows us to
consider

$\liminf_{m\to\infty}\frac{\chi(f,{\mathop{{\rm
Sym\,}}}^{m}{\mathcal{F}})}{m^{r+1}}$

(or similar limits) to define local relative Chern classes (see, e.g., [La2]
for one example of use of such definition).

Let us recall that a quotient surface singularity is a quotient of the
spectrum of a regular $2$-dimensional ring by a linear action of a finite
group. The following result follows from [La1, Theorem 5.1].

######  Theorem 3.19.

Assume that $X={\mathop{{\rm Spec\,}}}A$, where $A$ is the henselization of a
local ring of a quotient surface singularity. Then Conjecture 3.18 holds for
any desingularization of $X$. Moreover, for any vector bundle ${\mathcal{F}}$
on $\tilde{X}$ the number $c_{2}(f,{\mathcal{F}})$ is rational.

See [La1] for further discussion of Conjecture 3.18 and its proof in some
other cases.

##  4 Chern classes of reflexive sheaves on normal surfaces

Let $X$ be a proper normal variety defined over an algebraically closed field
$k$. For any coherent ${\mathcal{O}}_{X}$-module ${\mathcal{E}}$ of rank $r$
the sheaf $\det{\mathcal{E}}=(\bigwedge^{r}{\mathcal{E}})^{**}$ is a reflexive
coherent ${\mathcal{O}}_{X}$-module of rank $1$. So we can define
$c_{1}({\mathcal{E}})\in A^{1}(X)$ as the class of a divisor $D$ such that
$\det{\mathcal{E}}={\mathcal{O}}_{X}(D)$.

In this section we assume that that $\dim X=2$.

###  4.1 Second Chern class

Let $f:\tilde{X}\to X$ be any resolution of singularities and let $E$ be its
exceptional locus. For every $x\in f(E)$ we consider the map
$\nu_{x}:{\mathop{{\rm Spec\,}}}{\mathcal{O}}_{X,x}^{h}\to X$ from the
spectrum of the henselization of the local ring of $X$ at $x$ and the base
change $f_{x}=\nu_{x}^{*}f:\tilde{X}_{x}\to{\mathop{{\rm
Spec\,}}}{\mathcal{O}}_{X,x}^{h}$ of $f$ via $\nu_{x}$. Note that the group of
divisors on $\tilde{X}_{x}$ that are supported on the exceptional locus
$E_{x}$ of $f_{x}$ embeds into $A_{1}(\tilde{X})$. For any vector bundle
${\mathcal{F}}$ on $\tilde{X}$ we write $c_{1}(f_{x},{\mathcal{F}})\in
A_{1}(\tilde{X})\otimes{\mathbb{Q}}$ for the image of
$c_{1}(f_{x},{\mathcal{F}}|_{\tilde{X}_{x}})$. Then for any
${\mathcal{F}}\in{\mathop{{\rm Vect\,}}}(\tilde{X})$ we have

$c_{1}({\mathcal{F}})=f^{*}c_{1}(f_{[*]}{\mathcal{F}})+\sum_{x\in
f(E)}c_{1}(f_{x},{\mathcal{F}})$

in $A_{1}(\tilde{X})\otimes{\mathbb{Q}}$. So if
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ and we
choose some vector bundle ${\mathcal{F}}$ on $\tilde{X}$ such that
$f_{[*]}{\mathcal{F}}\simeq{\mathcal{E}}$ then

$f^{*}c_{1}({\mathcal{E}})=c_{1}({\mathcal{F}})-\sum_{x\in
f(E)}c_{1}(f_{x},{\mathcal{F}})$

treated as an element of $A_{1}(\tilde{X})\otimes{\mathbb{Q}}.$

######  Definition 4.1.

If ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ then
we choose some vector bundle ${\mathcal{F}}$ on $\tilde{X}$ such that
$f_{[*]}{\mathcal{F}}\simeq{\mathcal{E}}$. Then we use the homomorphism
$f_{*}:A_{0}(\tilde{X})\to A_{0}(X)$ to define the _second Chern class_ of
${\mathcal{E}}$ as

$c_{2}({\mathcal{E}}):=f_{*}c_{2}({\mathcal{F}})-\sum_{x\in
f(E)}c_{2}(f_{x},{\mathcal{F}})\,[x]$

treated as an element of $A_{0}(X)\otimes{\mathbb{R}}.$

Note that the function
$c_{2}:{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})\to
A_{0}(X)\otimes{\mathbb{R}}$ is well defined, i.e., the class
$f_{*}c_{2}({\mathcal{F}})-\sum_{x\in f(E)}c_{2}(f_{x},{\mathcal{F}})\,[x]$
does not depend on the choice of ${\mathcal{F}}$ and $f$. Namely, if we choose
another resolution of singularities $f^{\prime}:\tilde{X}^{\prime}\to X$ and
${\mathcal{F}}^{\prime}\in{\mathop{{\rm Vect\,}}}(\tilde{X}^{\prime})$ such
that $f^{\prime}_{[*]}{\mathcal{F}}^{\prime}\simeq{\mathcal{E}}$ then
$f_{*}c_{2}({\mathcal{F}}^{\prime})-f^{\prime}_{*}c_{2}({\mathcal{F}}^{\prime})$
is a well-defined $0$-cycle supported on $S=f(E)\cup f(E^{\prime})$ and
Proposition 3.7 implies equality of the corresponding degrees locally at each
point of $S$.

Using the above definition we can also define for
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ the
_discriminant_ $\Delta({\mathcal{E}})$ as

$\Delta({\mathcal{E}}):=2rc_{2}({\mathcal{E}})-(r-1)c_{1}({\mathcal{E}})^{2}\in
A_{0}(X)\otimes{\mathbb{R}},$

where $r$ is the rank of ${\mathcal{E}}$. Let us also recall that we have the
degree map $\int_{X}:A_{0}(X)\otimes{\mathbb{R}}\to{\mathbb{R}}$.

The following proposition summarizes some of the basic properties of Chern
classes on normal surfaces:

######  Proposition 4.2.

  1. 1.

For any rank $1$ reflexive coherent ${\mathcal{O}}_{X}$-module ${\mathcal{L}}$
on $X$ we have $c_{2}({\mathcal{L}})=0$.

  2. 2.

For any ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$
and any rank $1$ reflexive coherent ${\mathcal{O}}_{X}$-module ${\mathcal{L}}$
on $X$ we have
$\Delta({\mathcal{E}}\hat{\otimes}{\mathcal{L}})=\Delta({\mathcal{E}}).$

  3. 3.

For any vector bundle ${\mathcal{F}}$ on $\tilde{X}$ we have
$\int_{\tilde{X}}\Delta({\mathcal{F}})\geq\int_{X}\Delta(f_{[*]}{\mathcal{F}}).$

  4. 4.

If $\pi:Y\to X$ is a finite morphism from a normal surface $Y$ then for any
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ we have
$\int_{Y}c_{2}(\pi^{[*]}{\mathcal{E}})=\deg\pi\cdot\int_{X}c_{2}({\mathcal{E}}).$

###### Proof.

All the above properties follow easily from the corresponding properties of
relative local Chern classes listed in Proposition 3.2 and from properties of
Chern classes of vector bundles on smooth surfaces. ∎

######  Remark 4.3.

Note that for any two Weil divisors $D_{1}$ and $D_{2}$ on $X$ we have

$\int_{X}c_{2}({\mathcal{O}}_{X}(D_{1})\oplus{\mathcal{O}}_{X}(D_{2}))=D_{1}.D_{2}.$

In the local case this follows from [Wa, Proposition 2.5]. The global
assertion follows from this fact and definition of Mumford’s intersection
numbers on normal surfaces. Note that this equality implies that Mumford’s
intersection numbers behaves well under finite coverings (see Proposition 4.2,
(4)).

