# On holomorphic mappings with compact type range

A. Jiménez-Vargas  Departamento de Matemáticas, Universidad de Almería, 04120,
Almería, Spain<EMAIL_ADDRESS>,  D. Ruiz-Casternado  Departamento de
Matemáticas, Universidad de Almería, 04120, Almería, Spain
<EMAIL_ADDRESS>and  J. M. Sepulcre  Departamento de Matemáticas,
Universidad de Alicante, 03080 Alicante, Spain<EMAIL_ADDRESS>
###### Abstract.

Using Mujica’s linearization theorem, we extend to the holomorphic setting
some classical characterizations of compact (weakly compact, Rosenthal,
Asplund) linear operators between Banach spaces such as the Schauder,
Gantmacher and Gantmacher–Nakamura theorems and the
Davis–Figiel–Johnson–Pełczynski, Rosenthal and Asplund factorization theorems.

###### Key words and phrases:

Vector-valued holomorphic mapping, operator ideal, linearization,
factorization theorems, Schauder’s theorem.

###### 2020 Mathematics Subject Classification:

46E10, 46E15, 46G20

This paper was finished while the first author was visiting the Department of
Mathematics at Alicante University (Spain). He thanks this Department for its
kind hospitality. The first author was partially supported by project UAL-
FEDER grant UAL2020-FQM-B1858, by Junta de Andalucía grants P20$\\_$00255 and
FQM194, and by Ministerio de Ciencia e Innovación grant PID2021-122126NB-C31.
The third author was also supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF,
UE)

## Introduction

S. Dineen [6, p. 417] pointed out that a remarkable result due to K. F. Ng
[18] provides a Banach space $\mathcal{G}^{\infty}(U)$ whose dual is
isometrically isomorphic to the Banach space $\mathcal{H}^{\infty}(U)$ of all
bounded holomorphic complex-valued functions on an open subset $U$ of an
arbitrary complex Banach space $E$, endowed with the supremum norm.

J. Mujica presented in [16, Theorem 2.1] a linearization theorem for bounded
holomorphic mappings which is a refinement of Dineen’s idea. Specifically, he
proved that there exist a Banach space $\mathcal{G}^{\infty}(U)$ and a bounded
holomorphic mapping $\delta_{U}\colon U\to\mathcal{G}^{\infty}(U)$ satisfying
the following universal property: for each complex Banach space $F$ and each
bounded holomorphic mapping $f\colon U\to F$, there exists a unique bounded
linear operator $T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ such that
$T_{f}\circ\delta_{U}=f$. R. Ryan [23] had previously obtained a polynomial
version of Mujica’s theorem by using a tensor product approach. In [7], P.
Galindo, D. García and M. Maestre established a linearization theorem for
holomorphic mappings of bounded type. Moreover, J. Mujica and L. Nachbin also
obtained in [17] a linearization theorem for holomorphic functions between
locally convex spaces.

Applying Ng’s theorem, Mujica defined the space $\mathcal{G}^{\infty}(U)$ as
the norm-closed linear subspace of $\mathcal{H}^{\infty}(U)^{*}$ formed by
those functionals which are $\tau_{c}$-continuous when restricted to the
closed unit ball of $\mathcal{H}^{\infty}(U)$, where $\tau_{c}$ denotes the
compact-open topology on $\mathcal{H}^{\infty}(U)$. The correspondence
$f\mapsto T_{f}$ is an isometric isomorphism between the space
$\mathcal{H}^{\infty}(U,F)$ of all bounded holomorphic mappings from $U$ into
$F$ with the supremum norm, and the space $\mathcal{L}(G^{\infty}(U);F)$ of
all continuous linear operators from $\mathcal{G}^{\infty}(U)$ into $F$ with
the operator norm. The mapping $\delta_{U}\colon U\to\mathcal{G}^{\infty}(U)$
is defined by $\delta_{U}(x)=\delta(x)$, where $\delta(x)$ is the evaluation
functional at $x$ defined on $\mathcal{H}^{\infty}(U)$. The proof of Theorem
2.1 in [16] shows that the closed unit ball of $\mathcal{G}^{\infty}(U)$
coincides with the norm-closed absolutely convex hull of $\delta_{U}(U)$. In
particular, $\mathcal{G}^{\infty}(U)$ is the norm-closed linear hull of
$\delta_{U}(U)$ in $\mathcal{H}^{\infty}(U)^{*}$.

In Section 3 of his paper [16], Mujica briefly dealt with holomorphic mappings
that have compact type range. To be more precise, he showed that a mapping
$f\in\mathcal{H}^{\infty}(U,F)$ has relatively compact range (respectively,
relatively weakly compact range, finite rank) if and only if its linearization
$T_{f}\in\mathcal{L}(G^{\infty}(U);F)$ is a compact (respectively, weakly
compact, bounded finite-rank) operator. Our purpose in this paper is to
complete the study initiated by Mujica.

From a local point of view, the properties of compactness, weak compactness,
Rosenthal and Asplund for holomorphic mappings from $U$ into $F$ were
addressed by R. M. Aron and M. Schottenloher [2], R. Ryan [24], M. Lindström
[13] and N. Robertson [21], respectively. Let us recall that a mapping
$f\colon U\to F$ is said to be locally compact (respectively, locally weakly
compact, locally Rosenthal, locally Asplund) if every point $x\in U$ has a
neighborhood $V_{x}\subseteq U$ such that $f(V_{x})$ is relatively compact
(respectively, relatively weakly compact, Rosenthal, Asplund) in $F$. Clearly,
every mapping $f\colon U\to F$ having relatively compact range (respectively,
relatively weakly compact range, Rosenthal range, Asplund range) is locally
compact (respectively, locally weakly compact, locally Rosenthal, locally
Asplund), however, the converse is not true in general for mappings
$f\in\mathcal{H}^{\infty}(U,F)$ (see Example 3.2 in [16] for the first two
types of mappings).

We have organized this note as follows. In Section 1, we recall Mujica’s
linearization theorem and some of its consequences that will be needed to
establish our results. With the aid of the notion of transpose mapping of a
bounded holomorphic mapping, Section 2 is devoted to the analogues for bounded
holomorphic mappings of the results due to Schauder, Gantmacher and Nakamura
on the compactness and weak compactness of the adjoint of a bounded linear
operator between Banach spaces. We suggest the reader to compare our results
with Schauder and Gantmacher type theorems for holomorphic mappings of bounded
type established by M. González and J. M. Gutiérrez [8, 9, 10] and R. Ryan
[23, 24].

As a main result, we show that some factorization theorems for bounded linear
operators between Banach spaces can be extended to the holomorphic setting.
This is the case of the Davis–Figiel–Johnson–Pełczynski factorization theorem
[5] which states that every weakly compact operator factors through a
reflexive Banach space, the Rosenthal factorization theorem (see [1]) which
asserts that every Rosenthal operator factors through a Banach space not
containing $\ell_{1}$, and the Asplund factorization theorem which assures
that every Asplund operator factors through an Asplund space (see [4, Theorem
5.3.5]).

We refer to the book of R. E. Megginson [14] for a complete study on weak
topologies and linear operators on Banach spaces; to the monograph of J.
Mujica [15] for the theory of holomorphic mappings on Banach spaces; and to
the book of A. Pietsch [20] for the theory of operator ideals.

Notation. Through the paper, given a complex Banach space $E$, we denote by
$B_{E}$, $\stackrel{{\scriptstyle\circ}}{{B}}_{E}$, $S_{E}$ and $E^{*}$ the
closed unit ball, the open unit ball, the unit sphere and the dual space of
$E$, respectively. For a set $A\subseteq E$, $\mathrm{lin}(A)$,
$\overline{\mathrm{lin}}(A)$, $\mathrm{co}(A)$, $\overline{\mathrm{co}}(A)$,
$\mathrm{aco}(A)$ and $\overline{\mathrm{aco}}(A)$ stand for the linear hull,
the norm-closed linear hull, the convex hull, the norm-closed convex hull, the
absolutely convex hull, the norm-closed absolutely convex hull of $A$ in $E$,
respectively. If $E$ and $F$ are locally convex Hausdorff spaces,
$\mathcal{L}(E;F)$ denotes the vector space of all continuous linear operators
from $E$ into $F$. Unless stated otherwise, if $E$ and $F$ are Banach spaces,
we will understand that they are endowed with the norm topology. Given
$T\in\mathcal{L}(E;F)$, $T^{*}\colon F^{*}\to E^{*}$ denotes the adjoint
operator of $T$.

We will sometimes use the following notation: for each $x\in E$ and $x^{*}\in
E^{*}$, $\left\langle x^{*},x\right\rangle$ is defined to be $x^{*}(x)$.

##  1\. Preliminaries

Let $U$ be an open subset of a complex Banach space $E$. We know that
$\mathcal{H}^{\infty}(U)$ is a Banach space under the supremum norm and it is
actually a dual Banach space. In fact, there are different ways to construct a
predual of $\mathcal{H}^{\infty}(U)$. The most straightforward one is as the
norm-closed linear subspace of $\mathcal{H}^{\infty}(U)^{*}$ generated by the
functionals $\delta(x)\in\mathcal{H}^{\infty}(U)^{*}$ with $x\in U$, defined
by

$\left\langle\delta(x),f\right\rangle=f(x)\qquad\left(f\in\mathcal{H}^{\infty}(U)\right).$

Theorem 2.1 in [16] justifies the following notation.

######  Definition 1.1.

Let $U$ be an open subset of a complex Banach space $E$. The space
$\mathcal{G}^{\infty}(U)$ is the norm-closed linear subspace of
$\mathcal{H}^{\infty}(U)^{*}$ given by
$\overline{\mathrm{lin}}\left\\{\delta(x)\colon x\in U\right\\}$.

For consistency with the preceding notation, we may consider the mapping
$\delta_{U}\colon U\to\mathcal{G}^{\infty}(U)$ defined by
$\delta_{U}(x)=\delta(x)$ for $x\in U$.

In [16], Mujica established the following properties of
$\mathcal{G}^{\infty}(U)$ and $\delta_{U}$.

######  Theorem 1.2.

[16, Theorem 2.1] Let $E$ be a complex Banach space and let $U$ be an open set
in $E$.

  1. (1)

$\delta_{U}\colon U\to\mathcal{G}^{\infty}(U)$ is a holomorphic mapping with
$\left\|\delta(x)\right\|=1$ for all $x\in U$.

  2. (2)

For every complex Banach space $F$ and every mapping
$f\in\mathcal{H}^{\infty}(U,F)$, there exists a unique operator
$T_{f}\in\mathcal{L}(\mathcal{G}^{\infty}(U);F)$ such that
$T_{f}\circ\delta_{U}=f$, that is, the diagram

$U$$\mathcal{G}^{\infty}(U)$$F$$f$$\delta_{U}$$T_{f}$

commutes. Furthermore, $\left\|T_{f}\right\|=\left\|f\right\|_{\infty}$.

  3. (3)

For every complex Banach space $F$, the mapping $J_{U,F}\colon f\mapsto T_{f}$
is an isometric isomorphism from $\mathcal{H}^{\infty}(U,F)$ onto
$\mathcal{L}(\mathcal{G}^{\infty}(U);F)$. Its inverse $J^{-1}_{U,F}$ is the
mapping $T\mapsto T\circ\delta_{U}$ from
$\mathcal{L}(\mathcal{G}^{\infty}(U);F)$ onto $\mathcal{H}^{\infty}(U,F)$.

  4. (4)

$\mathcal{H}^{\infty}(U)$ is isometrically isomorphic to
$\mathcal{G}^{\infty}(U)^{*}$, via the mapping
$J_{U}:=J_{U,\mathbb{C}}\colon\mathcal{H}^{\infty}(U)\to\mathcal{G}^{\infty}(U)^{*}$
given by

$J_{U}(f)=T_{f}\qquad(f\in\mathcal{H}^{\infty}(U)).$

As a consequence, we have

$\left\langle
J_{U}(f),\delta_{U}(x)\right\rangle=f(x)\qquad(f\in\mathcal{H}^{\infty}(U),\;x\in
U).$

  5. (5)

The closed unit ball of $\mathcal{G}^{\infty}(U)$ coincides with the norm-
closed absolutely convex hull of $\delta_{U}(U)$.$\hfill\Box$

From Theorem 1.2 (4), we immediately deduce that, on bounded subsets of
$\mathcal{H}^{\infty}(U)$, the weak* topology agrees with the topology of
pointwise convergence.

######  Corollary 1.3.

Let $U$ be an open subset of a complex Banach space $E$. Let $(f_{i})$ be a
net in $\mathcal{H}^{\infty}(U)$ and $f\in\mathcal{H}^{\infty}(U)$.

  1. (1)

If $(f_{i})\to f$ weak* in $\mathcal{H}^{\infty}(U)$, then $(f_{i})\to f$
pointwise on $U$.

  2. (2)

It $(f_{i})$ is bounded in $\mathcal{H}^{\infty}(U)$ and $(f_{i})\to f$
pointwise on $U$, then $(f_{i})\to f$ weak* in
$\mathcal{H}^{\infty}(U)$.$\hfill\Box$

Using Corollary 1.3, we obtain the following result.

######  Corollary 1.4.

Let $E$ and $F$ be complex Banach spaces, let $U$ and $V$ be open subsets of
$E$ and $F$, respectively, and let $f\colon U\to V$ be a holomorphic mapping.
Then there exists a unique operator
$\widehat{f}\in\mathcal{L}(\mathcal{G}^{\infty}(U);\mathcal{G}^{\infty}(V))$
such that $\widehat{f}\circ\delta_{U}=\delta_{V}\circ f$. Furthermore,
$||\widehat{f}||=1$.

