# Ramified covering maps and stability of pulled back bundles

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>and  A. J. Parameswaran  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>
###### Abstract.

Let $f\,:\,C\,\longrightarrow\,D$ be a nonconstant separable morphism between
irreducible smooth projective curves defined over an algebraically closed
field. We say that $f$ is genuinely ramified if ${\mathcal{O}}_{D}$ is the
maximal semistable subbundle of $f_{*}{\mathcal{O}}_{C}$ (equivalently, the
homomorphism $f_{*}\,:\,\pi_{1}^{\rm et}(C)\,\longrightarrow\,\pi_{1}^{\rm
et}(D)$ is surjective). We prove that the pullback
$f^{*}E\,\longrightarrow\,C$ is stable for every stable vector bundle $E$ on
$D$ if and only if $f$ is genuinely ramified.

###### Key words and phrases:

Genuinely ramified map, stable bundle, maximal semistable subbundle, socle.

###### 2010 Mathematics Subject Classification:

14H30, 14H60, 14E20

###### Contents

  1. 1 Introduction
  2. 2 Genuinely ramified morphism
  3. 3 Properties of genuinely ramified morphisms
  4. 4 Pullback of stable bundles and genuinely ramified maps
  5. 5 Characterizations of genuinely ramified maps

##  1\. Introduction

Let $k$ be an algebraically closed field. Let $f\,:\,C\,\longrightarrow\,D$ be
a nonconstant separable morphism between irreducible smooth projective curves
defined over $k$. For any semistable vector bundle $E$ on $D$, the pullback
$f^{*}E$ is also semistable. However, $f^{*}E$ need not be stable for every
stable vector bundle $E$ on $D$. Our aim here is to characterize all $f$ such
that $f^{*}E$ remains stable for every stable vector bundle $E$ on $D$. It
should be mentioned that $E$ is stable (respectively, semistable) if $f^{*}E$
is stable (respectively, semistable).

For any $f$ as above the following five conditions are equivalent:

  1. (1)

The homomorphism between étale fundamental groups

$f_{*}\,:\,\pi_{1}^{\rm et}(C)\,\longrightarrow\,\pi_{1}^{\rm et}(D)$

induced by $f$ is surjective.

  2. (2)

The map $f$ does not factor through some nontrivial étale cover of $D$ (in
particular, $f$ is not nontrivial étale).

  3. (3)

The fiber product $C\times_{D}C$ is connected.

  4. (4)

$\dim H^{0}(C,\,f^{*}f_{*}{\mathcal{O}}_{C})\,=\,1$.

  5. (5)

The maximal semistable subbundle of the direct image $f_{*}{\mathcal{O}}_{C}$
is ${\mathcal{O}}_{D}$.

The map $f$ is called genuinely ramified if any (hence all) of the above five
conditions holds. Proposition 2.6 and Definition 2.5 show that the above
statements (1), (2) and (5) are equivalent; in Lemma 3.1 it is shown that the
statements (3), (4) and (5) are equivalent.

We prove the following (see Theorem 5.3):

######  Theorem 1.1.

Let $f\,:\,C\,\longrightarrow\,D$ be a nonconstant separable morphism between
irreducible smooth projective curves defined over $k$. The map $f$ is
genuinely ramified if and only if $f^{*}E\,\longrightarrow\,C$ is stable for
every stable vector bundle $E$ on $D$.

The key technical step in the proof of Theorem 1.1 is the following (see
Proposition 3.5):

######  Proposition 1.2.

Let $f\,:\,C\,\longrightarrow\,D$ be a genuinely ramified Galois morphism, of
degree $d$, between irreducible smooth projective curves defined over $k$.
Then

$f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})\,\subset\,\bigoplus_{i=1}^{d-1}{\mathcal{L}}_{i}\,,$

where each ${\mathcal{L}}_{i}$ is a line bundle on $D$ of negative degree.