###  4.2 Riemann–Roch theorem on normal surfaces

For ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ we
write $a(x,{\mathcal{E}})$ for $a(x,\nu_{x}^{*}{\mathcal{E}})$ (see the
previous subsection for the notation). Then we have the following Riemann–Roch
type theorem:

######  Theorem 4.4.

Let $X$ be a normal proper algebraic surface defined over an algebraically
closed field $k$. Then for any
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ we have

$\chi(X,{\mathcal{E}})=\frac{1}{2}c_{1}({\mathcal{E}}).(c_{1}({\mathcal{E}})-K_{X})-\int_{X}c_{2}({\mathcal{E}})+{\mathop{\rm
rk\,}}{\mathcal{E}}\cdot\chi(X,{\mathcal{O}}_{X})+\sum_{x\in{\mathop{{\rm
Sing\,}}}X}a(x,{\mathcal{E}}).$

###### Proof.

Let ${\mathcal{F}}$ be a vector bundle on $\tilde{X}$. By the Leray spectral
sequence we have

$\chi(X,f_{*}{\mathcal{F}})=\chi(\tilde{X},{\mathcal{F}})+h^{0}(X,R^{1}f_{*}{\mathcal{F}}).$

This shows that

$\chi(X,f_{[*]}{\mathcal{F}})=\chi(\tilde{X},{\mathcal{F}})+\sum_{x\in
f(E)}\chi(f_{x},{\mathcal{F}}).$

For any ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$
we can find ${\mathcal{F}}$ such that ${\mathcal{E}}\simeq
f_{[*]}{\mathcal{F}}$ (e.g., one can take
${\mathcal{F}}=f^{[*]}{\mathcal{E}}$). Then the required formula follows from
the Riemann–Roch theorem for ${\mathcal{F}}$ on $\tilde{X}$, Definition 3.6
and definitions of $c_{1}({\mathcal{E}})$ and $c_{2}({\mathcal{E}})$. ∎

###  4.3 Second Chern class in positive characteristic

The following theorem shows that in positive characteristic the second Chern
class is uniquely determined by two very simple properties.

######  Theorem 4.5.

Let $X$ be a normal proper algebraic surface defined over an algebraically
closed field of characteristic $p>0$. Then there exists a uniquely determined
function
$\int_{X}c_{2}:{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})\to{\mathbb{R}}$,
${\mathcal{E}}\to\int_{X}c_{2}({\mathcal{E}})$, such that the following
conditions are satisfied:

  1. 1.

There exists a function $\varphi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$ such
that for every
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ of rank
$r$ we have

$\left|\chi(X,{\mathcal{E}})-\left(\frac{1}{2}c_{1}({\mathcal{E}}).(c_{1}({\mathcal{E}})-K_{X})-\int_{X}c_{2}({\mathcal{E}})+r\chi(X,{\mathcal{O}}_{X})\right)\right|\leq\varphi(r).$

  2. 2.

For every ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$
we have
$\int_{X}c_{2}(F_{X}^{[m]}{\mathcal{E}})=p^{2m}\int_{X}c_{2}({\mathcal{E}}).$

###### Proof.

Existence of $\int_{X}c_{2}$ satisfying the above conditions was already
proven. Namely, by Theorem 4.4 we have

$\chi(X,{\mathcal{E}})-\left(\frac{1}{2}c_{1}({\mathcal{E}}).(c_{1}({\mathcal{E}})-K_{X})-\int_{X}c_{2}({\mathcal{E}})+r\chi(X,{\mathcal{O}}_{X})\right)=a({\mathcal{E}}),$

where $a({\mathcal{E}}):=\sum_{x\in{\mathop{{\rm
Sing\,}}}X}a(x,{\mathcal{E}})$. Since $X$ has finitely many singular points,
(1) follows from Theorem 3.12. The second condition follows from Proposition
4.2, (4).

Now let us prove that $\int_{X}c_{2}$ is uniquely determined by the conditions
(1) and (2). Note that
$c_{1}(F_{X}^{[m]}{\mathcal{E}})=p^{m}c_{1}({\mathcal{E}})$. So using (1) we
get

$\displaystyle\left|\chi(X,F_{X}^{[m]}{\mathcal{E}})-\left(\frac{1}{2}p^{m}c_{1}({\mathcal{E}}).(p^{m}c_{1}({\mathcal{E}})-K_{X})-p^{2m}\int_{X}c_{2}({\mathcal{E}})+r\chi(X,{\mathcal{O}}_{X})\right)\right|$
$\displaystyle\leq C_{r}.$

Dividing the above inequality by $p^{2m}$ and passing to the limit we get

$\int_{X}c_{2}({\mathcal{E}})=\frac{1}{2}c_{1}({\mathcal{E}})^{2}-\lim_{m\to\infty}\frac{\chi(X,F_{X}^{[m]}{\mathcal{E}})}{p^{2m}}.$

∎

######  Remark 4.6.

We can also talk about the second Chern character ${\mathop{\rm
ch\,}}_{2}({\mathcal{E}})$, which is defined as usual by setting ${\mathop{\rm
ch\,}}_{2}({\mathcal{E}})=\frac{1}{2}c_{1}({\mathcal{E}})^{2}-c_{2}({\mathcal{E}})$.
By the above, we see that

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})=\lim_{m\to\infty}\frac{\chi(X,F_{X}^{[m]}{\mathcal{E}})}{p^{2m}}.$

Existence of this limit is the main reason why we can define the second Chern
character for higher dimensional varieties in positive characteristic (see
Theorem 5.2).

###  4.4 Second Chern class in characteristic zero

Let us assume that the base field $k$ has characteristic $0$. The following
result follows from the Riemann–Roch theorem on resolution of singularities of
$X$ and from Theorem 3.19.

######  Theorem 4.7.

Let $X$ be a normal proper algebraic surface with at most quotient
singularities. Then for any
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ of rank
$r$ we have $\int_{X}c_{2}({\mathcal{E}})\in{\mathbb{Q}}$ and

$\chi(X,{\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}})=\frac{m^{r+1}}{r!}\left(c_{1}({\mathcal{E}})^{2}-\int_{X}c_{2}({\mathcal{E}})\right)+O(m^{r}).$

##  5 Chern classes in higher dimensions

In this section we develop the theory of the second Chern classes and
characters for reflexive sheaves on higher dimensional normal varieties. This
works well in case of positive characteristic. In characteristic zero in
dimensions $>2$ the theory depends heavily on Conjecture 3.18, so we can do it
only for varieties with at most quotient singularities in codimension $2$.

In this section we assume that $n\geq 2$.

###  5.1 The second Chern character in positive characteristic

Let us assume that the base field $k$ has positive characteristic $p$.

######  Proposition 5.1.

Let ${\mathcal{E}}$ be a reflexive coherent ${\mathcal{O}}_{X}$-module and let
$L_{1},...,L_{n-2}$ be line bundles on $X$. Then the sequence

$\left(\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}{p^{2m}}\right)_{m\in{\mathbb{N}}}$

is convergent to some real number.