###### Proof.

Let $C_{f}\colon\mathcal{H}^{\infty}(V)\to\mathcal{H}^{\infty}(U)$ be the
composition operator defined by

$C_{f}(g)=g\circ f\qquad(g\in\mathcal{H}^{\infty}(V)).$

Clearly,
$C_{f}\in\mathcal{L}(\mathcal{H}^{\infty}(V);\mathcal{H}^{\infty}(U))$ with
$||C_{f}||=1$. We claim that $J_{U}\circ C_{f}\circ J^{-1}_{V}$ is weak*-to-
weak* continuous from $\mathcal{G}^{\infty}(V)^{*}$ into
$\mathcal{G}^{\infty}(U)^{*}$. Let $(\phi_{i})$ be a net in
$\mathcal{G}^{\infty}(V)^{*}$ and $\phi\in\mathcal{G}^{\infty}(V)^{*}$. Assume
that $(\phi_{i})\to\phi$ weak* in $\mathcal{G}^{\infty}(V)^{*}$. By [14,
Corollary 2.6.10], $(\phi_{i})$ is norm-bounded in
$\mathcal{G}^{\infty}(V)^{*}$. Clearly, $(J^{-1}_{V}(\phi_{i}))\to
J^{-1}_{V}(\phi)$ weak* in $\mathcal{H}^{\infty}(V)$. Hence
$(J^{-1}_{V}(\phi_{i}))\to J^{-1}_{V}(\phi)$ pointwise on $V$ by Corollary 1.3
(1). In particular, $(J^{-1}_{V}(\phi_{i})\circ f)\to J^{-1}_{V}(\phi)\circ f$
pointwise on $U$. Since the net $(J^{-1}_{V}(\phi_{i})\circ f)$ is norm-
bounded since $\left\|J^{-1}_{V}(\phi_{i})\circ
f\right\|_{\infty}\leq\left\|J^{-1}_{V}(\phi_{i})\right\|_{\infty}=\left\|\phi_{i}\right\|$
for all $i$, Corollary 1.3 (2) guarantees that $(J^{-1}_{V}(\phi_{i})\circ
f)\to J^{-1}_{V}(\phi)\circ f$ weak* in $\mathcal{H}^{\infty}(U)$, that is,
$(C_{f}(J^{-1}_{V}(\phi_{i})))\to C_{f}(J^{-1}_{V}(\phi))$ weak* in
$\mathcal{H}^{\infty}(U)$. Finally, $(J_{U}(C_{f}(J^{-1}_{V}(\phi_{i}))))\to
J_{U}(C_{f}(J^{-1}_{V}(\phi)))$ weak* in $\mathcal{G}^{\infty}(U)^{*}$, and
this proves our claim.

Now, by [14, Corollaries 3.1.11 and 3.1.5], there is a unique operator
$\widehat{f}\in\mathcal{L}(\mathcal{G}^{\infty}(U);\mathcal{G}^{\infty}(V))$
such that $(\widehat{f})^{*}=J_{U}\circ C_{f}\circ J^{-1}_{V}$. Clearly,
$||\widehat{f}||=||(\widehat{f})^{*}||=||C_{f}||=1$. Given
$g\in\mathcal{H}^{\infty}(V)$ and $x\in U$, we have

$\left\langle(\widehat{f})^{*}(J_{V}(g)),\delta_{U}(x)\right\rangle=\left\langle
J_{V}(g)\circ\widehat{f},\delta_{U}(x)\right\rangle=\left\langle
J_{V}(g),\widehat{f}(\delta_{U}(x))\right\rangle$

and

$\displaystyle\left\langle(J_{U}\circ C_{f}\circ
J^{-1}_{V})(J_{V}(g)),\delta_{U}(x)\right\rangle$ $\displaystyle=\left\langle
J_{U}(C_{f}(g)),\delta_{U}(x)\right\rangle=\left\langle J_{U}(g\circ
f),\delta_{U}(x)\right\rangle$ $\displaystyle=g(f(x))=\left\langle
J_{V}(g),\delta_{V}(f(x))\right\rangle.$

From above we infer that

$\left\langle J_{V}(g),\widehat{f}(\delta_{U}(x))\right\rangle=\left\langle
J_{V}(g),\delta_{V}(f(x))\right\rangle\qquad(g\in\mathcal{H}^{\infty}(V),\;x\in
U),$

the surjectivity of
$J_{V}\colon\mathcal{H}^{\infty}(V)\to\mathcal{G}^{\infty}(V)^{*}$ yields

$\left\langle\phi,\widehat{f}(\delta_{U}(x))\right\rangle=\left\langle\phi,\delta_{V}(f(x))\right\rangle\qquad(\phi\in\mathcal{G}^{\infty}(V)^{*},\;x\in
U),$

and this implies that $\widehat{f}\circ\delta_{U}=\delta_{V}\circ f$. ∎

We finish this section with some results related to the transpose of a bounded
holomorphic mapping.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and let $f\in\mathcal{H}^{\infty}(U,F)$. Given $\varphi\in
F^{*}$, it is clear that $\varphi\circ f\colon U\to\mathbb{C}$ is holomorphic
with

$\left|(\varphi\circ
f)(x)\right|\leq\left\|\varphi\right\|\left\|f(x)\right\|\leq\left\|\varphi\right\|\left\|f\right\|_{\infty}$

for all $x\in U$. Hence $\varphi\circ f\in\mathcal{H}^{\infty}(U)$ with
$\left\|\varphi\circ
f\right\|_{\infty}\leq\left\|\varphi\right\|\left\|f\right\|_{\infty}$. This
justifies the following.

######  Definition 1.5.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and $f\in\mathcal{H}^{\infty}(U,F)$. We will call the transpose
mapping $f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ defined by
$f^{t}(\varphi)=\varphi\circ f$ the transpose of f.

Clearly, $f^{t}$ is linear and continuous with
$||f^{t}||\leq\left\|f\right\|_{\infty}$. In fact,
$||f^{t}||=\left\|f\right\|_{\infty}$. Indeed, for
$0<\varepsilon<\left\|f\right\|_{\infty}$, take $x\in U$ such that
$\left\|f(x)\right\|>\left\|f\right\|_{\infty}-\varepsilon$. By Hahn–Banach
theorem, there exists $\phi\in F^{*}$ with $\left\|\phi\right\|=1$ such that
$\phi(f(x))=\left\|f(x)\right\|$. We have

$\displaystyle\left\|f^{t}\right\|$ $\displaystyle\geq\sup_{0\neq\varphi\in
F^{*}}\frac{\left\|f^{t}(\varphi)\right\|_{\infty}}{\left\|\varphi\right\|}\geq\frac{\left\|\phi\circ
f\right\|_{\infty}}{\left\|\phi\right\|}=\left\|\phi\circ f\right\|_{\infty}$
$\displaystyle\geq\left|\phi(f(x))\right|=\left\|f(x)\right\|>\left\|f\right\|_{\infty}-\varepsilon.$

Letting $\varepsilon\to 0$, one obtains
$||f^{t}||\geq\left\|f\right\|_{\infty}$, as desired. Moreover, note that

$\displaystyle\left\langle(J_{U}\circ f^{t})(\varphi),\delta(x)\right\rangle$
$\displaystyle=\left\langle J_{U}(\varphi\circ f),\delta(x)\right\rangle$
$\displaystyle=\varphi(f(x))=\left\langle\varphi,T_{f}(\delta(x))\right\rangle$
$\displaystyle=\left\langle(T_{f})^{*}(\varphi),\delta(x)\right\rangle$

for all $\varphi\in F^{*}$ and $x\in U$, and since
$\mathcal{G}^{\infty}(U)=\overline{\mathrm{lin}}(\delta_{U}(U))$, we deduce
that $J_{U}\circ f^{t}=(T_{f})^{*}$, that is,
$f^{t}=J^{-1}_{U}\circ(T_{f})^{*}$. We have proved the following.

######  Proposition 1.6.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and let $f\in\mathcal{H}^{\infty}(U,F)$. Then
$f^{t}\in\mathcal{L}(F^{*};\mathcal{H}^{\infty}(U))$ with
$||f^{t}||=\left\|f\right\|_{\infty}$. Furthermore,
$f^{t}=J^{-1}_{U}\circ(T_{f})^{*}$. $\hfill\Box$

We next see that the mapping $f\mapsto f^{t}$ identifies
$\mathcal{H}^{\infty}(U,F)$ with the subspace of
$\mathcal{L}(F^{*};\mathcal{H}^{\infty}(U))$ formed by all weak*-to-weak*
continuous linear operators from $F^{*}$ into $\mathcal{H}^{\infty}(U)$ (see
[14, Corollary 3.1.12]).

######  Proposition 1.7.

Let $U$ be an open subset of a complex Banach space $E$ and let $F$ be a
complex Banach space. The mapping $f\mapsto f^{t}$ is an isometric isomorphism
from $\mathcal{H}^{\infty}(U,F)$ onto
$\mathcal{L}((F^{*},w^{*});(\mathcal{H}^{\infty}(U),w^{*}))$.

###### Proof.

Let $f\in\mathcal{H}^{\infty}(U,F)$. Clearly,
$f^{t}=J^{-1}_{U}\circ(T_{f})^{*}\in\mathcal{L}((F^{*},w^{*});(\mathcal{H}^{\infty}(U),w^{*}))$
by Theorem 1.2 (2) and [14, Theorem 3.1.11]. Moreover, we have
$||f^{t}||=\left\|f\right\|_{\infty}$ by Proposition 1.6. It remains to show
the surjectivity of the mapping in the statement. Take
$T\in\mathcal{L}((F^{*},w^{*});(\mathcal{H}^{\infty}(U),w^{*}))$. Then
$J_{U}\circ
T\in\mathcal{L}((F^{*},w^{*});(\mathcal{G}^{\infty}(U)^{*},w^{*}))$ and, by
[14, Theorem 3.1.11], there is a $S\in\mathcal{L}(\mathcal{G}^{\infty}(U);F)$
such that $S^{*}=J_{U}\circ T$. By Theorem 1.2 (3), there exists
$f\in\mathcal{H}^{\infty}(U,F)$ such that $T_{f}=S$. Hence
$T=J^{-1}_{U}\circ(T_{f})^{*}=f^{t}$, as desired. ∎

##  2\. Linearization of holomorphic mappings with compact type range

Let us recall that a bounded linear operator between Banach spaces $T\colon
E\to F$ is said to be compact (separable, weakly compact, Rosenthal, Asplund)
if $T(B_{E})$ is relatively compact (respectively, separable, relatively
weakly compact, Rosenthal, Asplund) in $F$.

We denote by $\mathcal{F}(E,F)$, $\overline{\mathcal{F}}(E,F)$,
$\mathcal{K}(E,F)$, $\mathcal{S}(E,F)$, $\mathcal{W}(E,F)$, $\mathcal{R}(E,F)$
and $\mathcal{A}(E,F)$ the linear spaces of bounded finite-rank linear
operators, approximable linear operators (i.e., operators which are the norm
limits of bounded finite-rank operators), compact linear operators, bounded
separable linear operators, weakly compact linear operators, Rosenthal linear
operators and Asplund linear operators from $E$ into $F$, respectively. The
following inclusions are known:

$\displaystyle\mathcal{F}(E,F)$
$\displaystyle\subseteq\overline{\mathcal{F}}(E,F)\subseteq\mathcal{K}(E,F)\subseteq\mathcal{W}(E,F)\subseteq\mathcal{R}(E,F)\cap\mathcal{A}(E,F),$
$\displaystyle\mathcal{K}(E,F)$ $\displaystyle\subseteq\mathcal{S}(E,F).$

Our aim is to study the following holomorphic variants of these concepts. If
$U$ is an open subset of a complex Banach space $E$ and $F$ is a complex
Banach space, we will consider bounded holomorphic mappings $f\colon U\to F$
that have a range $f(U)\subseteq F$ satisfying an algebraic or topological
property as, for instance, finite-dimensional range, relatively compact range,
separable range, relatively weakly compact range, Rosenthal range or Asplund
range.

Note that if $T\in\mathcal{L}(E,F)$, then $T$ is a compact (respectively,
separable, weakly compact, Rosenthal, Asplund) linear operator if and only if
the holomorphic mapping $T|_{\stackrel{{\scriptstyle\circ}}{{B}}_{E}}$ has
relatively compact (respectively, separable, relatively weakly compact,
Rosenthal, Asplund) range.

The study of the connections between these compactness properties of a mapping
$f\in\mathcal{H}^{\infty}(U,F)$ and its corresponding associated operator
$T_{f}\in\mathcal{L}(\mathcal{G}^{\infty}(U);F)$ was initiated by Mujica in
Propositions 3.1 and 3.4 of [16]. Apparently, these results are the only known
on this question and they have been included here with their proofs for the
convenience of the reader. We have divided our study for the different types
of holomorphic mappings considered.

###  2.1. Bounded finite-rank holomorphic mappings

Let us recall (see [16, p. 72]) that a mapping $f\colon U\to F$ has finite
rank if $\mathrm{lin}(f(U))$ is a finite dimensional subspace of $F$ in which
case this dimension is called the rank of $f$ and denoted by
$\mathrm{rank}(f)$. Let $\mathcal{H}^{\infty}_{\mathcal{F}}(U,F)$ denote the
linear space of all bounded finite-rank holomorphic mappings from $U$ to $F$.

######  Theorem 2.1.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and let $f\in\mathcal{H}^{\infty}(U,F)$. The following are
equivalent:

  1. (1)

$f\colon U\to F$ has finite rank.

  2. (2)

$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ has finite rank.

  3. (3)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ has finite rank.

In that case,
$\mathrm{rank}(f)=\mathrm{rank}(T_{f})=\mathrm{rank}((T_{f})^{*})=\mathrm{rank}(f^{t})$.

###### Proof.

Recall that $f(U)=T_{f}(\delta_{U}(U))$ by Theorem 1.2 (2).

$(1)\Leftrightarrow(2)$ ([16, Proposition 3.1 (a)]): If $f$ has finite rank,
then $\mathrm{lin}(f(U))$ is finite dimensional and therefore closed in $F$.
We have

$\displaystyle T_{f}(\mathcal{G}^{\infty}(U))$
$\displaystyle=T_{f}(\overline{\mathrm{lin}}(\delta_{U}(U)))\subseteq\overline{T_{f}(\mathrm{lin}(\delta_{U}(U)))}$
$\displaystyle=\overline{\mathrm{lin}}(T_{f}(\delta_{U}(U)))=\overline{\mathrm{lin}}(f(U))=\mathrm{lin}(f(U))$

and hence $T_{f}$ has finite rank. Conversely, if $T_{f}$ has finite rank,
then $f$ has finite rank since

$\displaystyle\mathrm{lin}(f(U))$
$\displaystyle=\mathrm{lin}(T_{f}(\delta_{U}(U)))=T_{f}(\mathrm{lin}(\delta_{U}(U)))$
$\displaystyle\subseteq
T_{f}(\overline{\mathrm{lin}}(\delta_{U}(U)))=T_{f}(\mathcal{G}^{\infty}(U)).$

Furthermore, in this case we have $\mathrm{rank}(f)=\mathrm{rank}(T_{f})$.