When $k\,=\,{\mathbb{C}}$, a vector bundle $F$ on a smooth complex projective
curve is stable if and only if $F$ admits an irreducible flat projective
unitary connection [NS]. From this characterization of stable vector bundles
it follows immediately that given a nonconstant map
$f\,:\,C\,\longrightarrow\,D$ between irreducible smooth complex projective
curves, $f^{*}E$ is stable for every stable vector bundle $E$ on $D$ if the
homomorphism of topological fundamental groups induced by $f$

$f_{*}\,:\,\pi_{1}(C,\,x_{0})\,\longrightarrow\,\pi_{1}(D,\,f(x_{0}))$

is surjective.

Theorem 4.4 was stated in [PS] (it is [PS, p. 524, Lemma 3.5(b)]) without a
complete proof. In the proof of Lemma 3.5(b), which is given in three
sentences in [PS, p. 524], it is claimed that the socle of a semistable bundle
descends under any ramified covering map (the first sentence).

##  2\. Genuinely ramified morphism

The base field $k$ is assumed to be algebraically closed; there is no
restriction on the characteristic of $k$.

Let $V$ be a vector bundle on an irreducible smooth projective curve $X$
defined over $k$. If

$0\,=\,V_{0}\,\subset\,V_{1}\,\subset\,\cdots\,\subset\,V_{n-1}\,\subset\,V_{n}\,=\,V$

is the Harder-Narasimhan filtration of $V$ [HL, p. 16, Theorem 1.3.4], then
define

$\mu_{\rm
max}(V)\,:=\,\mu(V_{1})\,:=\,\frac{\text{deg}(V_{1})}{\text{rk}(V_{1})}\ \
\text{ and }\ \ \mu_{\rm min}(V)\,:=\,\mu(V/V_{n-1})\,.$

Furthermore, the above subbundle $V_{1}\,\subset\,V$ is called the maximal
semistable subbundle of $V$.

If $V$ and $W$ are vector bundles on $X$, and
$\beta\,\in\,H^{0}(X,\,\text{Hom}(V,\,W))\setminus\\{0\\}$, then it can be
shown that

$\mu_{\rm max}(W)\,\geq\,\mu_{\rm min}(V)\,.$ (2.1)

Indeed, we have

$\mu_{\rm min}(V)\,\leq\,\mu(\beta(V))\,\leq\,\mu_{\rm max}(W)\,.$

######  Remark 2.1.

Let $f\,:\,X\,\longrightarrow\,Y$ be a nonconstant separable morphism between
irreducible smooth projective curves, and let $F$ be a semistable vector
bundle on $Y$. Then it is known that $f^{*}F$ is also semistable. Indeed,
fixing a nonconstant separable morphism $h\,:\,Z\,\longrightarrow\,X$, where
$Z$ is an irreducible smooth projective curve and $f\circ h$ is Galois, we see
that $(f\circ h)^{*}F$ is semistable, because its maximal semistable
subbundle, being $\text{Gal}(f\circ h)$ invariant, descends to a subbundle of
$F$. The semistability of $(f\circ h)^{*}F\,=\,h^{*}f^{*}F$ immediately
implies that $f^{*}F$ is semistable.

######  Lemma 2.2.

Let $f\,:\,X\,\longrightarrow\,Y$ be a nonconstant separable morphism of
irreducible smooth projective curves. Then for any semistable vector bundle
$E$ on $X$,

$\mu_{\rm max}(f_{*}E)\,\leq\,\mu(E)/{\rm deg}(f)\,.$

More generally, for any vector bundle $E$ on $X$,

$\mu_{\rm max}(f_{*}E)\,\leq\,\mu_{\rm max}(E)/{\rm deg}(f)\,.$

###### Proof.

Let $E$ be any vector bundle on $X$. The coherent sheaf $f_{*}E$ on $Y$ is
locally free, because it is torsion-free. We have

$H^{0}(Y,\,{\rm Hom}(F,\,f_{*}E))\,\cong\,H^{0}(X,\,{\rm Hom}(f^{*}F,\,E))$
(2.2)

for any vector bundle $F$ on $Y$; see [Ha, p. 110]. Setting $F$ in (2.2) to be
a semistable subbundle $V$ of $f_{*}E$ we see that $H^{0}(X,\,{\rm
Hom}(f^{*}V,\,E))\,\not=\,0$. The pullback $f^{*}V$ is semistable because $V$
is semistable and $f$ is separable; see Remark 2.1.