###### Proof.

First we use similar arguments to that from the proof of Proposition 2.3 to
reduce to the case, when all $L_{i}$ are very ample. For any line bundle
$M_{1}$ on $X$, Lemma 1.5, (1) and Lemma 1.7 imply that

$c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}}=c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot({\mathcal{O}}_{X}^{\oplus
r}-[{\mathcal{O}}_{-c_{1}(F_{X}^{[m]}{\mathcal{E}})}]),$

where $r$ denotes the rank of ${\mathcal{E}}$. Since
$c_{1}(F_{X}^{[m]}{\mathcal{E}})=p^{m}c_{1}({\mathcal{E}})$, Lemma 1.6 shows
that
$[{\mathcal{O}}_{-c_{1}(F_{X}^{[m]}{\mathcal{E}})}]+p^{m}[{\mathcal{O}}_{c_{1}({\mathcal{E}})}]\in
K_{n-2}(X)$. So Lemma 1.5, (1) gives

$c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{-c_{1}(F_{X}^{[m]}{\mathcal{E}})}]=p^{m}c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{c_{1}({\mathcal{E}})}]$

and we get

$\displaystyle\lim_{m\to\infty}\frac{\chi(X,c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}{p^{2m}}=0.$

But by Lemma 1.5 (3) we have

$\displaystyle{\chi(X,c_{1}(L_{1}\otimes
M_{1})c_{1}(L_{2})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}={\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}$
$\displaystyle+\chi(X,c_{1}(M_{1})c_{1}(L_{2})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})-{\chi(X,c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}.$

So if the limits

$\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}{p^{2m}}$

and

$\lim_{m\to\infty}\frac{\chi(X,c_{1}(M_{1})c_{1}(L_{2})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}{p^{2m}}$

exist, then

$\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1}\otimes
M_{1})c_{1}(L_{2})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}{p^{2m}}$

also exists and it is equal to their sum.

Since any line bundle $L$ can be written as $A\otimes B^{-1}$ for some very
ample line bundles $A$ and $B$ and the formula in the sequence is symmetric in
$(L_{1},...,L_{n-2})$, it is sufficient to prove convergence of the sequence
assuming that all line bundles $L_{i}$ are very ample.

Let $K$ be any uncountable algebraically closed field containing $k$ and let
$X_{K}\to X$ be the base change. Since

$\chi(X_{K},c_{1}((L_{1})_{K})...c_{1}((L_{n-2})_{K})\cdot{F_{X_{K}}^{[m]}{\mathcal{E}}_{K}})=\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}}),$

it is sufficient to prove convergence of the considered sequence after base
change to $K$. So in the following we can assume that our base field $k$ is
uncountable.

By Theorem 1.4 a general divisor $H_{1}\in|{\mathcal{L}}_{1}|$ is normal and
irreducible. Since $F_{X}^{[m]}{\mathcal{E}}$ is reflexive, $H_{1}$ is
$F_{X}^{[m]}{\mathcal{E}}$-regular and the restriction
$(F_{X}^{[m]}{\mathcal{E}})|_{H_{1}}$ is torsion free as
${\mathcal{O}}_{H_{1}}$-module (see, e.g., [HL, Lemma 1.1.13]). By [HL,
Corollary 1.1.14], for fixed $m$ and general $H_{1}$ the restriction
$(F_{X}^{[m]}{\mathcal{E}})|_{H_{1}}$ is also reflexive. Therefore, since $k$
is uncountable, there exists a divisor $H_{1}\in|{\mathcal{L}}_{1}|$ such that
$H_{1}$ is normal, irreducible and the restriction
$(F_{X}^{[m]}{\mathcal{E}})|_{H_{1}}$ is reflexive for all non-negative
integers $m$. Since ${\mathcal{E}}$ is locally free outside of a closed subset
of codimension $\geq 2$ in $X$, there exists a point $x\in H_{1}$ such that
${\mathcal{E}}_{x}$ is a free ${\mathcal{O}}_{X,x}$-module of some rank $r$.
Therefore ${\mathcal{E}}|_{H_{1}}$ has the same rank $r$ as ${\mathcal{E}}$
and we have
$(F_{X}^{[m]}{\mathcal{E}})|_{H_{1}}=F_{H_{1}}^{[m]}({\mathcal{E}}|_{H_{1}}).$
In particular, we have $c_{1}(L_{1})\cdot
F_{X}^{[m]}{\mathcal{E}}=F_{H_{1}}^{[m]}({\mathcal{E}}|_{H_{1}})$ in $K(X)$.

Proceeding in the same way, we can construct a sequence of divisors
$H_{i}\in|{\mathcal{L}}_{i}|$, $i=1,...,n-2$, such that for all non-negative
integers $m$ we have

  1. 1.

the intersection $X_{i}:=\bigcap_{j\leq i}H_{j}$ is normal and irreducible,

  2. 2.

the restriction $(F_{X}^{[m]}{\mathcal{E}})|_{X_{i}}$ is reflexive of rank
$r$,

  3. 3.

we have
$F_{X_{i}}^{[m]}({\mathcal{E}}|_{X_{i}})=c_{1}(L_{1})...c_{1}(L_{i})\cdot
F_{X}^{[m]}{\mathcal{E}}.$

In particular, $S=X_{n-2}$ is a normal surface and by the Riemann–Roch theorem
on $S$ (see Remark 4.6) we obtain

$\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}{p^{2m}}=\lim_{m\to\infty}\frac{\chi(S,F_{S}^{[m]}({\mathcal{E}}|_{S}))}{p^{2m}}=\int_{S}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}|_{S}).$

∎

######  Theorem 5.2.

Let us fix
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ and
consider the map $\int_{X}{\mathop{\rm
ch\,}}_{2}:N^{1}(X)^{\times(n-2)}\to{\mathbb{R}}$ sending
$(L_{1},...,L_{n-2})$ to

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}:=\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})}{p^{2m}}.$

This map satisfies the following properties:

  1. 1.

It is ${\mathbb{Z}}$-linear in all $L_{i}$.

  2. 2.

It is symmetric in $L_{1},...,L_{n-2}$.

  3. 3.

If ${\mathcal{E}}$ is a vector bundle on $X$ then

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}=\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})\cap c_{1}(L_{1})\cap...\cap c_{1}(L_{n-2})\cap[X].$

  4. 4.

If $k\subset K$ is an algebraically closed field extension then

$\int_{X_{K}}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}_{K})(L_{1})_{K}...(L_{n-2})_{K}=\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}.$

  5. 5.

If $n>2$ and $L_{1}$ is very ample then for a very general hypersurface
$H\in|L_{1}|$ we have

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}=\int_{H}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}|_{H})L_{2}|_{H}...L_{n-2}|_{H}.$

###### Proof.

The map is well defined by Proposition 5.1. (1) follows from the first part of
the proof of this proposition. (2) follows from the definition and Lemma 1.5,
(2). (4) follows from the definition and the fact that Euler characteristic
does not change under base field extension. To prove (5) let us first note
that by Bertini’s theorem general $H\in|L_{1}|$ is normal and irreducible. So
both sides of the equality are well defined. Moreover, by [HL, Corollary
1.1.14] for fixed $m$ and general $H\in|L_{1}|$, the restriction
$(F_{X}^{[m]}{\mathcal{E}})|_{H}$ is reflexive and hence isomorphic to
${F_{H}^{[m]}({\mathcal{E}}|_{H})}$ (if ${\mathcal{E}}|_{H}$ is also
reflexive). So for very general $H\in|L_{1}|$ we have

$\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{F_{X}^{[m]}{\mathcal{E}}})=\chi(H,c_{1}(L_{2}|_{H})...c_{1}(L_{n-2}|_{H})\cdot{F_{H}^{[m]}({\mathcal{E}}|_{H})})$

for all $m$ at the same time. Dividing the above equality by $p^{2m}$ and
passing to the limit gives (5). Finally, (3) follows from (4), (5) and the
analogous fact in the surface case. ∎

######  Definition 5.3.

For any reflexive coherent ${\mathcal{O}}_{X}$-module ${\mathcal{E}}$ of rank
$r$ and any line bundles $L_{1},...,L_{n-2}$ we set:

$\int_{X}c_{1}^{2}({\mathcal{E}})L_{1}...L_{n-2}:=c_{1}({\mathcal{E}})^{2}.L_{1}...L_{n-2},$
$\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}:=\frac{1}{2}\int_{X}c_{1}^{2}({\mathcal{E}})L_{1}...L_{n-2}-\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2},$
$\int_{X}\Delta({\mathcal{E}})HL_{1}...L_{n-2}:=2r\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}-(r-1)\int_{X}c_{1}^{2}({\mathcal{E}})L_{1}...L_{n-2}.$

If $D$ is a Weil divisor on $X$ then ${\mathcal{O}}_{X}(D)$ is a reflexive
${\mathcal{O}}_{X}$-module of rank $1$. In this case our definitions agree and
since $F_{X}^{[m]}{\mathcal{O}}_{X}(D)={\mathcal{O}}_{X}(p^{m}D)$, we have

$\displaystyle\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{O}}_{X}(D))L_{1}...L_{n-2}$
$\displaystyle=\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{\mathcal{O}}_{X}(p^{m}D))}{p^{2m}}$
$\displaystyle=\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{p^{m}D}])}{(p^{m})^{2}}$
$\displaystyle=\frac{1}{2}D^{2}.L_{1}...L_{n-2}=\frac{1}{2}\int_{X}c_{1}^{2}({\mathcal{O}}_{X}(D))L_{1}...L_{n-2}.$

In particular, $\int_{X}c_{2}({\mathcal{O}}_{X}(D))L_{1}...L_{n-2}=0$ and
$\int_{X}\Delta({\mathcal{O}}_{X}(D))L_{1}...L_{n-2}=0$.