$(2)\Leftrightarrow(3)$: Since the space $\mathcal{F}(E,F)$ of bounded finite-
rank linear operators is a completely symmetric operator ideal (see [20,
4.4.7]), we have

$\displaystyle T_{f}\in\mathcal{F}(\mathcal{G}^{\infty}(U),F)$
$\displaystyle\Leftrightarrow(T_{f})^{*}\in\mathcal{F}(F^{*},\mathcal{G}^{\infty}(U)^{*})$
$\displaystyle\Leftrightarrow
f^{t}=J^{-1}_{U}\circ(T_{f})^{*}\in\mathcal{F}(F^{*},\mathcal{H}^{\infty}(U)).$

In this case we now have
$\mathrm{rank}(T_{f})=\mathrm{rank}((T_{f})^{*})=\mathrm{rank}(f^{t})$. ∎

###  2.2. Holomorphic mappings with relatively compact range

We denote by $\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$ the linear space of all
holomorphic mappings from $U$ to $F$ that have relatively compact range. The
equivalence $(1)\Leftrightarrow(3)$ of the next result is a version for
holomorphic mappings with relatively compact range of the classical Schauder’s
theorem on the relationship of the compactness of a bounded linear operator
between Banach spaces and its adjoint.

######  Theorem 2.2.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and let $f\in\mathcal{H}^{\infty}(U,F)$. The following are
equivalent:

  1. (1)

$f\colon U\to F$ has relatively compact range.

  2. (2)

$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ is compact.

  3. (3)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ is compact.

  4. (4)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ is bounded-weak*-to-norm
continuous.

  5. (5)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ is compact and bounded-weak*-to-
weak continuous.

  6. (6)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ is compact and weak*-to-weak
continuous.

###### Proof.

$(1)\Leftrightarrow(2)$ ([16, Proposition 3.1 (b)]): Since
$\delta_{U}(U)\subseteq B_{\mathcal{G}^{\infty}(U)}$, if
$T_{f}\in\mathcal{K}(\mathcal{G}^{\infty}(U),F)$, then $T_{f}(\delta_{U}(U))$
must be relatively compact in $F$, and therefore
$f\in\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$ since
$f(U)=T_{f}(\delta_{U}(U))$ by Theorem 1.2 (2). Conversely, since
$B_{\mathcal{G}^{\infty}(U)}=\overline{\mathrm{aco}}(\delta_{U}(U))$ by
Theorem 1.2 (5), one has

$T_{f}(B_{\mathcal{G}^{\infty}(U)})=T_{f}(\overline{\mathrm{aco}}(\delta_{U}(U)))\subseteq\overline{\mathrm{aco}}(T_{f}(\delta_{U}(U)))\subseteq\overline{\mathrm{aco}}(\overline{T_{f}(\delta_{U}(U))}).$

So, if $f\in\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$, then
$T_{f}(\delta_{U}(U))$ is relatively compact in $F$, hence
$\overline{\mathrm{aco}}(\overline{T_{f}(\delta_{U}(U))})$ is compact in $F$
by Mazur’s compactness theorem ([14, Theorem 2.8.15]) and the fact that
$\mathrm{aco}(A)=\mathrm{co}(\mathbb{D}A)$ for any subset $A$ of a normed
space $E$, where $\mathbb{D}$ denotes the closed unit disc of $\mathbb{C}$.
Therefore $T_{f}(B_{\mathcal{G}^{\infty}(U)})$ is relatively compact in $F$,
which means that $T_{f}\in\mathcal{K}(\mathcal{G}^{\infty}(U),F)$.

$(2)\Leftrightarrow(3)$: Applying Schauder’s theorem [14, Theorem 3.4.15] and
[14, Proposition 3.4.10], we have

$\displaystyle T_{f}\in\mathcal{K}(\mathcal{G}^{\infty}(U),F)$
$\displaystyle\Leftrightarrow(T_{f})^{*}\in\mathcal{K}(F^{*},\mathcal{G}^{\infty}(U)^{*})$
$\displaystyle\Leftrightarrow
f^{t}=J^{-1}_{U}\circ(T_{f})^{*}\in\mathcal{K}(F^{*},\mathcal{H}^{\infty}(U)),$

$(2)\Leftrightarrow(4)$: Similarly, one obtains

$\displaystyle T_{f}\in\mathcal{K}(\mathcal{G}^{\infty}(U),F)$
$\displaystyle\Leftrightarrow(T_{f})^{*}\in\mathcal{L}((F^{*},bw^{*});\mathcal{G}^{\infty}(U)^{*})$
$\displaystyle\Leftrightarrow
f^{t}=J^{-1}_{U}\circ(T_{f})^{*}\in\mathcal{L}((F^{*},bw^{*});\mathcal{H}^{\infty}(U)),$

by [14, Theorem 3.4.16], where $bw^{*}$ denotes the bounded weak* topology.

$(4)\Leftrightarrow(5)\Leftrightarrow(6)$ follows directly from [12,
Proposition 3.1]. ∎

Next, we identify $\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$ with the subspace
of $\mathcal{L}((F^{*},w^{*});(\mathcal{G}^{\infty}(U),w^{*}))$ consisting of
all bounded-weak*-to-norm continuous linear operators from $F^{*}$ into
$\mathcal{H}^{\infty}(U)$.

######  Proposition 2.3.

Let $U$ be an open subset of a complex Banach space $E$ and let $F$ be a
complex Banach space. The mapping $f\mapsto f^{t}$ is an isometric isomorphism
from $\mathcal{H}_{\mathcal{K}}^{\infty}(U,F)$ onto
$\mathcal{L}((F^{*},bw^{*});\mathcal{H}^{\infty}(U))$.

###### Proof.

Let $f\in\mathcal{H}_{\mathcal{K}}^{\infty}(U,F)$. Then
$f^{t}\in\mathcal{L}((F^{*},bw^{*});\mathcal{H}^{\infty}(U))$ by Theorem 2.2
and $||f^{t}||=\left\|f\right\|_{\infty}$ by Proposition 1.6. To prove the
surjectivity, take $T\in\mathcal{L}((F^{*},bw^{*});\mathcal{H}^{\infty}(U))$.
Then $J_{U}\circ T\in\mathcal{L}((F^{*},bw^{*});\mathcal{G}^{\infty}(U)^{*})$.
If $Q_{\mathcal{G}^{\infty}(U)}$ denotes the natural injection from
$\mathcal{G}^{\infty}(U)$ into $\mathcal{G}^{\infty}(U)^{**}$, then
$Q_{\mathcal{G}^{\infty}(U)}(\gamma)\circ J_{U}\circ
T\in\mathcal{L}((F^{*},bw^{*});\mathbb{C})$ for all
$\gamma\in\mathcal{G}^{\infty}(U)$ and, by [14, Theorem 2.7.8],
$Q_{\mathcal{G}^{\infty}(U)}(\gamma)\circ J_{U}\circ
T\in\mathcal{L}((F^{*},w^{*});\mathbb{C})$ for all
$\gamma\in\mathcal{G}^{\infty}(U)$, that is, $J_{U}\circ
T\in\mathcal{L}((F^{*},w^{*});(\mathcal{G}^{\infty}(U)^{*},w^{*}))$ by [14,
Corollary 2.4.5]. Hence $J_{U}\circ T=S^{*}$ for some
$S\in\mathcal{L}(\mathcal{G}^{\infty}(U);F)$ by [14, Theorem 3.1.11]. Note
that $S^{*}\in\mathcal{L}((F^{*},bw^{*});\mathcal{G}^{\infty}(U)^{*})$ and
this means that $S\in\mathcal{K}(\mathcal{G}^{\infty}(U),F)$ by [14, Theorem
3.4.16]. Now, $S=T_{f}$ for some $f\in\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$
by Theorem 1.2 (3) and Theorem 2.2. Finally, we have $T=J^{-1}_{U}\circ
S^{*}=J^{-1}_{U}\circ(T_{f})^{*}=f^{t}$. ∎

###  2.3. Bounded holomorphic mappings with separable range

We denote by $\mathcal{H}^{\infty}_{\mathcal{S}}(U,F)$ the space of all
mappings $f\in\mathcal{H}^{\infty}(U,F)$ such that $f(U)$ is separable.
Clearly, $\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$ is contained in
$\mathcal{H}^{\infty}_{\mathcal{S}}(U,F)$.

######  Theorem 2.4.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and let $f\in\mathcal{H}^{\infty}(U,F)$. Consider the following
assertions:

  1. (1)

$f\colon U\to F$ has separable range.

  2. (2)

$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ has separable range.

  3. (3)

$(T_{f})^{*}\colon F^{*}\to\mathcal{G}^{\infty}(U)^{*}$ has separable range.

  4. (4)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ has separable range.

Then $(1)\Leftrightarrow(2)\Leftarrow(3)\Leftrightarrow(4)$.

###### Proof.

$(1)\Leftrightarrow(2)$: We next use some inclusions obtained in the proof of
Theorem 2.1. If $f(U)$ is separable, then so is
$\overline{\mathrm{lin}}(f(U))$, and since
$T_{f}(\mathcal{G}^{\infty}(U))\subseteq\overline{\mathrm{lin}}(f(U))$, we
deduce that $T_{f}(\mathcal{G}^{\infty}(U))$ is separable. Conversely, if
$T_{f}(\mathcal{G}^{\infty}(U))$ is separable, then so is $f(U)$ since
$f(U)\subseteq\mathrm{lin}(f(U))\subseteq T_{f}(\mathcal{G}^{\infty}(U))$.

$(3)\Leftrightarrow(4)$: We apply that the space $\mathcal{S}(E,F)$ of all
bounded linear operators between Banach spaces having separable range is an
operator ideal (see [20, 1.8.2]).

$(3)\Rightarrow(2)$: We apply that if $T\in\mathcal{L}(E;F)$ and
$T^{*}\in\mathcal{L}(F^{*};E^{*})$, then $T\in\mathcal{S}(E,F)$ (see [20,
4.4.8]). ∎

###  2.4. Approximable bounded holomorphic mappings

We now enlarge the set of bounded finite-rank holomorphic mappings as follows.
We say that a mapping $f\in\mathcal{H}^{\infty}(U,F)$ is approximable if it is
the limit in the uniform norm of a sequence of mappings of
$\mathcal{H}_{\mathcal{F}}^{\infty}(U,F)$. The set of such mappings will be
denoted by $\mathcal{H}^{\infty}_{\overline{\mathcal{F}}}(U,F)$.

Next, we see that every approximable mapping $f\in\mathcal{H}^{\infty}(U,F)$
has relatively compact range.

######  Proposition 2.5.

Let $U$ be an open subset of a complex Banach space $E$ and let $F$ be a
complex Banach space. Then
$\mathcal{H}^{\infty}_{\overline{\mathcal{F}}}(U,F)\subseteq\mathcal{H}_{\mathcal{K}}^{\infty}(U,F)$.

###### Proof.

Let $f\in\mathcal{H}^{\infty}_{\overline{\mathcal{F}}}(U,F)$. Hence there is a
sequence $(f_{n})_{n\in\mathbb{N}}$ in
$\mathcal{H}_{\mathcal{F}}^{\infty}(U,F)$ such that
$\left\|f_{n}-f\right\|_{\infty}\to 0$ as $n\to\infty$. Since
$T_{f_{n}}\in\mathcal{F}(G^{\infty}(U),F)$ by Theorem 2.1,
$\mathcal{F}(G^{\infty}(U),F)\subseteq\mathcal{K}(\mathcal{G}^{\infty}(U),F)$
and
$\left\|T_{f_{n}}-T_{f}\right\|=\left\|T_{f_{n}-f}\right\|=\left\|f_{n}-f\right\|_{\infty}$
for all $n\in\mathbb{N}$, we deduce that
$T_{f}\in\mathcal{K}(\mathcal{G}^{\infty}(U),F)$ by [14, Corollary 3.4.9], and
so $f\in\mathcal{H}_{\mathcal{K}}^{\infty}(U,F)$ by Theorem 2.2. ∎

An application of the principle of local reflexivity obtained by C. V. Hutton
[11] shows that an operator $T\in\mathcal{L}(E;F)$ can be approximated by
bounded finite-rank linear operators from $E$ into $F$ if and only if $T^{*}$
can be approximated by bounded finite-rank linear operators from $F^{*}$ into
$E^{*}$. We now invoke Hutton’s theorem to obtain the following.

######  Theorem 2.6.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and $f\in\mathcal{H}^{\infty}(U,F)$. The following are
equivalent:

  1. (1)

$f\colon U\to F$ can be approximated by bounded finite-rank holomorphic
mappings.

  2. (2)

$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ can be approximated by bounded
finite-rank linear operators.

  3. (3)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ can be approximated by bounded
finite-rank linear operators.

###### Proof.

We have

$\displaystyle f\in\mathcal{H}^{\infty}_{\overline{\mathcal{F}}}(U,F)$
$\displaystyle\Leftrightarrow
T_{f}\in\overline{\mathcal{F}}(\mathcal{G}^{\infty}(U),F)$
$\displaystyle\Leftrightarrow(T_{f})^{*}\in\overline{\mathcal{F}}(F^{*},\mathcal{G}^{\infty}(U)^{*})$
$\displaystyle\Leftrightarrow
f^{t}=J^{-1}_{U}\circ(T_{f})^{*}\in\overline{\mathcal{F}}(F^{*},\mathcal{H}^{\infty}(U))$

by Theorem 1.2 (3) and Theorem 2.1 for the first equivalence, Hutton’s theorem
[11, Theorem 2.1] for the second one, and the operator ideal property of
$\overline{\mathcal{F}}(E,F)$ for the third one [20, 4.2.2]. ∎

###  2.5. Holomorphic mappings with relatively weakly compact range

The Davis–Figiel–Johnson–Pełczynski factorization theorem [5] asserts that any
weakly compact linear operator between Banach spaces factors through a
reflexive Banach space. We now extend this result to holomorphic mappings with
relatively weakly compact range and give also the analogs of Gantmacher and
Gantmacher–Nakamura theorems for such mappings.