First take $E$ to be semistable. Hence for any nonzero homomorphism
$\beta\,:\,f^{*}V\,\longrightarrow\,E$,

${\rm deg}(f)\cdot\mu(V)\,=\,\mu(f^{*}V)\,\leq\,\mu(E)$ (2.3)

(see (2.1)). Now setting $V$ in (2.3) to be the maximal semistable subbundle
of $f_{*}E$ we conclude that

$\mu_{\rm max}(f_{*}E)\,\leq\,\mu(E)/{\rm deg}(f).$ (2.4)

To prove the general (second) statement, for any vector bundle $E$ on $X$, let

$0\,=\,E_{0}\,\subset\,E_{1}\,\subset\,\cdots\,\subset\,E_{n-1}\,\subset\,E_{n}\,=\,E$

be the Harder–Narasimhan filtration of $E$ [HL, p. 16, Theorem 1.3.4].
Consider the filtration of subbundles

$0\,=\,f_{*}E_{0}\,\subset\,f_{*}E_{1}\,\subset\,\cdots\,\subset\,f_{*}E_{n-1}\,\subset\,f_{*}E_{n}\,=\,f_{*}E\,.$
(2.5)

From (2.4) we know that

$\mu_{\rm max}((f_{*}E_{i})/(f_{*}E_{i-1}))\,=\,\mu_{\rm
max}(f_{*}(E_{i}/E_{i-1}))\,\leq\,\mu(E_{i}/E_{i-1})/{\rm deg}(f)$
$\leq\,\mu(E_{1})/{\rm deg}(f)\,=\,\mu_{\rm max}(E)/{\rm deg}(f)$ (2.6)

for all $1\,\leq\,i\,\leq\,n$. Observe that using (2.1) and the filtration in
(2.5) it follows that

$\mu_{\rm max}(f_{*}E)\,\leq\,{\rm Max}\\{\mu_{\rm
max}((f_{*}E_{i})/(f_{*}E_{i-1}))\\}_{i=1}^{n},$

while from (2.6) we have

${\rm Max}\\{\mu_{\rm
max}((f_{*}E_{i})/(f_{*}E_{i-1}))\\}_{i=1}^{n}\,\leq\,\mu_{\rm max}(E)/{\rm
deg}(f).$

Therefore, $\mu_{\rm max}(f_{*}E)\,\leq\,\mu_{\rm max}(E)/{\rm deg}(f)$, and
this completes the proof. ∎

The following lemma characterizes the étale maps among the separable
morphisms.

######  Lemma 2.3.

Let $f\,:\,X\,\longrightarrow\,Y$ be a nonconstant separable morphism between
irreducible smooth projective curves. Then the following three conditions are
equivalent:

  1. (1)

The map $f$ is étale.

  2. (2)

The degree of $f_{*}{\mathcal{O}}_{X}$ is zero.

  3. (3)

The vector bundle $f_{*}{\mathcal{O}}_{X}$ is semistable.

###### Proof.

We have $\mu_{\rm max}(f_{*}{\mathcal{O}}_{X})\,\geq\,0$, because
${\mathcal{O}}_{Y}\,\subset\,f_{*}{\mathcal{O}}_{X}$ (see (2.2) and (2.1)). On
the other hand, from Lemma 2.2 it follows that $\mu_{\rm
max}(f_{*}{\mathcal{O}}_{X})\,\leq\,0$, so

$\mu_{\rm max}(f_{*}{\mathcal{O}}_{X})\,=\,0\,.$ (2.7)

From (2.7) it follows immediately that $f_{*}{\mathcal{O}}_{X}$ is semistable
if and only if $\text{deg}(f_{*}{\mathcal{O}}_{X})\,=\,0$. Therefore,
statements (2) and (3) are equivalent.