###  5.2 Properties of Chern classes in positive characteristic

In this subsection we assume that $k$ has positive characteristic. Apart from
Theorem 5.2, the second Chern classes have the following properties analogous
to that from Proposition 4.2. We can assume in this proposition that $n>2$.

######  Proposition 5.4.

For any line bundles $L_{1},...,L_{n-2}$ on $X$ the following conditions are
satisfied:

  1. 1.

For any rank $1$ reflexive coherent ${\mathcal{O}}_{X}$-module ${\mathcal{L}}$
on $X$ we have

$\int_{X}c_{2}({\mathcal{L}})L_{1}...L_{n-2}=0.$

  2. 2.

For any ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$
and any rank $1$ reflexive coherent ${\mathcal{O}}_{X}$-module ${\mathcal{L}}$
on $X$ we have

$\int_{X}\Delta({\mathcal{E}}\hat{\otimes}{\mathcal{L}})L_{1}...L_{n-2}=\int_{X}\Delta({\mathcal{E}})L_{1}...L_{n-2}.$

  3. 3.

If $f:\tilde{X}\to X$ is a resolution of singularities and it has separably
generated residue field extensions then for any vector bundle ${\mathcal{F}}$
on $\tilde{X}$ we have

$\int_{\tilde{X}}\Delta({\mathcal{F}})f^{*}L_{1}...f^{*}L_{n-2}\geq\int_{X}\Delta(f_{[*]}{\mathcal{F}})L_{1}...L_{n-2}.$

  4. 4.

Let $\pi:Y\to X$ be a finite morphism from a normal projective variety $Y$.
Assume that either $\pi$ has separably generated residue field extensions or
it is the Frobenius morphism (or composition of such maps). Then for any
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ we have

$\int_{Y}c_{2}(\pi^{[*]}{\mathcal{E}})\pi^{*}L_{1}...\pi^{*}L_{n-2}=\deg\pi\cdot\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}.$

###### Proof.

Let us first assume that $k$ has positive characteristic. Passing to the base
change we can assume that the base field is uncountable. By Theorem 5.2, (4)
it is sufficient to check properties (1) and (2) after restricting to a very
general complete intersection surface. In this case these properties follow
from Proposition 4.2, (1) and (2). If $\pi$ is the Frobenius morphism then (4)
follows from the definition of $\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}.$
Apart from that case, we use Theorem 1.4 and Theorem 5.2, (4) to reduce (3)
and (4) to the surface case where these properties follow from Proposition
4.2, (3) and (4). For example in (4), Theorem 1.4 says that for general
$H\in|L_{1}|$ both $H$ and $\pi^{-1}(H)\in|\pi^{*}L_{1}|$ are normal and
irreducible, so we can reduce the statement to lower dimension. ∎

One of the most important results that follow from our theory is the following
Riemann–Roch type inequality:

######  Theorem 5.5.

Let $L_{1},...,L_{n-2}$ be very ample line bundles on $X$. Then there exists a
constant $C$ such that for all
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ we have

$\displaystyle\left\lvert\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{\mathcal{E}})-r\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{\mathcal{O}}_{X})-\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}\right.$
$\displaystyle\left.+\frac{1}{2}c_{1}({\mathcal{E}}).(K_{X}+L_{1}+...+L_{n-2}).L_{1}...L_{n-2}\right\rvert\leq
C\cdot r^{2},$

where $r$ is the rank of ${\mathcal{E}}$.

###### Proof.

Here we use notation from the proof of Lemma 2.6. Note that for any
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$ we have
by Theorem 4.4

$\displaystyle\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{\mathcal{E}})=$
$\displaystyle\chi({\mathcal{S}}_{K},{\mathcal{E}}_{K})=\int_{{\mathcal{S}}_{K}}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}_{K})-\frac{1}{2}c_{1}({\mathcal{E}}_{K}).K_{{\mathcal{S}}_{K}}$
$\displaystyle+r\chi({\mathcal{S}}_{K},{\mathcal{O}}_{{\mathcal{S}}_{K}})+\sum_{x\in{\mathop{{\rm
Sing\,}}}{\mathcal{S}}_{K}}a(x,{\mathcal{E}}_{K}).$

By adjunction we have
$K_{{\mathcal{S}}_{K}}=K_{X_{K}}+(L_{1})_{K}+...+(L_{n-2})_{K}$. Therefore

$c_{1}({\mathcal{E}}_{K}).K_{{\mathcal{S}}_{K}}=c_{1}({\mathcal{E}}).(K_{X}+L_{1}+...+L_{n-2}).L_{1}...L_{n-2}.$

We have
$\chi({\mathcal{S}}_{K},{\mathcal{O}}_{{\mathcal{S}}_{K}})=\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{\mathcal{O}}_{X}).$
By Theorem 5.2, (4) we also have

$\int_{{\mathcal{S}}_{K}}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}_{K})=\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}.$

Now the required inequality follows from Theorem 3.12 applied to singularities
of ${\mathcal{S}}_{K}$. ∎

######  Remark 5.6.

By definition of $\int_{X}c_{2}$ we immediately have

$\int_{Y}c_{2}(F_{X}^{[*]}{\mathcal{E}})L_{1}...L_{n-2}=p^{2}\cdot\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}$

for all ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$.
As in Theorem 4.5
$\int_{X}c_{2}:{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})\times
N^{1}(X)^{\times(n-2)}\to{\mathbb{R}}$ is uniquely determined by this property
and inequality from Theorem 5.5.

###  5.3 The second Chern class in characteristic zero

Here we assume that $k$ is an algebraically closed field of characteristic
$0$. In this subsection we assume that $X$ is a normal projective variety with
at most quotient singularities in codimension $2$. This means that any general
complete intersection surface in $X$ has at most quotient singularities.

######  Proposition 5.7.

Let ${\mathcal{E}}$ be a reflexive coherent ${\mathcal{O}}_{X}$-module of rank
$r$ and let $L_{1},...,L_{n-2}$ be line bundles on $X$. Then the sequence

$\left(\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{{\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}}})}{m^{r+1}}\right)_{m\in{\mathbb{N}}}$

is convergent to some rational number.