We will denote by $\mathcal{H}^{\infty}_{\mathcal{W}}(U,F)$ the linear space
of all holomorphic mappings from $U$ to $F$ that have relatively weakly
compact range. Clearly, $\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$ is contained
in $\mathcal{H}^{\infty}_{\mathcal{W}}(U,F)$.

######  Theorem 2.7.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and let $f\in\mathcal{H}^{\infty}(U,F)$. The following are
equivalent:

  1. (1)

$f\colon U\to F$ has relatively weakly compact range.

  2. (2)

$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ is weakly compact.

  3. (3)

There exist a reflexive complex Banach space $G$, an operator
$T\in\mathcal{L}(G;F)$ and a mapping $g\in\mathcal{H}^{\infty}(U,G)$ such that
$f=T\circ g$.

  4. (4)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ is weakly compact.

  5. (5)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ is weak*-to-weak continuous.

###### Proof.

$(1)\Leftrightarrow(2)$ ([16, Proposition 3.1 (b)]): It follows with a similar
proof to that of the equivalence $(1)\Leftrightarrow(2)$ of Theorem 2.2 by
taking into account that the norm closure and weak closure of the convex hull
of a subset of a normed space coincide [14, Corollary 2.5.18] and that the
norm-closed convex hull of a weakly compact subset of a Banach space is itself
weakly compact [14, Theorem 2.8.14].

$(2)\Rightarrow(3)$: Applying the Davis–Figiel–Johnson–Pełczynski theorem,
there exist a reflexive complex Banach space $G$ and operators
$T\in\mathcal{L}(G;F)$ and $S\in\mathcal{L}(\mathcal{G}^{\infty}(U);G)$ such
that $T_{f}=T\circ S$. Taking
$g:=S\circ\delta_{U}\in\mathcal{H}^{\infty}(U,G)$, we conclude that
$f=T_{f}\circ\delta_{U}=T\circ S\circ\delta_{U}=T\circ g$.

$(3)\Rightarrow(1)$: $f(U)=T(g(U))$ is relatively weakly compact because $T$
is weak-to-weak continuous by [14, Theorem 2.5.11] and $g(U)$ is relatively
weakly compact in $G$ since it is a bounded subset of the reflexive Banach
space $G$ (see [14, Theorem 2.8.2]).

$(2)\Leftrightarrow(4)$: We have

$\displaystyle T_{f}\in\mathcal{W}(\mathcal{G}^{\infty}(U),F)$
$\displaystyle\Leftrightarrow(T_{f})^{*}\in\mathcal{W}(F^{*},\mathcal{G}^{\infty}(U)^{*})$
$\displaystyle\Leftrightarrow
f^{t}=J^{-1}_{U}\circ(T_{f})^{*}\in\mathcal{W}(F^{*},\mathcal{H}^{\infty}(U)),$

by Gantmacher’s theorem [14, Theorem 3.5.13] and [14, Proposition 3.5.11].

$(2)\Leftrightarrow(5)$: We have

$\displaystyle T_{f}\in\mathcal{W}(\mathcal{G}^{\infty}(U),F)$
$\displaystyle\Leftrightarrow(T_{f})^{*}\in\mathcal{L}((F^{*},w^{*});(\mathcal{G}^{\infty}(U)^{*},w))$
$\displaystyle\Leftrightarrow
f^{t}=J^{-1}_{U}\circ(T_{f})^{*}\in\mathcal{L}((F^{*},w^{*});(\mathcal{H}^{\infty}(U),w))$

by Gantmacher–Nakamura’s theorem [14, Theorem 3.5.14] and [14, Corollary
2.5.12]. ∎

We next identify $\mathcal{H}^{\infty}_{\mathcal{W}}(U,F)$ with the subspace
of $\mathcal{L}((F^{*},w^{*});(\mathcal{G}^{\infty}(U),w^{*}))$ formed by all
weak*-to-weak continuous linear operators from $F^{*}$ into
$\mathcal{H}^{\infty}(U)$.

######  Proposition 2.8.

Let $U$ be an open subset of a complex Banach space $E$ and let $F$ be a
complex Banach space. The mapping $f\mapsto f^{t}$ is an isometric isomorphism
from $\mathcal{H}^{\infty}_{\mathcal{W}}(U,F)$ onto
$\mathcal{L}((F^{*},w^{*});(\mathcal{H}^{\infty}(U),w))$.

###### Proof.

In view of Theorem 2.7 and Proposition 1.6, we only need to show that the
mapping in the statement is surjective. Let
$T\in\mathcal{L}((F^{*},w^{*});(\mathcal{H}^{\infty}(U),w))$. Then $J_{U}\circ
T\in\mathcal{L}((F^{*},w^{*});(\mathcal{G}^{\infty}(U)^{*},w))$ by [14,
Theorem 2.5.11], and this last set is contained in
$\mathcal{L}((F^{*},w^{*});(\mathcal{G}^{\infty}(U)^{*},w^{*}))$ since the
weak topology is stronger than the weak* topology on the dual of a normed
space. It follows that $J_{U}\circ T=S^{*}$ for some
$S\in\mathcal{L}(\mathcal{G}^{\infty}(U);F)$ by [14, Theorem 3.1.11]. Hence
$S^{*}\in\mathcal{L}((F^{*},w^{*});(\mathcal{G}^{\infty}(U)^{*},w))$ and, by
Gantmacher–Nakamura’s theorem [14, Theorem 3.5.14],
$S\in\mathcal{W}(\mathcal{G}^{\infty}(U),F)$. Now, $S=T_{f}$ for some
$f\in\mathcal{H}_{\mathcal{W}}^{\infty}(U,F)$ by Theorem 1.2 (3) and Theorem
2.7. Finally, $T=J^{-1}_{U}\circ S^{*}=J^{-1}_{U}\circ(T_{f})^{*}=f^{t}$, as
desired. ∎

###  2.6. Bounded holomorphic mappings with Rosenthal range

Given Banach spaces $E$ and $F$, a set $A\subseteq E$ is called Rosenthal (or
conditionally weakly compact) if every sequence in $A$ admits a weak Cauchy
subsequence, and a bounded linear operator $T\colon E\to F$ is called a
Rosenthal operator if $T(B_{E})$ is a Rosenthal subset of $F$. It is known
(see, for example, [1]) that $T\colon E\to F$ is Rosenthal if and only if it
factors through a Banach space that does not contain an isomorphic copy of
$\ell_{1}$.

We now prove a similar result for holomorphic mappings with Rosenthal range,
and the Rosenthal factorization theorem then allows us to factor these
mappings through a Banach space not containing an isomorphic copy of
$\ell_{1}$. We denote by $\mathcal{H}^{\infty}_{\mathcal{R}}(U,F)$ the linear
space of all bounded holomorphic mappings from $U$ to $F$ that have Rosenthal
range. Clearly, $\mathcal{H}^{\infty}_{\mathcal{W}}(U,F)$ is contained in
$\mathcal{H}^{\infty}_{\mathcal{R}}(U,F)$.

######  Theorem 2.9.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and let $f\in\mathcal{H}^{\infty}(U,F)$. The following are
equivalent:

  1. (1)

$f\colon U\to F$ has Rosenthal range.

  2. (2)

$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ is Rosenthal.

  3. (3)

There exist a complex Banach space $G$ which does not contain an isomorphic
copy of $\ell_{1}$, an operator $T\in\mathcal{L}(G;F)$ and a mapping
$g\in\mathcal{H}^{\infty}(U,G)$ such that $f=T\circ g$.

###### Proof.

$(1)\Leftrightarrow(2)$: It can be proved similarly as the same equivalence in
Theorem 2.2 taking into account that the norm-closed absolutely convex hull of
a Rosenthal subset of a Banach space is itself Rosenthal (see [13, p. 357]).

$(2)\Rightarrow(3)$: Rosenthal factorization theorem (see e.g. [1]) gives a
complex Banach space $G$ not containing an isomorphic copy of $\ell_{1}$ and
operators $T\in\mathcal{L}(G;F)$ and
$S\in\mathcal{L}(\mathcal{G}^{\infty}(U);G)$ such that $T_{f}=T\circ S$. If we
write $g:=S\circ\delta_{U}\in\mathcal{H}^{\infty}(U,G)$, then
$f=T_{f}\circ\delta_{U}=T\circ S\circ\delta_{U}=T\circ g$.

$(3)\Rightarrow(1)$: Notice that $f(U)=T(g(U))$ where $T$ is weak-to-weak
continuous [14, Theorem 2.5.11] and $g(U)$ is Rosenthal in $G$ by Rosenthal’s
$\ell_{1}$-theorem [22]. ∎

###  2.7. Holomorphic mappings with Asplund range

By [4, Definition 5.1.2], a bounded set $D\subseteq E$ is said to be Asplund
or to have the Asplund property if every convex continuous function $f\colon
E\to\mathbb{R}$ is $D$-differentiable on a residual subset of $E$. A Banach
space $E$ is called an Asplund space if every convex continuous function
$f\colon E\to\mathcal{R}$ is Fréchet differentiable on a dense $G_{\delta}$
set in $E$. This definition is due to E. Asplund [3] under the name strong
differentiability space. We refer to R. D. Bourgin [4, Theorem 5.2.11] and R.
R. Phelps [19, Theorem 2.34] for some equivalent formulations of the concept
of Asplund set in a Banach space. In particular, we remark that the Asplund
spaces are the Banach spaces for which each separable subspace has a separable
dual. We will also use the paper [25] by C. Stegall for the properties of
Asplund sets and Asplund operators in Banach spaces (see also Theorem 5.5.4 in
[4]).

Following [25, Definition 1.2], we say that a closed, bounded, convex subset
$K$ of a Banach space $E$ has the Radon–Nikodym property (RNP) if for any
finite measure space $(\Omega,\Sigma,\mu)$, every $m\colon\Sigma\to E$ that is
$\mu$-continuous, countably additive, of finite variation, with average range
$\\{\mu(E)^{-1}m(E)\colon E\in\Sigma,\,\mu(E)>0\\}\subseteq K$, is
representable by a Bochner integrable function. A Banach space $E$ has RNP if
every closed, bounded and convex subset of $E$ has RNP. By [25, Theorem 2.8],
a Banach space $E$ is an Asplund space if and only if $E^{*}$ has RNP.

Given two Banach spaces $E$ and $F$, we say that an operator
$T\in\mathcal{L}(E;F)$ is Radon–Nikodým if $T$ factors through a Banach space
$W$ with $W$ having RNP. We denote by $\mathcal{RN}(E,F)$ the linear space of
all Radon–Nikodým operators from $E$ into $F$. According to [25, Theorem
2.11], we have that $T\in\mathcal{A}(E,F)$ if and only if
$T^{*}\in\mathcal{RN}(F^{*},E^{*})$.

We will denote by $\mathcal{H}^{\infty}_{\mathcal{A}}(U,F)$ the linear space
of all holomorphic mappings from $U$ to $F$ such that $f(U)$ is an Asplund
subset of $F$. Clearly, $\mathcal{H}^{\infty}_{\mathcal{W}}(U,F)$ is contained
in $\mathcal{H}^{\infty}_{\mathcal{A}}(U,F)$.

######  Theorem 2.10.

Let $U$ be an open subset of a complex Banach space $E$, let $F$ be a complex
Banach space and let $f\in\mathcal{H}^{\infty}(U,F)$. The following are
equivalent:

  1. (1)

$f\colon U\to F$ has Asplund range.

  2. (2)

$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ is Asplund.

  3. (3)

There exist a complex Asplund space $G$, an operator $T\in\mathcal{L}(G;F)$
and a mapping $g\in\mathcal{H}^{\infty}(U,G)$ such that $f=T\circ g$.

  4. (4)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ is Radon–Nikodým.

###### Proof.

$(1)\Leftrightarrow(2)$: Since $f(U)$ is an Asplund set and
$T_{f}(B_{\mathcal{G}^{\infty}(U)})\subseteq\overline{\mathrm{aco}}(f(U))$,
then $T_{f}(B_{\mathcal{G}^{\infty}(U)})$ is an Asplund set by Lemmas 1.3 and
1.4 in [25] and Theorem 5.5.4 in [4].

$(2)\Rightarrow(3)$: If $T_{f}\in\mathcal{A}(\mathcal{G}^{\infty}(U),F)$, then
Theorem 2.11 in [25] assures that $T_{f}=T\circ S$ with
$S\in\mathcal{L}(\mathcal{G}^{\infty}(U);G)$ and $T\in\mathcal{L}(G;F)$, where
$G$ is a complex Asplund space. Taking
$g:=S\circ\delta_{U}\in\mathcal{H}^{\infty}(U,G)$, we have
$f=T_{f}\circ\delta_{U}=T\circ S\circ\delta_{U}=T\circ g$.

$(3)\Rightarrow(1)$: $f(U)=T(g(U))$ is an Asplund set by Theorem 2.8, Theorem
1.13 and Lemma 1.7 in [25].

$(2)\Rightarrow(4)$: By [25, Theorem 2.11], [20, 24.2] and Proposition 1.6, we
have

$\displaystyle T_{f}\in\mathcal{A}(\mathcal{G}^{\infty}(U),F)$
$\displaystyle\Leftrightarrow(T_{f})^{*}\in\mathcal{RN}(F^{*},\mathcal{G}^{\infty}(U)^{*})$
$\displaystyle\Leftrightarrow
f^{t}=J^{-1}_{U}\circ(T_{f})^{*}\in\mathcal{RN}(F^{*},\mathcal{H}^{\infty}(U)).$

∎

Combining Theorem 1.2 (3) with the results stated above, we see that the
isometric isomorphism $f\mapsto T_{f}$ from $\mathcal{H}^{\infty}(U,F)$ onto
$\mathcal{L}(\mathcal{G}^{\infty}(U);F)$ induces the following
identifications.