Let ${\mathcal{R}}$ be the ramification divisor on $X$ for $f$; define the
effective divisor $B\,:=\,f_{*}{\mathcal{R}}$ on $Y$. We know that

$(\det f_{*}{\mathcal{O}}_{X})^{\otimes 2}\,=\,{\mathcal{O}}_{Y}(-B)$

(see [Ha, p. 306, Ch. IV, Ex. 2.6(d)], [Se]). Therefore, $f$ is étale (meaning
$B\,=\,0$) if and only if ${\rm deg}(f_{*}{\mathcal{O}}_{X})\,=\,0$. So
statements (1) and (2) are equivalent. ∎

Let $f\,:\,X\,\longrightarrow\,Y$ be a nonconstant separable morphism between
irreducible smooth projective curves. The algebra structure of
${\mathcal{O}}_{X}$ produces an ${\mathcal{O}}_{Y}$–algebra structure on the
direct image $f_{*}{\mathcal{O}}_{X}$.

######  Lemma 2.4.

Let ${\mathcal{V}}\,\subset\,f_{*}{\mathcal{O}}_{X}$ be the maximal semistable
subbundle. Then ${\mathcal{V}}$ is a sheaf of ${\mathcal{O}}_{Y}$–subalgebras
of $f_{*}{\mathcal{O}}_{X}$.

###### Proof.

The action of ${\mathcal{O}}_{Y}$ on $f_{*}{\mathcal{O}}_{X}$ is the standard
one. Let

${\mathbf{m}}\,:\,(f_{*}{\mathcal{O}}_{X})\otimes(f_{*}{\mathcal{O}}_{X})\,\longrightarrow\,f_{*}{\mathcal{O}}_{X}$

be the ${\mathcal{O}}_{Y}$–algebra structure on the direct image
$f_{*}{\mathcal{O}}_{X}$ given by the algebra structure of the coherent sheaf
${\mathcal{O}}_{X}$. We need to show that

${\mathbf{m}}({\mathcal{V}}\otimes{\mathcal{V}})\,\subset\,{\mathcal{V}}\,,$
(2.8)

where $\mathcal{V}$ is the maximal semistable subbundle of
$f_{*}{\mathcal{O}}_{X}$.

Since $\mathcal{V}$ is semistable of degree zero (see (2.7)), and $\mu_{\rm
max}((f_{*}{\mathcal{O}}_{X})/{\mathcal{V}})\,<\,0$, using (2.1) we conclude
that in order to prove (2.8) it suffices to show that
${\mathcal{V}}\otimes{\mathcal{V}}$ is semistable of degree zero. Indeed,
there is no nonzero homomorphism from ${\mathcal{V}}\otimes{\mathcal{V}}$ to
$(f_{*}{\mathcal{O}}_{X})/{\mathcal{V}}$, if
${\mathcal{V}}\otimes{\mathcal{V}}$ is semistable of degree zero.

We have $\text{deg}({\mathcal{V}}\otimes{\mathcal{V}})\,=\,0$, because
$\text{deg}({\mathcal{V}})\,=\,0$. So ${\mathcal{V}}\otimes{\mathcal{V}}$ is
semistable if it does not contain any coherent subsheaf of positive degree. As

${\mathcal{V}}\otimes{\mathcal{V}}\,\subset\,(f_{*}{\mathcal{O}}_{X})\otimes(f_{*}{\mathcal{O}}_{X})\,,$

if ${\mathcal{V}}\otimes{\mathcal{V}}$ contains a subsheaf of positive degree,
then $(f_{*}{\mathcal{O}}_{X})\otimes(f_{*}{\mathcal{O}}_{X})$ also contains a
subsheaf of positive degree.

Therefore, to prove the lemma it is enough to show that
$(f_{*}{\mathcal{O}}_{X})\otimes(f_{*}{\mathcal{O}}_{X})$ does not contain any
subsheaf of positive degree.