###### Proof.

Proof is similar to the proof of Proposition 5.1. Namely, for any line bundle
$M_{1}$ on $X$, Lemma 1.5, (1) and Lemma 1.7 imply that

$c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{{\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}}}=c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot({\mathcal{O}}_{X}^{\oplus\binom{m+r-1}{m}}-[{\mathcal{O}}_{-c_{1}({\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}})}]).$

Since $c_{1}({\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}})=\binom{m+r-1}{r}c_{1}({\mathcal{E}})$, Lemma 1.5
(1) and Lemma 1.6 give

$c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{-c_{1}({\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}})}]=-\binom{m+r-1}{r}c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot[{\mathcal{O}}_{c_{1}({\mathcal{E}})}].$

Therefore

$\lim_{m\to\infty}\frac{\chi(X,c_{1}(M_{1})c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{{\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}}})}{m^{r+1}}=0.$

Now the same arguments at that in the proof of Proposition 5.1 reduce the
assertion to the case when all $L_{i}$ are very ample. Similarly as before we
reduce to the case when the base field $k$ is uncountable and then restrict to
a very general complete intersection surface
$S\in|L_{1}|\cap...\cap|L_{n-2}|$. Then we get

$\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{{\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}}})}{m^{r+1}}=\lim_{m\to\infty}\frac{\chi(S,{\mathop{{\rm
Sym\,}}}^{[m]}({\mathcal{E}}|_{S}))}{m^{r+1}},$

which by Theorem 4.7 exists and is a rational number. ∎

The method of proof of Proposition 5.7 works for any normal projective variety
$X$ in characteristic $0$ for which we know Conjecture 3.18 for any general
complete intersection surface in $X$. As in the previous subsections, the
above proposition allows us to define
$\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}$ by

$\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}:=c_{1}({\mathcal{E}})^{2}.L_{1}...L_{n-2}-r!\lim_{m\to\infty}\frac{\chi(X,c_{1}(L_{1})...c_{1}(L_{n-2})\cdot{{\mathop{{\rm
Sym\,}}}^{[m]}{\mathcal{E}}})}{m^{r+1}}.$

We can use this to define $\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}$ and
$\int_{X}\Delta({\mathcal{E}})L_{1}...L_{n-2}$.

######  Theorem 5.8.

Let us fix
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$. The map
$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}):N^{1}(X)^{\times(n-2)}\to{\mathbb{Q}}$, sending
$(L_{1},...,L_{n-2})$ to $\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}$, satisfies the following properties:

  1. 1.

It is ${\mathbb{Z}}$-linear and symmetric.

  2. 2.

If ${\mathcal{E}}$ is a vector bundle on $X$ then

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}=\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})\cap c_{1}(L_{1})\cap...\cap c_{1}(L_{n-2})[X].$

  3. 3.

If $n>2$ and $L_{1}$ is very ample then for a very general hypersurface
$H\in|L_{1}|$ we have

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}=\int_{H}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}|_{H})L_{2}|_{H}...L_{n-2}|_{H}.$

  4. 4.

There exists some $N\in{\mathbb{N}}$ such that the image of
$\int_{X}{\mathop{\rm ch\,}}_{2}$ is contained in
$\frac{1}{N}{\mathbb{Z}}\subset{\mathbb{Q}}$.

###### Proof.

Properties (1)–(3) can be proven in the same way as Theorem 5.2. To prove (4)
it is sufficient to note that this holds on normal surfaces with only quotient
singularities and use the proof of Theorem 5.5 to reduce to this case. ∎

######  Remark 5.9.

Let $(X,D)$ be a projective klt pair defined over an algebraically closed
field of characteristic zero. In that case $X$ is well known to have quotient
singularities in codimension $2$. Then for fixed
${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$, the
multilinear forms ${\mathop{\rm Pic\,}}X^{\times(n-2)}\to{\mathbb{Q}}$ given
by sending $(L_{1},...,L_{n-2})$ to
$c_{1}({\mathcal{E}})^{2}.L_{1}...L_{n-2}$,
$\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}$ or $\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}$ coincide (modulo passing to
numerical equivalence classes) with analogous forms considered in [GKPT,
Theorem 3.13]. This follows, e.g., from [La1, Theorem 5.1] and [GKPT,
(3.13.2)]. The construction of these forms in [GKPT] uses Mumford’s Chern
classes for ${\mathbb{Q}}$-bundles on ${\mathbb{Q}}$-varieties and it does not
generalize to varieties that do not have quotient singularities in codimension
$2$. Moreover, this construction works well only in the characteristic zero
case.

###  5.4 Properties of Chern classes in characteristic zero

In this subsection we keep the notation form previous subsection. Then we have
the following proposition analogous to Proposition 4.2.

######  Proposition 5.10.

For any line bundles $L_{1},...,L_{n-2}$ on $X$ the following conditions are
satisfied:

  1. 1.

For any rank $1$ reflexive coherent ${\mathcal{O}}_{X}$-module ${\mathcal{L}}$
on $X$ we have

$\int_{X}c_{2}({\mathcal{L}})L_{1}...L_{n-2}=0.$

  2. 2.

For any ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$
and any rank $1$ reflexive coherent ${\mathcal{O}}_{X}$-module ${\mathcal{L}}$
on $X$ we have

$\int_{X}\Delta({\mathcal{E}}\hat{\otimes}{\mathcal{L}})L_{1}...L_{n-2}=\int_{X}\Delta({\mathcal{E}})L_{1}...L_{n-2}.$

  3. 3.

If $f:\tilde{X}\to X$ is a resolution of singularities then for any vector
bundle ${\mathcal{F}}$ on $\tilde{X}$ we have

$\int_{\tilde{X}}\Delta({\mathcal{F}})f^{*}L_{1}...f^{*}L_{n-2}\geq\int_{X}\Delta(f_{[*]}{\mathcal{F}})L_{1}...L_{n-2}.$

  4. 4.

If $\pi:Y\to X$ is a finite morphism from a normal projective variety $Y$ then
for any ${\mathcal{E}}\in{\mathop{\operatorname{Ref}\,}}({\mathcal{O}}_{X})$
we have

$\int_{Y}c_{2}(\pi^{[*]}{\mathcal{E}})\pi^{*}L_{1}...\pi^{*}L_{n-2}=\deg\pi\cdot\int_{X}c_{2}({\mathcal{E}})L_{1}...L_{n-2}.$

###### Proof.

The proof is the same as that of Proposition 5.4 except that our assumption on
morphisms to have separably generated residue field extensions is
automatically satisfied in characteristic zero. ∎

######  Remark 5.11.

In case $k$ has characteristic $0$ and $f:Y\to X$ is a quasi-étale morphism of
klt pairs the property (4) was proven in [GKPT, Lemma 3.16]. Note that our
assertion is much stronger as it does not require $f$ to be quasi-étale.

In characteristic $0$ the Riemann–Roch type inequality analogous to that from
Theorem 5.5 is also satisfied but we will not use it in the following.

###  5.5 K-theoretic formulation

Let $X$ be a normal projective variety defined over an algebraically closed
field $k$. We define the _Grothendieck group $K^{{\mathop{\operatorname{\rm
ref}}}}(X)$ of reflexive sheaves on $X$_ as the free abelian group on the
isomorphism classes $[{\mathcal{E}}]$ of coherent reflexive
${\mathcal{O}}_{X}$-modules modulo the relations
$[{\mathcal{E}}_{2}]=[{\mathcal{E}}_{1}]+[{\mathcal{E}}_{3}]$ for each locally
split short exact sequence

$0\to{\mathcal{E}}_{1}\to{\mathcal{E}}_{2}\to{\mathcal{E}}_{3}\to 0$

of coherent reflexive ${\mathcal{O}}_{X}$-modules. Note that
$K^{{\mathop{\operatorname{\rm ref}}}}(X)$ is a ring with unity
$[{\mathcal{O}}_{X}]$ and multiplication given by

$[{\mathcal{E}}_{1}]\cdot[{\mathcal{E}}_{2}]=[{\mathcal{E}}_{1}\hat{\otimes}{\mathcal{E}}_{2}].$

We have well defined ${\mathbb{Z}}$-linear maps ${\mathop{\rm
ch\,}}_{1}:K^{{\mathop{\operatorname{\rm ref}}}}(X)\to A^{1}(X)$,
$[{\mathcal{E}}]\to c_{1}({\mathcal{E}})$ and
$\chi:K^{{\mathop{\operatorname{\rm ref}}}}(X)\to{\mathbb{Z}}$,
$[{\mathcal{E}}]\to\chi(X,{\mathcal{E}})$.