######  Corollary 2.11.

Let $U$ be an open subset of a complex Banach space $E$ and let $F$ be a
complex Banach space. The mapping $f\mapsto T_{f}$ is an isometric isomorphism
from $\mathcal{H}^{\infty}_{\mathcal{I}}(U,F)$ onto
$\mathcal{I}(\mathcal{G}^{\infty}(U),F)$ in the cases
$\mathcal{I}=\mathcal{F},\mathcal{K},\mathcal{S},\overline{\mathcal{F}},\mathcal{W},\mathcal{R},\mathcal{A}$.
$\hfill\Box$

Acknowledgements. The authors are grateful to the referees for their helpful
comments that have improved this paper.

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# A $\pi_{1}$ obstruction to having finite index monodromy and an unusual
subgroup of infinite index in $\textrm{Mod}(\Sigma_{g})$

Ishan Banerjee

(April 2023)

###### Abstract

Let $X$ be an algebraic surface with $\mathcal{L}$ an ample line bundle on
$X$. Let $\Gamma(X,\mathcal{L})$ be the _geometric monodromy_ group a
associated to family of nonsingular curves in $X$ that are zero loci of
sections of $\mathcal{L}$. We provide obstructions to $\Gamma(X,\mathcal{L})$
being finite index in the mapping class group. We also show that for any
$k\geq 0$, the image of monodromy is finite index in appropriate subgroups of
the quotient of the mapping class group by the $k$th term of the Johnson
filtration assuming that $\mathcal{L}$ is sufficiently ample. This enables us
to construct several subgroups of the mapping class group with unusual
properties, in some cases providing the first examples of subgroups with those
properties.

##  1 Introduction

Let $X$ be a smooth complex projective surface. Let $\mathcal{L}$ be an ample
line bundle on $X$. Let $H^{0}(X,\mathcal{L})$ be the space of global sections
of $\mathcal{L}$. Let $\mathbb{P}H^{0}(X,\mathcal{L})$ be its projective
space. Since two sections of $\mathcal{L}$ are scalar multiples of each other
if and only if their zero loci are the same, we may identify
$\mathbb{P}H^{0}(X,\mathcal{L})$ with the space of complex curves in $X$ which
arise as zeroes of sections of $\mathcal{L}$.

Let

$\bar{E}(X,\mathcal{L})=\\{(p,f)\in
X\times\mathbb{P}H^{0}(X,\mathcal{L})|f(p)=0\\}.$

Let $\pi:\bar{E}(X,\mathcal{L})\to\mathbb{P}H^{0}(X,\mathcal{L})$ be the
projection map. The fibres of $\pi$ are curves in $X$.

Let

$U(X,\mathcal{L})=\\{f\in\mathbb{P}H^{0}(X,\mathcal{L})|\not\exists p\in
X,f(p)=0\textrm{ and }df(p)=0\\}.$

The space $U(X,\mathcal{L})$ parametrizes _nonsingular_ curves in $X$ that are
zero loci of sections of $\mathcal{L}$. It is an open subset of
$\mathbb{P}H^{0}(X,\mathcal{L}).$ Let

$E(X,\mathcal{L})=\pi^{-1}(U(\mathcal{L})).$

The map $\pi:\bar{E}(X,\mathcal{L})\to\mathbb{P}H^{0}(X,\mathcal{L})$
restricts to a map $E(X,\mathcal{L})\to U(X,\mathcal{L})$. It is a proper
submersion, so by Ehresmann’s fibration theorem it is a $C^{\infty}$ fibre
bundle. Let $f\in U(X,\mathcal{L})$, and let $C\subseteq X$ be the zero locus
of $f$. Associated to this fiber bundle is a monodromy representation
$\rho:\pi_{1}(U(X,\mathcal{L}),f)\to\textrm{Mod}(C)$, where $\textrm{Mod}(C)$
is the mapping class group of $C$. We will denote the image of $\rho$ by
$\Gamma(X,\mathcal{L})$.

In the case when $X$ is a toric surface the group $\Gamma(X,\mathcal{L})$ is
completely understood by work of Salter and Calderon- Salter. This is done in
the papers [11] and [1]. However a toric surface is incredibly special and
part of the proof relies on constructing families of vanishing cycles in an
extremely combinatorial fashion (called A- cycles and B-cycles in those paper)
that does not seem likely to generalise to other kids of surfaces.

In this paper we will prove some results about the group
$\Gamma(X,\mathcal{L})$ under various _topological_ hypotheses on $X$,
specifically hypotheses on $\pi_{1}(X)$. Our results will be far weaker than
those in [11] and [1]. However the class of surfaces that we deal with is far
larger.

Our main theorems are as follows:

######  Theorem 1.

Let $i:C\to X$ be the inclusion map. Let $i_{*}:\pi_{1}(C)\to\pi_{1}(X)$
denote the induced map on $\pi_{1}$. The group $\Gamma(X,\mathcal{L})$ is
contained in the group $\mathcal{A}[i_{*}]$ where
$\mathcal{A}[i_{*}]\subseteq\textrm{Mod}(C)$ is the subgroup generated by Dehn
twists about nonseparating curves in $\ker i_{*}$. If $\pi_{1}X$ is torsion-
free and infinite, or if $H^{1}(X,\mathbb{Q})\neq 0$, $\mathcal{A}[i_{*}]$ is
of infinite index in $\textrm{Mod}(C)$.

The reader may be concerned that it may be the case that $\pi_{1}X$ is
infinite and the map $i_{*}$ is $0$, in which case $\mathcal{A}[i_{*}]$ would
be all of $\textrm{Mod}(C)$. We note here that by the Lefschetz hyperplane
theorem $i_{*}$ is surjective, so this situation can not arise.

######  Theorem 2.

Let $k>0,r\gg k$. Let $C$ be the zero locus of a section of
$\mathcal{L}^{\otimes r}$ in $X$. Let $i:C\to X$ be the inclusion. Let
$V=i^{*}(H^{1}(X))\subseteq H^{1}(C)$. Let
$\mathcal{A}[V]\subseteq\textrm{Mod}(C)$ be the subgroup generated by Dehn
twists about nonseparting simple closed curves in $C$ whose associated
homology class in $H^{1}(C)$ is in $V^{\perp}$. Let
$J_{k}\subseteq\textrm{Mod}(C)$ denote the $k$th term of the Johnson
filtration. Then

$\Gamma(X,\mathcal{L}^{\otimes r})/J_{k}\cap\Gamma(X,\mathcal{L}^{\otimes r})$

is a finite index subgroup of $\mathcal{A}[V]/J_{k}\cap\mathcal{A}[V].$

There is a tension between the results of Theorems 1 and 2. For instance,
suppose $\pi_{1}X$ is infinite and torsion free and $H^{1}(X,\mathbb{Q})=0$.
In these cases the group $\Gamma(X,\mathcal{L}^{\otimes r})$ is infinite index
in $\textrm{Mod}(C)$ but for $r\gg k$ , the group is finite index in
$\textrm{Mod}(C)/J_{k}$. Thus the fact that $\Gamma(X,\mathcal{L}^{\otimes
r})$ is infinite index must involve it not containing lots of elements deep in
the Johnson filtration.

In order to establish Theorem 2 we establish some of the properties of the
higher Johnson-Morita homomorphisms when restricted to $\mathcal{A}[V]$. This
is covered in Section 5 . We believe this to be of independent interest.

As a result of the above Theorems and also Theorem 3, we construct subgroups
of $\textrm{Mod}(C)$ with unusual properties.

######  Corollary 1.

Let $k>0$. Let $X$ be an smooth projective surface such that $\pi_{1}X$ is a
torsion free lattice in $Sp_{4}(\mathbb{Z})$. Let $\mathcal{L}$ be an ample
line bundle on $X$. Let $r\gg k$. Let $C$ be the zero locus of a section of
$\mathcal{L}^{\otimes r}$. Then,

  1. 1.

$\Gamma(X,\mathcal{L}^{\otimes r})$ is infinite index in $\textrm{Mod}(C)$.

  2. 2.

$\Gamma(X,\mathcal{L}^{\otimes r})/\Gamma(X,\mathcal{L}^{\otimes r})J_{k}$ is
finite index in $\textrm{Mod}(C)/J_{k}$.

######  Corollary 2.

Let $k>0$. Let $X$ be an smooth projective surface such that $\pi_{1}X$ is the
product of a lattice in $Sp_{4}(\mathbb{Z})$ and $\mathbb{Z}^{2m}$. Let
$\mathcal{L}$ be an ample line bundle on $X$. Let $r\gg k$. Let $C$ be the
zero locus of a section of $\mathcal{L}^{\otimes r}$. Let
$V=i^{*}(H^{1}(X))\subseteq H^{1}(C)$ Then for $r\gg k$,

  1. 1.

$\Gamma(X,\mathcal{L}^{\otimes r})$ is infinite index in $\mathcal{A}[V]$.

  2. 2.

$\Gamma(X,\mathcal{L}^{\otimes r})/\Gamma(X,\mathcal{L}^{\otimes r})\cap
J_{k}$ is finite index in $\mathcal{A}[V]/\mathcal{A}[V]\cap J_{k}$.

We note that it is well-known that surfaces with such fundamental groups
exist.

We note that in the above Theorems and Corollaries, we could have instead of
taking a sufficiently large power of a fixed ample line bundle, taken a line
bundle that is sufficiently ample in some appropriate sense, i.e. some fixed
cohomology groups vanish. Essentially all one has to do is to make assumptions
on $\mathcal{L}$ that enable us to prove analogues of Proposition 7.

###  1.1 Vanishing cycles and their relation to monodromy

In this section we will provide context and motivation for our results.
Suppose that we have a $C^{\infty}$ map
$f:\mathbb{D}\to\mathbb{P}(H^{0}(X,\mathcal{L}))$ such that
$f(\mathbb{D}\setminus\\{0\\})\subseteq U(X,\mathcal{L})$.

Suppose furthermore that $f(1)=C$ and that $f^{*}\pi$ defines a Lefschetz
fibration over $\mathbb{D}$, with the fiber over $0$, $C_{0}$ being the only
singular fiber. We assume $C_{0}$ has a unique nodal point.

In this case the theory of Lefschetz fibrations tells us that the topology of
the family $f^{*}\pi$ may be described in terms of a simple closed curve
$\alpha\subseteq C$ that is ’pinched’ under parallel transport to the nodal
fiber $C_{0}$. Such an $\alpha$ is called a vanishing cycle. A question of
Donaldson [4] asks which nonseparating simple closed curves in $C$ arise as
vanishing cycles.

This question is related to (and is in fact almost equivalent to) the problem
of determining the monodromy group $\Gamma(X,\mathcal{L})$ associated to
$\mathcal{L}$.

In our situation, it is well known that the monodromy group
$\Gamma(X,\mathcal{L})$ is generated by Dehn twists about vanishing cycles.
More precisely we have the following propositions.

######  Proposition 1.

Let $X,\mathcal{L},C$ be as above. Assume $\mathcal{L}$ is very ample. Suppose
that a nonseparating simple closed curve $\alpha\subseteq C$ is a vanishing
cycle. Then the Dehn twist $T_{\alpha}\in\Gamma(X,\mathcal{L})$. Furthermore,
$\Gamma(X,\mathcal{L})$ is generated by Dehn twists about nonseparating
vanishing cycles.

######  Proposition 2.

Let $X,\mathcal{L},C$ be as above. Assume $\mathcal{L}$ is very ample. Let
$\Delta$ be the set of nonseparating simple closed curves arising as vanishing
cycles. Then if $\phi\in\Gamma(X,\mathcal{L})$ and $\delta\in\Delta$,
$\phi(\delta)\in\Delta$. Thus, $\Gamma(X,\mathcal{L})$ acts on $\Delta$.
Furthermore this action is transitive.

A reference for the above two Propositions may be found in [12] Section 3.2.3,
note that the reference only deals with homology classes of vanishing cycles,
but the proofs go through to establish the above propositions after the
obvious adjustments.

While we will not prove Propositions 1 and 2 here, we will explain some of the
ideas that go into the proofs.

It is a general fact that $\pi_{1}(\mathbb{P}^{n}\setminus D,x)$ where $D$ is
some irreducible hypersurface and $x\in\mathbb{P}^{n}\setminus D$ is generated
by meridianal loops that are all conjugate to each other. A meridianal loop is
defined as follows:

  1. 1.

Let $y$ be a point in the smooth locus of $D$.

  2. 2.

Let $\gamma$ be a path starting at $x$ and ending at $y$ that lies in
$\mathbb{P}^{n}\setminus D$ except at the endpoint.

  3. 3.

The meridianal loop $M_{\gamma}$ is the loop that starts at $x$, follows
$\gamma$ till it reaches a small neighbourhood of $y$, goes counterclockwise
around the divisor $D$ and returns along $\gamma^{-1}$.

In our case, one can argue that the monodromy of these meridianal loops are
Dehn twists about nonseparating simple closed curves in $C$. The fact that the
meridianal loops are Dehn twists about _nonseparating_ curves is a consequence
of the fact that a smooth point in $\Sigma(\mathcal{L})$ defines an
irreducible nodal curve. The fact that these meridians are all conjugate leads
to the transitivity of the group $\Gamma(X,\mathcal{L})$ on $\Delta$.

As a result of the above two Propositions, every cycle is a vanishing cycle if
and only if $\Gamma(X,\mathcal{L})=\textrm{Mod}(C)$.

###  1.2 Existing work and a general question

As mentioned earlier, in the case when $X$ is a toric surface and
$\mathcal{L}$ is an ample line bundle, Salter [11] proves that the monodromy
group is either hyperelliptic or finite index in the entire mapping class
group. Later work of Calderon-Salter [1] proves that the monodromy group is
either hyperelliptic or the subgroup of the mapping class group preserving a
higher spin structure( see [11] for further details on what a higher spin
structure is). Furthermore if we fix an ample line bundle $\mathcal{L}$, and
look at $\Gamma(X,\mathcal{L}^{\otimes r})$ for $r$ sufficiently large, then
the monodromy group is not hyperelliptic, and hence is finite index. This
leads us to the following question.

######  Question 1.

Let X be a smooth projective complex surface. Let $\mathcal{L}$ be an ample
line bundle. Is $\Gamma(X,\mathcal{L}^{\otimes r})$ finite index for $r$
sufficiently large?

Theorem 1 asserts that if $H^{1}(X,\mathbb{Q})\neq 0$ or if $\pi_{1}(X)$ is
torsionfree and infinite, then the answer is no. Theorem 2 asserts that while
the answer to the above question is no, a weakening of the question has a
positive answer, namely $\Gamma(X,\mathcal{L}^{\otimes r})$ surjects onto a
finite index subgroup of the image of $\mathcal{A}(V)$ in the quotient of the
mapping class group by a term of the Johnson filtration.

Even in the case when $X$ is simply connected or has finite fundamental group,
Theorem 1 and Theorem 2 shut down one avenue of attack on Question 1, i.e. to
prove that the monodromy groups $\Gamma(X,\mathcal{L})$ are of finite index.
Namely one could hope that subgroups satisfying properties akin to those in
Theorem 2 and that of Lemma 1 are automatically finite index in the mapping
class group, indeed this was the hope of the author going into this project.
One could imagine that if a subgroup of the mapping class group contains
subgroups that are mapping class groups of sufficiently large subsurfaces of
$C$ and contain enough Dehn twists to virtually surject onto $Sp(H_{1}(C)),$
then the subgroup would be forced to be finite index. However this is simply
not the case.

###  1.3 Acknowledgements

I would like to thank Aaron Calderon and Nick Salter for helpful discussions
about spin and framed mapping class groups. I’d like to thank Nick Salter for
going through an early draft of this paper. I’d like to thank Peter Huxford
for helpful comments.

##  2 Containment

In this section we will establish that the monodromy group
$\Gamma(X,\mathcal{L})$ is contained in $\mathcal{A}[i_{*}]$.

We will first study stabilizer subgroups.

######  Definition 1.

Let $G$ be a group. Let $\phi:\pi_{1}(C)\to G$ be a surjective homomorphism.
Let $S$ be the set of conjugacy classes of surjective homomorphisms from
$\pi_{1}(C)$ to G. There is a right action of $\textrm{Mod}(C)$ on $S$ arising
from precomposition.

We define $\textrm{Stab}[\phi]$ to be the stabilizer subgroup of $\phi$ under
this action.

The following proposition characterises the Dehn twists about non-separating
simple closed curves that lie in $\textrm{Stab}[\phi]$.

######  Proposition 3.

Let $\alpha\subseteq\Sigma_{g}$ be a non-separating simple closed curve,
disjoint from a basepoint $*$. Let $T_{\alpha}$ denote the Dehn twist about
$\alpha$. Then $T_{\alpha}$ stabilizes $\phi$ up to conjugacy if and only if
$\alpha\in\ker\phi$.

###### Proof.

Suppose $\alpha\in\ker\phi$. Then one can explicitly find a model of
$\Sigma_{g}$ with a basepoint $*$, and a geometric basis of curves
$\\{a_{1},b_{1},\dots a_{g},b_{g}\\}$ passing through $*$ such that $\alpha$
is freely isotopic to $a_{1}$. The element $a_{1}$ is mapped to $1$ under
$\phi$. Consider an explicit representative curve $\bar{\alpha}$ of $\alpha$
such that

  1. 1.

$\bar{\alpha}$ is disjoint from $*$.

  2. 2.

$\bar{\alpha}$ is disjoint from $\\{a_{1},\dots,a_{g},b_{2},\dots b_{g}\\}.$

  3. 3.

$\bar{\alpha}$ intersects $b_{1}$ once transversally.

. Then $T_{\bar{\alpha}}(a_{i})=a_{i}$ for all $i$,
$T_{\bar{\alpha}}(b_{i})=b_{i}$ for $i\neq 1$ and
$T_{\bar{\alpha}}(b_{1})b_{1}^{-1}$ is conjugate to $a_{1}$. Hence, $\phi\circ
T_{\bar{\alpha}}=\phi$ which is what we wanted to prove.

Suppose $T_{\alpha}$ stabilizes $\phi$ up to conjugacy. Then there exists a
lift $T$ of $T_{\alpha}$ in $\textrm{Mod}(\Sigma_{g},*)$( the mapping class
group of $\Sigma_{g}$ preserving the marked point $*$) that stabilizes $\phi$
exactly, i.e. $\phi\circ T=i_{*}$. However any lift of $T_{\alpha}$ must be of
the form $T_{\bar{\alpha}}$ where $\bar{\alpha}$ is a simple closed curve,
disjoint from $*$, freely (i.e. without necessarily respecting the basepoint
$*$) isotopic to $\alpha$. But then we may again choose a geometric basis of
$\Sigma_{g}$, $\\{a_{1},b_{1},\dots,a_{g},b_{g}\\}$ such that:

  1. 1.

$\bar{\alpha}$ is disjoint from $\\{a_{1},\dots,a_{g},b_{2},\dots b_{g}\\}.$

  2. 2.

$\bar{\alpha}$ intersects $b_{1}$ once.

  3. 3.

$\bar{\alpha}$ is freely isotopic to $a_{1}$.

Then $T_{\bar{\alpha}}(b_{1})b_{1}^{-1}$ is conjugate to $a_{1}$ and hence
$\phi(T_{\bar{\alpha}}(b_{1}))=\phi(b_{1})$ if and only if $a_{1}\in\ker
i_{*}$ which is what we wanted to prove.

∎

Now we establish that our monodromy group lies in the stabilizer subgroup.

######  Proposition 4.

Let $X$ be a smooth projective surface, $\mathcal{L}$ an ample line bundle,
$C$ a smooth zero locus of a section of $\mathcal{L}$. Let $i:C\to X$ denote
the inclusion map. Let $\Gamma(X,\mathcal{L})\subseteq\textrm{Mod}(C)$ denote
the monodromy group. Then
$\Gamma(X,\mathcal{L})\subseteq\textrm{Stab}[i_{*}]$, and furthermore
$\Gamma(X,\mathcal{L})\subseteq\mathcal{A}[i_{*}]$.

###### Proof.

Let $\phi\in\Gamma(X,\mathcal{L})$. Let $i:C\to X$ be the inclusion map. Then
$i$ is homotopic to $i\circ\phi$ essentially by construction, the parallel
transport involved in defining the monodromy gives a homotopy between the two
maps. Hence they induce the same homomorphism between $\pi_{1}(C)$ and
$\pi_{1}(X)$ up to conjugation (the fact that we need to deal with
homomorphisms up to conjugation is due to the fact that the homotopy need not
preserve a base point).

Hence $\phi\in\textrm{Stab}[i_{*}]$. If a simple nonseparating closed curve
$\alpha$ is such that $T_{\alpha}$ in $\textrm{Stab}[i]$, Then $\alpha\in\ker
i_{*}$ by Proposition 3. Hence, $T_{\alpha}\in\mathcal{A}[i]$. Since
$\Gamma(X,\mathcal{L})$ is generated by such $T_{\alpha}$,
$\Gamma(X,\mathcal{L})\subseteq\mathcal{A}[i]$. ∎

##  3 Infinite index

In this section we will establish that under the assumptions mentioned in
Theorem 1, $\mathcal{A}[i_{*}]$ is of infinite index in the mapping class
group.

Let $g>0$. Let $\Sigma_{g}$ be a Riemann surface of genus $g$. Let
$\gamma_{1},\dots\gamma_{r}$ be simple closed curves in $\Sigma_{g}$, Let
$N\subseteq\pi_{1}(\Sigma_{g})$ be the normal subgroup generated by
$\gamma_{1},\dots\gamma_{r}$. Let $G=\pi_{1}(\Sigma_{g})/N$. Let
$\rho:\pi_{1}(\Sigma_{g})\to G$ denote the quotient map. Let $S$ be the set
$\textrm{Hom}(\pi_{1}(\Sigma_{g}),G)/conj$, i.e the set of homomorphisms from
$\pi_{1}(\Sigma_{g})$ to $G$, up to conjugation. The group
$\textrm{Mod}(\Sigma_{g})$ acts on $S$ and $\rho$ is an element of $S$.

######  Proposition 5.

Suppose that $G\neq 0$ is torsion-free. The stabilizer subgroup of $\rho$ in
$\textrm{Mod}(\Sigma_{g})$ is of infinite index.

###### Proof.

We first establish that there is some pair of essential simple closed curves
$\alpha,\beta$ on $\Sigma_{g}$, where $\alpha$ and $\beta$ have geometric
intersection number 1, with $\alpha\in\ker\rho$ and $\beta\not\in\ker\rho$.
Let $\mathcal{C}$ be the graph whose vertices are essential simple closed
curves in $\Sigma_{g}$ and whose edges consist of curves with geometric
intersection number 1. The graph $\mathcal{C}$ is connected. If there are no
pairs $\alpha,\beta$ of geometric intersection number 1, with
$\alpha\in\ker\rho$ and $\beta\not\in\ker\rho$, then $\mathcal{C}$ would not
be connected and $\mathcal{C}$ could be separated into two disjoint subgraphs
$\mathcal{C}_{1},\mathcal{C}_{2}$ each consisting of curves in the kernel of
$\rho$ and not in the kernel of $\rho$ respectively.

Let $\alpha,\beta$ on $\Sigma_{g}$, where $\alpha$ and $\beta$ have geometric
intersection number 1, with $\alpha\in\ker\rho$ and $\beta\not\in\ker\rho$.
Let $T_{\beta}$ be the Dehn twist about $\beta$. Then
$\rho(T_{\beta}^{k}(\alpha))=\rho(\beta)^{k}\neq 1$. Hence no power of
$T_{\beta}$ stabilizes $\rho$ as otherwise $\rho(T_{\beta}^{k}(\alpha))$ would
be conjugate to $\rho(\alpha)=1$. Hence the stabilizer subgroup is not finite
index. ∎

######  Proposition 6.

Let $C,X,\mathcal{L},i$ be as above. If $H^{1}(X,\mathbb{Q})\neq 0$,
$\textrm{Stab}[i_{*}]$ is infinite index in the mapping class group.

###### Proof.

Let $\alpha\in H^{1}(X,\mathbb{Q})$ be a nonzero class. The Lefschetz
hyperplane theorem implies that $i^{*}\alpha\neq 0$. Then for any
$\phi\in\textrm{Stab}[i_{*}]$, $\phi^{*}(i^{*}\alpha)=i^{*}\alpha$. Let
$mi^{*}\alpha$ be a multiple of $i^{*}\alpha$ that is integral.

Consider the surjective map $\textrm{Mod}(C)\to Sp(H^{1}(C),\mathbb{Z})$ The
above implies that $\textrm{Stab}[i_{*}]$ is mapped into the stabilizer
subgroup of the vector $mi^{*}(\alpha)$. But this stabilizer subgroup is not
finite index in $Sp(H^{1}(C,\mathbb{Z}))$ and hence $\textrm{Stab}[i_{*}]$ is
not finite index in $\textrm{Mod}(C)$.

∎

With the above Propositions we can now prove Theorem 1.

###### Proof of Theorem 1.

The group $\Gamma(X,\mathcal{L})$ is contained in $\mathcal{A}[i_{*}]$ by
Proposition 4.

The group $\mathcal{A}[i_{*}]\subseteq\textrm{Stab}[i_{*}]$. The group
$\textrm{Stab}[i_{*}]$in the cases when $H^{1}(X,\mathbb{Q})\neq 0$ or
$\pi_{1}X$ is torsion free and infinite, is of infinite index in
$\textrm{Mod}(C)$ by Propositions 5 and 6. Hence $\mathcal{A}[i_{*}]$ is also
infinite index. ∎

##  4 $\Gamma(X,\mathcal{L}^{\otimes r})$ contains framed mapping class groups
of subsurfaces

######  Lemma 1.

Let $X,\mathcal{L}$ be as above. Let $r>0$. Let $C$ be the zero locus a
section of $\mathcal{L}^{\otimes r}.$ Let $i:C\to X$ be the inclusion. Suppose
$i^{*}(H^{1}(X,\mathbb{Z}))=V\subseteq H^{1}(C,\mathbb{Z})$. Then for $r$
sufficiently large:

  1. 1.

$\Gamma(X,\mathcal{L}^{\otimes r})$ surjects onto a finite index subgroup of
$Sp(V^{\perp},\mathbb{Z})$.

  2. 2.

Fix a plane curve singularity $f:\mathbb{C}^{2}\to\mathbb{C}$. Let $C_{0}$ be
a Milnor fiber of $f$. Then one can embed $C_{0}$ as a subsurface of $C$. Let
$\Gamma_{v}$ be the versal deformation monodromy subgroup of
$\textrm{Mod}(C_{0})$. Let $\phi:\textrm{Mod}(C_{0})\to\textrm{Mod}(C)$ be the
homomorphism induced by inclusion. Then
$\phi(\Gamma_{v})\subseteq\Gamma(X,\mathcal{L}^{\otimes r})$.

For details on what the versal monodromy subgroup is, see [3]. We note that
the requirement on $r$ depends on the choice of plane curve singularity $f$,
one needs larger vales of $r$ for more complicated singularities.

We will actually prove something stronger than (2)- we will show that that
there is a singular curve $C_{sing}\subseteq X$ that is the zero locus of a
section of $\mathcal{L}^{\otimes r}$. The curve $C_{sing}$ will have precisely
one singularity that is isomorphic to $f$ and further more all deformations of
$f$ can be realized as deformations of $C_{sing}$ in $X$. This will crucially
rely on the description of the versal deformation space in [5].

In this section we will establish Lemma 1. We will start by providing a
reference for part (1) of Lemma 1 as this was already known.

###### Proof of Part(1) of Lemma 1.

The fact that $\Gamma(X,\mathcal{L}^{\otimes r})$ surjects onto a finite index
subgroup of $Sp(V^{\perp},\mathbb{Z})$ is part of Jannsen’s thesis (Chapter 3)
and follows from Theorem 3.8 of [7]. This establishes (1). ∎

The proof of part (2) of Lemma 1 will involve embedding a versal deformation
space of an isolated plane curve singularity into
$\mathbb{P}H^{0}(X,\mathcal{L}^{\otimes r})$.