Let us assume that $k$ has positive characteristic $p$. Since the functor
$F_{X}^{[*]}$ is exact on locally split short exact sequences, we have a well
defined homomorphism of rings

$F_{X}^{[*]}:K^{{\mathop{\operatorname{\rm ref}}}}(X)\to
K^{{\mathop{\operatorname{\rm ref}}}}(X)$

sending $[{\mathcal{E}}]$ to $[F_{X}^{[*]}{\mathcal{E}}]$ (here we use that in
the definition of $K^{{\mathop{\operatorname{\rm ref}}}}(X)$ we consider only
locally split short exact sequences). If $n\geq 2$ and we fix some line
bundles $L_{1},...,L_{n-2}$ then $\int_{X}{\mathop{\rm
ch\,}}_{2}(\cdot)L_{1}...L_{n-2}$ defines a ${\mathbb{Z}}$-linear map
$K^{{\mathop{\operatorname{\rm ref}}}}(X)\to{\mathbb{R}}$ (this can be proven
as Corollary 3.17; see [La4, Lemma 2.1]).

If $X$ is a surface then we define the Chern character

${\mathop{\rm ch\,}}:K^{{\mathop{\operatorname{\rm ref}}}}(X)\to
A^{*}(X)\otimes{\mathbb{R}}$

by setting ${\mathop{\rm ch\,}}([{\mathcal{E}}]):={\mathop{\rm
rk\,}}{\mathcal{E}}+c_{1}({\mathcal{E}})+{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})$. Corollary 3.17 and our definitions imply that this
extends to a homomorphism of abelian groups. Note that
$A^{*}(X)\otimes{\mathbb{R}}$ is a ring and conjecturally, ${\mathop{\rm
ch\,}}$ is also a homomorphism of rings (see Remark 3.4).

##  6 Applications

Let $X$ be a normal projective variety of dimension $n$ defined over an
algebraically closed field $k$. Let ${\mathcal{O}}_{X}(1)$ be an ample line
bundle on $X$. All the results below hold when we replace
${\mathcal{O}}_{X}(1)$ by a collection of ample line bundles but proofs become
much more complicated and we deal with these results in [La4].

###  6.1 Boundedness on normal varieties

We will often write $H$ for a Cartier divisor such that
${\mathcal{O}}_{X}(1)={\mathcal{O}}_{X}(H)$. Let ${\mathcal{E}}$ be a torsion
free coherent ${\mathcal{O}}_{X}$-module of rank $r$. We will write
$H^{i}\cdot{\mathcal{E}}$ for the class
$c_{1}({\mathcal{O}}_{X}(1))^{i}\cdot{\mathcal{E}}$ in $K(X)$. By [Ko, Chapter
VI, Theorem 2.13] we have

$\chi(X,{\mathcal{E}}(m))=\sum_{i=0}^{n}\chi(X,H^{i}\cdot{\mathcal{E}})\binom{m+i-1}{i}.$

There are also uniquely determined integers
$a_{0}^{H}({\mathcal{E}}),...,a_{n}^{H}({\mathcal{E}})$ such that

$\chi(X,{\mathcal{E}}(m))=\sum_{i=0}^{n}a_{i}^{H}({\mathcal{E}})\binom{m+n-i}{n-i}.$

######  Lemma 6.1.

We have $a_{0}^{H}({\mathcal{E}})=rH^{n}$,

$a_{1}^{H}({\mathcal{E}})=c_{1}({\mathcal{E}}).H^{n-1}-\frac{r}{2}(K_{X}+(n-1)H).H^{n-1}-rH^{n}$

and

$a_{2}^{H}({\mathcal{E}})=\chi(X,H^{n-2}\cdot{\mathcal{E}})-c_{1}({\mathcal{E}}).H^{n-1}+\frac{r}{2}(K_{X}+(n-1)H).H^{n-1}.$

###### Proof.

Using Theorem 1.4 and [HL, Lemma 1.1.12 and Corollary 1.1.14]) we can
construct a sequence of divisors $H_{1},...,H_{n}\in|{\mathcal{O}}_{X}(1)|$
such that for all $i=1,...,n$ the following conditions are satisfied:

  1. 1.

the intersection $\bigcap_{j\leq i}H_{j}$ is normal,

  2. 2.

the restriction ${\mathcal{E}}|_{\bigcap_{j\leq i}H_{j}}$ is torsion free of
rank $r$.

From the definition we see that
$a_{0}^{H}({\mathcal{E}})=\chi({\mathcal{E}}|_{\bigcap_{j\leq n}H_{j}})$ and

$a_{i}^{H}({\mathcal{E}})=\chi({\mathcal{E}}|_{\bigcap_{j\leq
n-i}H_{j}})-\chi({\mathcal{E}}|_{\bigcap_{j\leq n-i+1}H_{j}})$

for $i>0$. So the equality $a_{0}^{H}({\mathcal{E}})=rH^{n}$ is clear. By the
above the curve $C:={\bigcap_{j\leq n-1}H_{j}}$ is smooth and by the
adjunction formula we have

$-2\chi({\mathcal{O}}_{C})=\deg\omega_{C}=\deg{\mathcal{O}}_{C}(K_{X}+H_{1}+...+H_{n-1})=(K_{X}+(n-1)H).H^{n-1}.$

This, together with the Riemann–Roch theorem for ${\mathcal{E}}|_{C}$, gives
the formula for $a_{1}^{H}({\mathcal{E}})$. The last formula follows easily
from the previous two formulas and equality
$a_{2}^{H}({\mathcal{E}})=\chi({\mathcal{E}}|_{\bigcap_{j\leq
n-2}H_{j}})-(a_{0}^{H}({\mathcal{E}})+a_{1}^{H}({\mathcal{E}}))$. ∎

We define a _slope_ of ${\mathcal{E}}$ with respect to $H$ as

$\mu_{H}({\mathcal{E}}):=\frac{c_{1}({\mathcal{E}}).H^{n-1}}{r}.$

This allows us to define slope $H$-semistability and the maximal
$H$-destabilizing slope $\mu_{\max,H}({\mathcal{E}})$ for any coherent torsion
free ${\mathcal{O}}_{X}$-module ${\mathcal{E}}$.

Let us recall the following special case of [La3, Theorem 4.4].

######  Theorem 6.2.

Let $(X,H)$ be as above and let us fix some integers $a_{0}$, $a_{1}$, $a_{2}$
and a rational number $\mu_{\max}$. Then the set of coherent reflexive
${\mathcal{O}}_{X}$-modules ${\mathcal{E}}$ with
$a_{0}^{H}({\mathcal{E}})=a_{0}$, $a_{1}^{H}({\mathcal{E}})=a_{1}$,
$a_{2}^{H}({\mathcal{E}})\geq a_{2}$ and
$\mu_{\max,H}({\mathcal{E}})\leq\mu_{\max}$ is bounded.

Using Lemma 6.1 we can rewrite the above theorem in the following way:

######  Corollary 6.3.

Let us fix some positive integer $r$, integers $c_{1}$ and $\chi$ and some
rational number $\mu_{\max}$. Then the set of coherent reflexive
${\mathcal{O}}_{X}$-modules ${\mathcal{E}}$ of rank $r$ with
$c_{1}({\mathcal{E}}).H^{n-1}=c_{1}$,
$\chi(X,H^{n-2}\cdot{\mathcal{E}})\geq\chi$ and
$\mu_{\max,H}({\mathcal{E}})\leq\mu_{\max}$ is bounded. In particular, the set
of slope $H$-semistable coherent reflexive ${\mathcal{O}}_{X}$-modules
${\mathcal{E}}$ of rank $r$ with $c_{1}({\mathcal{E}}).H^{n-1}=c_{1}$ and
$\chi(X,H^{n-2}\cdot{\mathcal{E}})\geq\chi$ is bounded.