Let $f:\mathbb{C}^{2}\to\mathbb{C}$ be a polynomial function. Let us assume
that $f^{-1}(0)$ has an isolated plane curve singularity at $(0,0)$, i.e. the
system of equations

$f(x,y)=0,f_{x}(x,y)=0,f_{y}(x,y)=0$

has an isolated solution at the origin. Let $\mathbb{C}[[x,y]]$ be the ring of
formal power series in $x$ and $y$. Let $I_{f}$ be the ideal of
$\mathbb{C}[[x,y]]$ generated by $f,f_{x}$ and $f_{y}$. Then it is known that
$\mathbb{C}[[x,y]]/I_{f}$ is finite dimensional.

Let $X$ be a smooth complex algebraic surface and with a line bundle
$\mathcal{L}$. Let $p\in X$. The completed local ring of $X$ at $p$ denoted
$\bar{\mathcal{O}}_{X,p}$ is isomorphic to $\mathbb{C}[[x,y]]$. Let $U$ be a
neighbourhood of $p$ such that $\mathcal{L}|_{U}$ and $T^{*}X|_{U}$ are
trivial and $U$ is biholomorphic to a disk. Then these trivialisations over
$U$ give a map $H^{0}(U,\mathcal{L})\to\bar{\mathcal{O}}_{X,p}$. Furthermore
given $f\in H^{0}(U,\mathcal{L})$ we can define
$I_{f}\subseteq\bar{\mathcal{O}}_{X,p}$ to be the ideal generated by
$(f,f_{x},f_{y})$, where $x$ and $y$ are analytic local parameters for $X$ at
$p$.

######  Proposition 7.

Let $f:\mathbb{C}^{2}\to\mathbb{C}$ define a plane curve with an isolated
singularity at the origin. Let $X$ be a smooth projective surface,
$\mathcal{L}$ an ample line bundle on $X$. Let $p\in X$. Fix an isomorphism of
$\bar{\mathcal{O}}_{X,p}$ with $\mathbb{C}[[x,y]]$. Then for $r$ sufficiently
large there exists:

  1. 1.

$\tilde{f}\in H^{0}(X,\mathcal{L}^{\otimes r})$, such that
$I_{\tilde{f}}=I_{f}$ (under the chosen identification of
$\bar{\mathcal{O}}_{X,p}$ with $\mathbb{C}[[x,y]]$) and the zero locus of
$\tilde{f}$ is smooth outside $p$.

  2. 2.

$g_{1},\dots g_{r}\in H^{0}(X,\mathcal{L}^{\otimes r})$ such that the set
$\\{g_{1},\dots,g_{r}\\}$ form a $\mathbb{C}$ basis of
$\bar{\mathcal{O}}_{X,p}/I_{\tilde{f}}$

###### Proof.

We will start with the proof of part (1). Let $k>0$. Let $I_{p}$ denote the
ideal sheaf of $p$. Let $N_{k}(p)$ denote the subscheme of $X$ defined by
$I_{p}^{k}$. By Serre Vanishing, the map $H^{0}(X,\mathcal{L}^{\otimes r})\to
H^{0}(N^{k}(p),\mathcal{L}^{\otimes r})$ is surjective, for $r$ big enough. We
may consider the restriction, $f|_{N^{k}(p)}$ as a section of
$H^{0}(N^{k}(p),\mathcal{L}^{\otimes r})$. Let $\mathfrak{m}$ denote the
maximal ideal of $\mathbb{C}[[x,y]]$. For $k$ big enough, the ideal $I_{f}$
contains $\mathfrak{m}^{k}$ and any $f_{1}$ such that
$f-f_{1}\in\mathfrak{m}^{k}$ satisfies $I_{f}=I_{f_{1}}$.

Consider the variety

$D=\\{(p^{\prime},h)\in X\setminus p\times H^{0}(X,\mathcal{L}^{\otimes
r})|h(p^{\prime})=0,dh(p^{\prime})=0,h|_{N^{k}(p)}=f|_{N^{k}(p)}\\}.$

The variety $D$ maps to

$W=\\{h\in H^{0}(X,\mathcal{L}^{\otimes r})|h|_{N^{k}(p)}=f|_{N^{k}(p)}\\}.$

It suffices to show that the map is not surjective. This will follow if we can
prove that the dimension of the source is less than that of the target. This
follows from the following computation, $D$ admits a projection $\phi$ to
$X\setminus\\{p\\}$. The fiber $\phi^{-1}(p^{\prime})$ consist of sections
$h\in H^{0}(X,\mathcal{L}^{\otimes r})$ such that
$h(p^{\prime})=0,dh(p^{\prime})=0,h|_{N^{k}(p)}=f|_{N^{k}(p)}$. Again Serre
Vanishing implies that for $r$ big enough, this is an affine space of
codimension $\dim X+1$ in $W$. Hence the dimension of $D\leq\dim W-\dim
X-1+\dim X=\dim W-1$. Hence $D$ does not surject onto $W$.

For part (2) we may apply Serre Vanishing to conclude that for $r$
sufficiently large, $H^{0}(X,\mathcal{L}^{\otimes r})$ surjects onto
$\bar{\mathcal{O}}_{X,p}/I_{\tilde{f}}$, hence we may take a preimage of a
basis of $\bar{\mathcal{O}}_{X,p}/I_{\tilde{f}}$ to obtain our
$\\{g_{1},\dots,g_{r}\\}.$ ∎

Fix a plane curve singularity $f$ and a point $p\in X$. Let
$\tilde{f},g_{1}\dots,g_{r}$ be as in Proposition 7. Let
$\mathbb{D}_{\epsilon}$ be the open $\epsilon$ ball in $\mathbb{C}^{r}$.
Consider the set

$V_{\epsilon}=\\{\tilde{f}+\sum
t_{i}g_{i}|\,(t_{1},\dots,t_{r})\in\mathbb{D}_{\epsilon}\\}.$

######  Proposition 8.

Let $B_{s}$ be open ball of radius $s$ centred at $p$. Let

$E_{V_{\epsilon}}=\\{(p,h)\in X\times V_{\epsilon}|\,h(p)=0,p\not\in
B_{s}\\}.$

Let $\pi:E_{V_{\epsilon}}\to V_{\epsilon}$ be the projection. Then for
$\epsilon$ small enough $\pi$ is a proper submersion and by Ehresmann’s
theorem defines a $C^{\infty}$ fiber bundle. Since $V_{\epsilon}$ is
contractible, it is a trivial fiber bundle.

###### Proof.

Since $\pi$ is obviously proper, it suffices to prove that it is a submersion.
Given $f+\sum t_{i}g_{i}\in V_{\epsilon}$, we have a section of the bundle
$(\mathcal{O}_{X}\oplus T^{*}X)\oplus\mathcal{L}^{\otimes r}$ given by
$(f+\sum t_{i}g_{i},df+\sum t_{i}dg_{i})$. It suffices to prove that for
$\epsilon$ small enough this section is non vanishing outside $B_{s}$. This
follows by continuity and the fact that $(f,df)$ is nonvanishing outside $B$.
∎

Propositions 7 and 8 are important because of the description of the versal
deformation space given in [5], essentially $V_{\epsilon}$ is a versal
deformation space of $f$.

Let $V^{\circ}_{\epsilon}=V\cap U(X,\mathcal{L}^{\otimes r})$. Let $C$ be the
zero locus of some $h\in V^{\circ}_{\epsilon}$. We have a monodromy
homomorphism $\rho:\pi_{1}(V^{\circ}_{\epsilon},h)\to\textrm{Mod}(C).$

######  Proposition 9.

The monodromy homomorphism
$\rho:\pi_{1}(V^{0}_{\epsilon},h)\to\textrm{Mod}(C)$ factorises as

$\pi_{1}(V^{\circ}_{\epsilon},h)\to\textrm{Mod}(C\cap\bar{B}_{s},C\cap\partial
B_{s})\to\textrm{Mod}(C).$

Here $\textrm{Mod}(C\cap\bar{B}_{s},C\cap\partial B_{s})$ denotes the mapping
class group of the surface with boundary $C\cap\bar{B}_{s}$ that is the
identity map on the boundary. The homomorphism
$\textrm{Mod}(C\cap\bar{B}_{s},C\cap\partial B_{s})\to\textrm{Mod}(C)$
corresponds to extending by the identity.

###### Proof.

The family $E(X,\mathcal{L}^{\otimes r})|_{V^{\circ}_{\epsilon}}\to
V^{\circ}_{\epsilon}$ is formed by gluing two bundles together, the trivial
bundle $E_{V_{\epsilon}}|_{V^{\circ}_{\epsilon}}$ and the nontrivial bundle
$\\{(f,x)\in V^{0}_{\epsilon}\times{B_{s}}|f(x)=0\\}.$ Let

$\rho_{0}:\pi_{1}(V^{\circ}_{\epsilon},h)\to\textrm{Mod}(C\cap\bar{B}_{s},C\cap\partial
B_{s})$

be the monodromy of the above bundle. Then the monodromy homomorphism
$\rho:\pi_{1}(V^{\circ}{})\to\textrm{Mod}(C)$ factorises through $\rho_{0}$
(as is always the case when a bundle is formed by gluing a trivial bundle in
this way).

∎

######  Proposition 10.

The image of the monodromy homomorphism
$\pi_{1}(V^{0}_{\epsilon},h)\to\textrm{Mod}(C\cap\bar{B}_{s},C\cap\partial
B_{s})$ is the monodromy group of the versal deformation space of the plane
curve singularity $f$.

###### Proof.

The space $V_{\epsilon}$ is a versal deformation space of $f$ (see for example
section 3 of [5]). Since it is a versal deformation space its monodromy group
is the monodromy group of the versal deformation space of the plane curve
singularity $f$. ∎

We are now in a position to prove part (2) of Lemma 1

###### Proof of part(2) of Lemma 1.

Part (2) of Lemma 1 follows immediately from Propositions 9 and 10. We note
that the Milnor fiber $C_{0}$ of Lemma 1 is just $C\cap\bar{B}_{s}$ of
Propositions9 and 10. ∎

##  5 The subgroups $\mathcal{A}(V)$ and Johnson homomorphisms

In this section we will discuss the groups $\mathcal{A}(V)$ and their image
under Johnson-Morita homomorphisms. Let $C$ be a Riemann surface. Let
$H=H^{1}(C,\mathbb{Q})$. Let $V\subseteq H$ be a symplectic subspace, and
$V^{\perp}$ its perpendicular subspace. Let $\rho:\textrm{Mod}(C)\to Sp(H)$
denote the representation of $\textrm{Mod}(C)$ arising from the action of
mapping classes on cohomology. Since $V\subseteq H$, $Sp(V^{\perp})\subseteq
Sp(H)$

Let $\mathcal{I}(V)=\rho^{-1}(Sp(V^{\perp}))$. Let $\mathcal{A}(V)$ be the
normal subgroup of $\mathcal{I}(V)$ generated by a single Dehn twist about a
nonseparating curve $\alpha$, with $[\alpha]\in V^{\perp}$.

######  Proposition 11.

$\mathcal{A}(V)$ is generated by the set of $T_{a}$ where $a$ is a
nonseparating curve, with $[a]\in V^{\perp}$.

###### Proof.

The Proposition immediately follows from the following claim. Claim:
$\mathcal{I}(V)$ acts transitively on the set of simple closed curves $a$,
satisfying $[a]\in V^{\perp}$.

We will now establish this claim.

Let $a,b$ be non- separating simple closed curves with $[a],[b]\in V^{\perp}.$
Then there exists $g\in Sp(V^{\perp})$ such that $g([a])=[b]$. Since
$\mathcal{I}(V)\to Sp(V^{\perp})$ is surjective, we may find
$\tilde{g}\in\mathcal{I}(V)$ lifting $g$. Then $\tilde{g}(a)$ and $b$ are
homologous. But by a well known theorem of Johnson [9] there is an element $h$
in the Torelli group such that $h\tilde{g}(a)=b$. But
$h\tilde{g}\in\mathcal{I}(V)$. This establishes the claim and hence the
Proposition. ∎

In what follows we will discuss the Malcev completion of groups and the
Johnson filtration of the mapping class group. We will be fairly brief, the
interested reader is advised to consult [6] and[8] for more details.

Given a group $G$, we will call the $k$th term of its lower central series by
$G_{k}$, with $G=G_{0}$. We will denote $G/G_{k}$ by $N_{k}(G)$ Consider the
group $N_{k}(\pi_{1}(C))$. It is a nilpotent group and as a result is embedded
in a $\mathbb{Q}$ algebraic nilpotent group called its Malcev completion,
which we will denote $L_{k}(C)$. Let $\mathfrak{l}_{k}(C)$ denote the Lie
algebra of $L_{k}(C)$. The functoriality of passing to Malcev completions and
taking Lie algebras means that we have a representation $\textrm{Mod}(C)\to
Out(\mathfrak{l}_{k}(C))$. Note that $\mathfrak{l}_{k}(C)$ is the quotient of
the free graded $k$-nilpotent Lie algebra on $H$ by the ideal generated by the
symplectic form.

We recall that the $k^{th}$ term of the Johnson filtration on
$\textrm{Mod}(C)$, $J_{k}$ is defined by $J_{k}:=\ker(\textrm{Mod}(C)\to
Out(\mathfrak{l}_{k}(C)))$.

Let $\mathfrak{l}_{k}(V^{\perp})$ denote the the free graded $k$ nilpotent Lie
algebra on $V^{\perp}$. Let $Aut^{s}(\mathfrak{l}_{k}(V^{\perp}))$ denote the
automorphism group of $\mathfrak{l}_{k}(V^{\perp})$ that fixes the symplectic
form $\omega|_{V^{\perp}}$.

Let $\mathfrak{l}_{k}(H)$ denote the the free graded $k$ nilpotent Lie algebra
on $H$. Let $Aut^{s}(\mathfrak{l}_{k}(H))$ denote the automorphism group of
$\mathfrak{l}_{k}(H)$ that fixes the symplectic form $\omega$. There is a
homomorphism $\phi:Aut^{s}(\mathfrak{l}_{k}(V^{\perp}))\to
Aut^{s}(\mathfrak{l}_{k}(H))$ defined as follows: Since $\mathfrak{l}_{k}(H)$
is a free graded nilpotent Lie algebra, to define a map it suffices to define
where the generators go. We define $\phi(g)(h)=h$ for $h\in V$ and
$\phi(g)(h)=g(h)$ for $h\in V^{\perp}$.

There is a homomorphism $\psi:Aut^{s}(\mathfrak{l}_{k}(H))\to
Aut(\mathfrak{l}_{k}(C))$ defined as follows: Since $\mathfrak{l}_{k}(C)$ is a
quotient of $\mathfrak{l}_{k}(H)$ by the ideal generated by $\omega$, we may
define $\psi(g)([a])=[g(a)]$, where $a\in\mathfrak{l}_{k}(H)$, $[a]$ is its
equivalence class in the quotient. This is well defined as $g(\omega)=\omega$.

Let $C_{0}\subseteq C$ denote a subsurface of $C$ with 1 boundary component
such that the image of $H_{1}(C_{0})$ in $H_{1}(C)$ is identified with
$V^{\perp}$ under Poincare duality.

######  Proposition 12.

Let $C_{0},C_{0}^{\prime}\subseteq C$ denote two connected subsurfaces, both
with 1 boundary component such that the image of $H_{1}(C_{0})$ and
$H^{1}(C_{0}^{\prime})$in $H_{1}(C)$ is identified with $V^{\perp}$ under
Poincare duality. Then there is a diffeomorphism of $C$ taking $C_{0}$ to
$C_{0}^{\prime}$, and $\phi$ may be chosen to lie in the Torelli group.

###### Proof.

By the classification of surfaces, one has a self map $C\to C$ that takes
$C_{0}$ to $C_{0}^{\prime}$. However as constructed $\phi$ may not lie in the
Torelli group. We note that the map on homology induced by $\phi$ preserves
$V$ and $V^{\perp}$. Let $\phi_{*}|_{V}=A_{1}$ and
$\phi_{*}|{V^{\perp}}=A_{2}$. Let $(\psi_{1},\psi_{2})$ be diffeomorphisms of
$C_{0}$ and $C\setminus C_{0}$ that are the identity on the boundary and whose
induced maps on homology are $A_{1}^{-1}$ and $A_{2}^{-1}$. Let $\psi$ be the
self diffeomorphism of $C$ formed by gluing $\psi_{1}$ and $\psi_{2}$. Then
one observes that $\phi\circ\psi$ is in the Torelli group and restricts to a
diffeomorphism between $C_{0}$ and $C_{0}^{\prime}$.

∎

Let us fix a point $*$ on the boundary of $C_{0}$. We have a commutative
diagram:

${J_{k}(C_{0})}$${\textrm{Mod}(C,*)}$${\textrm{Mod}(C)}$${\textrm{Aut}^{s}(\mathfrak{l}_{k}(V^{\perp}))}$${\textrm{Aut}^{s}(\mathfrak{l}_{k}(H))}$${\textrm{Out}(\mathfrak{l}_{k}(C))}$

######  Proposition 13.

The image of the composite map from $\textrm{Mod}(C_{0})\to
Out(\mathfrak{l}_{k}(C))$ does not depend on the choice of $C_{0}$.

The image of $\mathcal{A}(V)$ in $Out(\mathfrak{l}_{k}(C))$ is precisely the
image of the composite map from $\textrm{Mod}(C_{0})\to
Out(\mathfrak{l}_{k}(C))$.

###### Proof.

The fact that the image of the composite is independent of the choice of
$C_{0}$ is an immediate consequence of Proposition 12 .

To establish that the two images are the same, note that the image of
$\textrm{Mod}(C_{0})$ in $\textrm{Mod}(C)$ is contained in $\mathcal{A}[V]$
and that given a nonseparating simple closed curve $a$ with $[a]\in
V^{\perp}$, one can choose a $C_{0}$ satisfying the above properties with
$a\in C_{0}$. Since all $C_{0}$ have the same image, we have that the image of
$\mathcal{A}[V]$ is contained in the image of the composite map.

∎

######  Corollary 3.

Let $k\geq 2$. For $\dim V^{\perp}\ll k$, the group $\mathcal{A}(V)\cap
J_{k}/J_{k+1}$ is normally generated in $\mathcal{I}(V)$ by the image of
$J_{k}(C_{0})$.

###### Proof.

This immediately follows from the previous Proposition. ∎

We will also need the following related Proposition.

######  Proposition 14.

Let $C$ be a Riemann surface, $\mathcal{I}$ denote its Torelli group. Let
$V\subseteq H_{1}(C,\mathbb{Q})$ denote a symplectic subspace. Let
$\phi:\mathcal{I}\to\Lambda^{3}H_{1}(C,\mathbb{Z})/H_{1}(C,\mathbb{Z})$ denote
the Johnson homomorphism. Then

$\phi(\mathcal{I}\cap\mathcal{A}(V))\subseteq\Lambda^{3}V^{\perp}/V^{\perp}\subseteq\Lambda^{3}H_{1}(C,\mathbb{Z})/H_{1}(C,\mathbb{Z}).$

###### Proof.

Let $C_{0}\subseteq C$ be a subsurface with one boundary component such that
the image of $H_{1}(C_{0},\mathbb{Z})\to H_{1}(C,\mathbb{Z})$ is $V^{\perp}$.
We note that $\Lambda^{3}H_{1}(C,\mathbb{Z})/H_{1}(C,\mathbb{Z})$ is
isomorphic to $J_{1}(C)/J_{2}(C)$. As a result, by Corollary 3 the image of
$\mathcal{A}(V)\cap\mathcal{I}$ in
$\Lambda^{3}H_{1}(C,\mathbb{Z})/H_{1}(C,\mathbb{Z})$ agrees with that of
$J_{1}(C_{0})$. However, the naturality of the Johnson homomorphism implies
that the Johnson homomorphism $\phi:J_{1}(C_{0})\to J_{1}(C)/J_{2}(C)$
factorises as

$J_{1}(C_{0})\to\Lambda^{3}V^{\perp}\to\Lambda^{3}H_{1}(C,\mathbb{Z})/H_{1}(C,\mathbb{Z}),$

where the first map is the Johnson homomorphism for $C_{0}$ and the latter map
is the map induced by the inclusion
$\Lambda^{3}V^{\perp}\to\Lambda^{3}H_{1}(C,\mathbb{Z})$ followed by the
quotient map. ∎

##  6 Surjectivity of monodromy groups into quotients

The main purpose of this section is to prove the following Theorem.

######  Theorem 3.

Let $X,\mathcal{L}$ be an algebraic surface, $\mathcal{L}$ an ample line
bundle. Let $r>0$. Let $C$ be a Riemann surface embedded in $X$ as a section
of $\mathcal{L}^{\otimes r}$. Let $i:C\to X$ denote the inclusion. Let
$V=i^{*}(H^{1}(X,\mathbb{Q})).$ Let $k\geq 1$. Then for $r\gg k$,
$\Gamma(X,\mathcal{L}^{\otimes r})\cap J_{k}/J_{k+1}$ is finite index in
$A(V)\cap J_{k}/J_{k+1}$.

###### Proof.

The proof will be by the following steps:

  1. 1.

We prove that $\Gamma(X,\mathcal{L}^{\otimes r})$ contains a framed mapping
class group of a large subsurface $C^{\prime}$ contained in a subsurface of
the form $C_{0}$, where $C_{0}$ is as in Proposition 13.

  2. 2.

The image of $\Gamma(X,\mathcal{L}^{\otimes r})$ in $Sp(V^{\perp})$ is finite
index. Thus $\Gamma(X,\mathcal{L}^{\otimes r})\cap J_{k}/J_{k+1}$ is preserved
under the action of a finite index subgroup of $Sp(V^{\perp})$.

  3. 3.

We argue that (1) and (2) imply that $\Gamma(X,\mathcal{L}^{\otimes r})\cap
J_{k}/J_{k+1}$ contains a finite index subgroup of the image of
$J_{k}(C_{0})$.

  4. 4.

We use Proposition13 to argue that this then implies the result.

Part (1) is almost the same as part (2) of Lemma 1. We note that the
subsurface $C^{\prime}$ given to us by Lemma 1 is such that its homology has a
basis of vanishing cycles. As a result the map $H^{1}(X,\mathbb{Z})\to
H^{1}(C^{\prime},\mathbb{Z})$ is 0. Hence we may extend $C^{\prime}$ to a
maximal $C_{0}$ such that the pullback $H^{1}(X,\mathbb{Z})\to
H^{1}(C_{0},\mathbb{Z})$ is zero. This establishes (1).

To prove (2) note that by Lemma 1, $\Gamma$ surjects onto a finite index
subgroup of $Sp(V^{\perp})$ say $G$. Hence $\Gamma(X,\mathcal{L}^{\otimes
r})\cap J_{k}/J_{k+1}$ is preserved under the action of conjugation by
elements in $G$. This establishes (2).

We will establish (3) for $k\geq 2$ and $k=1$ separately. We first deal with
the case when $k\geq 2$. When $k\geq 2$, $J_{k}(C^{\prime})$ is contained in
the framed mapping class group of $C^{\prime}$.

We note that in the subsurface $C_{0}$, as long as $g(C^{\prime})\gg k$ the
orbit of the image of $J_{k}(C^{\prime})$ under a finite index subgroup of
$Sp(V^{\perp})$ is finite index in $J_{k}(C_{0})/J_{k+1}(C_{0}).$ This is due
to the following reason- by Theorem of [2], the $\textrm{Mod}(C_{0})$ orbits
of $J_{k}(C^{\prime})$ generates $J_{k}(C_{0})$. This implies that the
$Sp(V^{\perp})$ orbits of $J_{k}(C^{\prime})/J_{k}(C^{\prime})\cap
J_{k+1}(C_{0})$ generates $J_{k}(C_{0})/J_{k+1}(C_{0})$ and as a result the
orbits of $J_{k}(C^{\prime})/J_{k}(C^{\prime})\cap J_{k+1}(C_{0})$ under a
finite index subgroup of $Sp(V^{\perp})$ generates a finite index subgroup of
$J_{k}(C_{0})/J_{k+1}(C_{0})$. This establishes (3) in the case when $k\geq
2$.

When $k=1$, we note that according to [1] a framed mapping class group on the
surface $C^{\prime}$ contains the kernel $K(C^{\prime})$ of the Chillingworth
map $\mathcal{I}(C^{\prime})\to H_{1}(C^{\prime})$. This kernel contains an
element $\alpha$ such that the composite homomorphism

$\mathcal{I}(C^{\prime})\to\Lambda^{3}H_{1}(C^{\prime},\mathbb{Q})/H_{1}(C,\mathbb{Q})\to\Lambda^{3}H_{1}(C)/H_{1}(C,\mathbb{Q})$

sends $\alpha$ to a nonzero element, here the homomorphism is given by first
performing the Johnson homomorphism and composing it with the induced map
coming from $H_{1}(C^{\prime}\mathbb{Q})\to H_{1}(C,\mathbb{Q})$.

The image of such an $\alpha$ in $\Lambda^{3}H_{1}(C)/H_{1}(C,\mathbb{Q})$ is
forced to lie in the subgroup $\Lambda^{3}(V^{\perp})/V^{\perp}$. However,

$\Lambda^{3}(V^{\perp})/V^{\perp}\otimes\mathbb{Q}$

is an irreducible representation of $Sp(V^{\perp}\otimes\mathbb{Q})$ and as a
result, any nontrivial subgroup of

$\Lambda^{3}(V^{\perp})/V^{\perp}$

that is invariant under $Sp(V^{\perp})$ is finite index. As a result
$\Gamma\cap J_{1}(C)$ surjects onto a finite index subgroup of
$\Lambda^{3}(V^{\perp})/V^{\perp}$. As we have established in Proposition 14 ,
this implies that $\Gamma\cap J_{1}(C)/\Gamma\cap J_{2}(C)$ is finite index in
$\mathcal{A}(V)\cap J_{1}(C)/\mathcal{A}(V)\cap J_{2}(C).$

Due to (3), $\Gamma(X,\mathcal{L}^{\otimes r})\cap J_{k}/J_{k+1}$ contains a
finite index subgroup of the image of $J_{k}(C_{0})$. We then note that
$\mathcal{A}[V]\cap J_{k}$ and $J_{k}(C_{0})$ have the same image by
Proposition 13. This establishes (4).

∎

We now prove the two corollaries.

###### Proof of Corollary 1.

This is an immediate consequence of Theorems 1 and 2, using the fact that for
any lattice $G\subseteq Sp_{4}(\mathbb{Z})$, $H_{1}(G,\mathbb{Q})=0$ (this is
true for any lattice in a higher rank Lie group, by a theorem of Raghunathan
[10]). ∎

###### Proof of Corollary 2.

Part (2) is an immediate consequence of the fact that for any lattice
$G\subseteq Sp_{4}(\mathbb{Z})$, $H_{1}(G,\mathbb{Q})=0$ and Theorem 2.

Let $\gamma$ be a nonseparating simple closed curve such that $[\gamma]\in
V^{\perp}$ and $\gamma\not\in\ker i_{*}$. Then we claim that $T_{\gamma}^{k}$
is not in $\textrm{Stab}[i_{*}]$ and hence not in $\Gamma$ for all
$k\in\mathbb{Z}\setminus\\{0\\}$. The proof of the claim is as follows,
consider some nonseparating simple closed curve $\beta$, intersecting $\gamma$
at one point transversely such that $\beta\in\ker i_{*}$ (such a $\beta$
always exists by an argument identical to that in Proposition 5). Then if
$T_{\gamma}^{k}\in\textrm{Stab}[i_{*}]$, $i_{*}(T_{\gamma}^{k}\beta)$ is
conjugate to $i_{*}(\beta)=1$. But $i_{*}(T_{\gamma}^{k}\beta)$ is conjugate
to $i_{*}(\gamma)^{k}\neq 1$, since $G$ is torsion-free.

But the fact that $T_{\gamma}^{k}$ is never in $\Gamma$ for $k\neq 0$, and
$T_{\gamma}^{k}\in\mathcal{A}(V)$ for all $k$ implies that $\Gamma$ is of
infinite index in $\mathcal{A}(V).$ ∎

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