###  6.2 Boundedness and Bogomolov’s inequality on normal varieties in
positive characteristic

In this subsection we assume that the base field $k$ has positive
characteristic. In the following, we write $N_{H}(X)$ for the group $N_{L}(X)$
introduced in Subsection 2.2 in the case of
$(L_{1},...,L_{n-2})=({\mathcal{O}}_{X}(1),...,{\mathcal{O}}_{X}(1))$.

######  Theorem 6.4.

Let us fix some positive integer $r$ and some real numbers $c_{2}$ and
$\mu_{\max}$. Let us also fix some class $c_{1}\in N_{H}(X)$. Then the set
${\mathcal{A}}$ of coherent reflexive ${\mathcal{O}}_{X}$-modules
${\mathcal{E}}$ of rank $r$ with $[c_{1}({\mathcal{E}})]=c_{1}\in N_{H}(X)$,
$\int_{X}c_{2}({\mathcal{E}})H^{n-2}\leq c_{2}$ and
$\mu_{\max,H}({\mathcal{E}})\leq\mu_{\max}$ is bounded.

###### Proof.

By Theorem 5.5 we have

$\displaystyle\chi(X,H^{n-2}\cdot{\mathcal{E}})\leq\frac{1}{2}c_{1}.(c_{1}-(K_{X}+(n-2)H)).H^{n-2}-\int_{X}c_{2}({\mathcal{E}})H^{n-2}$
$\displaystyle+r\chi(X,H^{n-2}\cdot{\mathcal{O}}_{X})+C\cdot r.$

So our assertion follows from Corollary 6.3. ∎

The proof of the following theorem was motivated by a similar proof by
Maruyama in the smooth case (see the proof of [Ma, Corollary 2.10]).

######  Theorem 6.5.

Let us fix some positive integer $r$ and some non-negative rational number
$\alpha$. There exists some constant $\tilde{C}=\tilde{C}(X,H,r,\alpha)$
depending only on $X$, $H$, $r$ and $\alpha$ such that for every coherent
reflexive ${\mathcal{O}}_{X}$-module ${\mathcal{E}}$ of rank $r$ with
$\mu_{\max,H}({\mathcal{E}})-\mu_{H}({\mathcal{E}})\leq\alpha$ we have

$\int_{X}\Delta({\mathcal{E}})H^{n-2}\geq\tilde{C}.$

###### Proof.

Let us choose a basis $L_{1},....,L_{s}$ of $N_{H}(X)$ as a
${\mathbb{Z}}$-module (see Lemma 2.6) and let us write
$[c_{1}({\mathcal{E}})]=\sum a_{i}L_{i}$ for some $a_{i}\in{\mathbb{Z}}$.
There exist uniquely determined integers $q_{i}$ and $r_{i}$ such that
$a_{i}=q_{i}r+r_{i}$ and $0\leq r_{i}<r$. Since

$\int_{X}\Delta({\mathcal{E}})H^{n-2}=\int_{X}\Delta({\mathcal{E}}(-\sum
q_{i}L_{i}))H^{n-2}$

and there are only finitely many possibilities for $[c_{1}({\mathcal{E}}(-\sum
q_{i}L_{i}))]=\sum r_{i}L_{i}$, it is sufficient to prove existence of the
above constant assuming that $c_{1}=[c_{1}({\mathcal{E}})]\in N_{H}(X)$ is
fixed.

If $\int_{X}c_{2}({\mathcal{E}})H^{n-2}\geq 0$ then
$\int_{X}\Delta({\mathcal{E}})H^{n-2}\geq-(r-1)c_{1}^{2}.H^{n-2}$.

On the other hand, by Theorem 6.4 the set ${\mathcal{A}}$ of reflexive
coherent ${\mathcal{O}}_{X}$-modules ${\mathcal{E}}$ of rank $r$ with
$[c_{1}({\mathcal{E}})]=c_{1}\in N_{H}(X)$,
$\int_{X}c_{2}({\mathcal{E}})H^{n-2}\leq 0$ and

$\mu_{\max,H}({\mathcal{E}})\leq\mu_{\max}=\alpha+\frac{1}{r}c_{1}.H^{n-1}$

is bounded. So there exists some constant $D$ such that for every
${\mathcal{E}}\in{\mathcal{A}}$ and any ${\mathcal{E}}$-regular sequence
$H_{1},...,H_{n-2}\in|{\mathcal{O}}_{X}(1)|$ we have

$\chi(X,H^{n-2}\cdot{\mathcal{E}})=\chi(X,{\mathcal{E}}|_{\bigcap_{j\leq
n-2}H_{j}})\leq D.$

But Theorem 5.5 gives

$\displaystyle\chi(X,H^{n-2}\cdot{\mathcal{E}})\geq$
$\displaystyle\frac{1}{2r}c_{1}.(c_{1}-r(K_{X}+(n-2)H)).H^{n-2}-\frac{1}{2r}\int_{X}\Delta({\mathcal{E}})H^{n-2}$
$\displaystyle+r\chi(X,H^{n-2}\cdot{\mathcal{O}}_{X})-C\cdot r^{2}.$

Therefore for any ${\mathcal{E}}\in{\mathcal{A}}$ we have

$\displaystyle\int_{X}\Delta({\mathcal{E}})H^{n-2}\geq\max$
$\displaystyle(-(r-1)c_{1}^{2}.H^{n-2},c_{1}.(c_{1}-r(K_{X}+(n-2)H).H^{n-2}$
$\displaystyle+2r^{2}\chi(X,H^{n-2}\cdot{\mathcal{O}}_{X})-2C\cdot
r^{3}+2D\cdot r).$

∎

The above theorem easily implies Bogomolov’s inequality for strongly
semistable reflexive sheaves:

######  Corollary 6.6.

Let ${\mathcal{E}}$ be a coherent reflexive ${\mathcal{O}}_{X}$-module. If
${\mathcal{E}}$ is strongly slope $H$-semistable then

$\int_{X}\Delta({\mathcal{E}})H^{n-2}\geq 0.$

###### Proof.

Let us consider the set $\\{F_{X}^{[m]}{\mathcal{E}}\\}_{m\geq 0}$. Note that
by assumption each sheaf in this set is reflexive of rank $r$ and slope
$H$-semistable, i.e.,
$\mu_{\max,H}(F_{X}^{[m]}{\mathcal{E}})-\mu_{H}(F_{X}^{[m]}{\mathcal{E}})=0$.
So by Theorem 6.5 there exists a constant $\tilde{C}$ such that

$\int_{X}\Delta(F_{X}^{[m]}{\mathcal{E}})H^{n-2}=p^{2m}\int_{X}\Delta({\mathcal{E}})H^{n-2}\geq\tilde{C}$

for all $m\geq 0$. Dividing by $p^{2m}$ and passing with $m$ to infinity, we
get the required inequality. ∎

######  Corollary 6.7.

Let us fix some positive integer $r$, integer ${\mathop{\rm ch\,}}_{1}$ and
some real numbers ${\mathop{\rm ch\,}}_{2}$ and $\mu_{\max}$. Then the set
${\mathcal{B}}$ of coherent reflexive ${\mathcal{O}}_{X}$-modules
${\mathcal{E}}$ of rank $r$ with $\int_{X}{\mathop{\rm
ch\,}}_{1}({\mathcal{E}}).H^{n-1}={\mathop{\rm ch\,}}_{1}$,
$\int_{X}{\mathop{\rm ch\,}}_{2}({\mathcal{E}}).H^{n-2}\geq{\mathop{\rm
ch\,}}_{2}$ and $\mu_{\max,H}({\mathcal{E}})\leq\mu_{\max}$ is bounded.

###### Proof.

By Theorem 6.5 there exists a constant $\tilde{C}$ such that for all
${\mathcal{E}}\in{\mathcal{B}}$ we have

$\tilde{C}\leq\int_{X}\Delta({\mathcal{E}})H^{n-2}=c_{1}({\mathcal{E}})^{2}.H^{n-2}-2r\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}).H^{n-2}.$

Therefore $c_{1}({\mathcal{E}})^{2}.H^{n-2}\geq\tilde{C}+2r\,{\mathop{\rm
ch\,}}_{2}$. Let us write $[c_{1}({\mathcal{E}})]=\alpha[H]+D\in N_{H}(X)$,
where $\alpha=\frac{{\mathop{\rm ch\,}}_{1}}{H^{n}}$. Then $D.H^{n-1}=0$ and
$c_{1}({\mathcal{E}})^{2}.H^{n-2}=\alpha^{2}H^{n}+D^{2}.H^{n-2}$ and we have

$D^{2}.H^{n-2}\geq\tilde{C}+2r\,{\mathop{\rm ch\,}}_{2}-\alpha^{2}H^{n}.$

But by the Hodge index theorem (see Lemma 2.6) the intersection form is
negative definite on $H^{\perp}\subset N_{H}(X)$, so there are only finitely
many possibilities for $D$ and hence there are also finitely many
possibilities for the classes $[c_{1}({\mathcal{E}})]\in N_{H}(X)$. Now the
assertion follows from Theorem 6.4. ∎

###  6.3 Application to F-divided sheaves on normal varieties

In this section we use the above developed theory to reprove [ES, Theorem 2.1]
in the style well-known from the smooth varieties.

Let $X$ be a normal projective variety of dimension $\geq 1$ defined over an
algebraically closed field $k$. Let $j:U\hookrightarrow X$ be an open
embedding of a big open subset contained in the regular locus of $X$. Let
${\mathbb{E}}=({\mathcal{E}}_{m},\sigma_{m})_{m\geq 0}$ be an F-divided sheaf
on $U$ and let us set $\tilde{\mathcal{E}}_{m}:=j_{*}{\mathcal{E}}_{m}$.

######  Lemma 6.8.

For every ample divisor $H$ we have $[c_{1}(\tilde{\mathcal{E}}_{m})]=0\in
N_{H}(X)$ for all $m$ and
$\lim_{m\to\infty}\int_{X}c_{2}(\tilde{\mathcal{E}}_{m})H^{n-2}=0$.

###### Proof.

The isomorphisms $\sigma_{m}:{\mathcal{E}}_{m}\to
F_{X}^{*}{\mathcal{E}}_{m+1}$ extend uniquely to isomorphisms
$\tilde{\mathcal{E}}_{m}\to F_{X}^{[*]}\tilde{\mathcal{E}}_{m+1}$. In
particular, we see that
$c_{1}(\tilde{\mathcal{E}}_{0})=p^{m}c_{1}(\tilde{\mathcal{E}}_{m})$. By Lemma
2.6 there exists some $N$ such that for every Weil divisor $D$ on $X$ we have

$N\cdot c_{1}(\tilde{\mathcal{E}}_{0}).D.H^{n-2}=p^{m}N\cdot
c_{1}(\tilde{\mathcal{E}}_{m}).D.H^{n-2}\in p^{m}{\mathbb{Z}}.$

Since $N_{H}(X)$ is a free ${\mathbb{Z}}$-module, the class of
$c_{1}(\tilde{\mathcal{E}}_{0})$ in $N_{H}(X)$ must vanish. This also implies
vanishing of all classes $[c_{1}(\tilde{\mathcal{E}}_{m})]\in N_{H}(X)$. The
second assertion follows from equalities
$\int_{X}c_{2}(\tilde{\mathcal{E}}_{0})H^{n-2}=p^{2m}\int_{X}c_{2}(\tilde{\mathcal{E}}_{m})H^{n-2}$.
∎

The following corollary is due to Esnault and Srinivas (see [ES, Theorem 2.1]
for a slightly weaker statement).

######  Corollary 6.9.

There exists a bounded set of (slope $H$-semistable) sheaves ${\mathcal{S}}$
such that for every $r>0$ and every rank $r$ F-divided sheaf
${\mathbb{E}}=({\mathcal{E}}_{m},\sigma_{m})_{m\geq 0}$ on $U$ there exists
$m_{0}({\mathbb{E}})\in{\mathbb{Z}}_{\geq 0}$ such that for all $m\geq
m_{0}({\mathbb{E}})$ the sheaves ${\mathcal{E}}_{m}$ belong to the set
${\mathcal{S}}$.

###### Proof.

Let us fix a positive real number $c_{2}$ and let ${\mathcal{S}}$ denote the
set of slope $H$-semistable sheaves ${\mathcal{F}}$ on $X$ such that
$[c_{1}({\mathcal{F}})]=0\in N_{H}(X)$ and
$\int_{X}c_{2}({\mathcal{F}})H^{n-2}\leq c_{2}$. By Theorem 6.4 this set is
bounded.

Now fix ${\mathbb{E}}=({\mathcal{E}}_{m},\sigma_{m})_{m\geq 0}$ on $U$ and as
before set $\tilde{\mathcal{E}}_{m}:=j_{*}{\mathcal{E}}_{m}$. Since

$\mu_{\max,H}(\tilde{\mathcal{E}}_{m})\geq
p\mu_{\max,H}(\tilde{\mathcal{E}}_{m+1}),$

we have $\mu_{\max,H}(\tilde{\mathcal{E}}_{0})\geq
p^{m}\mu_{\max,H}(\tilde{\mathcal{E}}_{m})$. Since
$r!\cdot\mu_{\max,H}(\tilde{\mathcal{E}}_{m})\in{\mathbb{Z}}$, there exists
$m_{1}({\mathbb{E}})\in{\mathbb{Z}}_{\geq 0}$ such that for all $m\geq
m_{1}({\mathbb{E}})$ we have $\mu_{\max,H}(\tilde{\mathcal{E}}_{m})\leq 0$.
Since $\mu_{H}(\tilde{\mathcal{E}}_{m})=0$ we see that the sheaves
$\tilde{\mathcal{E}}_{m}=j_{*}{\mathcal{E}}_{m}$ are slope $H$-semistable for
all $m\geq m_{1}({\mathbb{E}})$. Now the above lemma implies that for any
${\mathbb{E}}$ there exists $m_{0}({\mathbb{E}})\in{\mathbb{Z}}_{\geq 0}$ such
that ${\mathcal{E}}_{m}\in{\mathcal{S}}$ for all $m\geq m_{0}({\mathbb{E}})$.
∎

######  Remark 6.10.

The above proof shows that one can choose $m_{0}$ that does not depend on
${\mathbb{E}}$ but only on the rank $r$ if and only if
$\int_{X}c_{2}(\tilde{\mathcal{E}}_{m})H^{n-2}=0$ for all $m\geq 0$. In
general, this would follow if we knew that the second relative Chern classes
of vector bundles defined in Subsection 3.1 are rational with bounded
denominators but this seems very unlikely. However, this happens, e.g., if $X$
has only quotient singularities in codimension $2$ (cf. Theorem 5.8, (5)).

## Acknowledgements

The author would like to thank M. Enokizono for pointing out [En].

A large part of the paper was written while the author was an External Senior
Fellow at Freiburg Institute for Advanced Studies (FRIAS), University of
Freiburg, Germany. The author would like to thank Stefan Kebekus for his
hospitality during the author’s stay in FRIAS.

The author was partially supported by Polish National Centre (NCN) contract
numbers 2018/29/B/ST1/01232 and 2021/41/B/ST1/03741. The research leading to
these results has received funding from the European Union’s Horizon 2020
research and innovation programme under the Maria Skłodowska-Curie grant
agreement No 754340.

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