The projection formula, [Ha, p. 124, Ch. II, Ex. 5.1(d)], [Se], says that

$(f_{*}{\mathcal{O}}_{X})\otimes_{{\mathcal{O}}_{Y}}(f_{*}{\mathcal{O}}_{X})\,\cong\,f_{*}(f^{*}(f_{*}{\mathcal{O}}_{X}))\,.$
(2.9)

Since ${\mathcal{O}}_{Y}\,\subset\,f_{*}{\mathcal{O}}_{X}$, we have

${\mathcal{O}}_{Y}\,=\,{\mathcal{O}}_{Y}\otimes_{{\mathcal{O}}_{Y}}{\mathcal{O}}_{Y}\,\subset\,(f_{*}{\mathcal{O}}_{X})\otimes_{{\mathcal{O}}_{Y}}(f_{*}{\mathcal{O}}_{X})\,,$

and hence $\mu_{\rm
max}((f_{*}{\mathcal{O}}_{X})\otimes_{{\mathcal{O}}_{Y}}(f_{*}{\mathcal{O}}_{X}))\,\geq\,0$.
Now from (2.9) it follows that

$\mu_{\rm max}(f_{*}(f^{*}(f_{*}{\mathcal{O}}_{X})))\,\geq\,0\,.$ (2.10)

Since $f$ is separable, the pullback, by $f$, of a semistable bundle on $Y$ is
semistable (see Remark 2.1), and consequently the Harder–Narasimhan filtration
of $f^{*}F$ is the pullback, by $f$, of the Harder–Narasimhan filtration of
$F$. Therefore, from (2.7) it follows that

$\mu_{\rm max}(f^{*}(f_{*}{\mathcal{O}}_{X}))\,=\,0\,.$

Now applying the second part of Lemma 2.2,

$0\,=\,\mu_{\rm
max}(f^{*}(f_{*}{\mathcal{O}}_{X}))/\text{deg}(f)\,\geq\,\mu_{\rm
max}(f_{*}(f^{*}(f_{*}{\mathcal{O}}_{X})))\,.$

This and (2.10) together imply that

$\mu_{\rm max}(f_{*}(f^{*}(f_{*}{\mathcal{O}}_{X})))\,=\,0\,.$

Therefore, using (2.9) it follows that

$\mu_{\rm
max}((f_{*}{\mathcal{O}}_{X})\otimes(f_{*}{\mathcal{O}}_{X}))\,=\,0\,.$

Hence $(f_{*}{\mathcal{O}}_{X})\otimes(f_{*}{\mathcal{O}}_{X})$ does not
contain any subsheaf of positive degree. It was shown earlier that the lemma
follows from the statement that
$(f_{*}{\mathcal{O}}_{X})\otimes(f_{*}{\mathcal{O}}_{X})$ does not contain any
subsheaf of positive degree. ∎

######  Definition 2.5.

A nonconstant separable morphism $f\,:\,X\,\longrightarrow\,Y$ between
irreducible smooth projective curves is called genuinely ramified if
${\mathcal{O}}_{Y}$ is the maximal semistable subbundle of
$f_{*}{\mathcal{O}}_{X}$.

######  Proposition 2.6.

Let $f\,:\,X\,\longrightarrow\,Y$ be a nonconstant separable morphism between
irreducible smooth projective curves. Then the following three conditions are
equivalent:

  1. (1)

The map $f$ is genuinely ramified.

  2. (2)

The map $f$ does not factor through any nontrivial étale cover of $Y$ (in
particular, $f$ is not nontrivial étale).

  3. (3)

The homomorphism between étale fundamental groups induced by $f$

$f_{*}\,:\,\pi_{1}^{\rm et}(X)\,\longrightarrow\,\pi_{1}^{\rm et}(Y)$

is surjective.

###### Proof.

(1) $\Longrightarrow$ (2): If $f$ factors through a nontrivial étale covering
$g\,:\,{\widetilde{Y}}\,\longrightarrow\,Y$, then
$g_{*}{\mathcal{O}}_{\widetilde{Y}}$ is semistable of degree zero (see Lemma
2.3) and its rank coincides with the degree of $g$. Since

$g_{*}{\mathcal{O}}_{\widetilde{Y}}\,\subset\,f_{*}{\mathcal{O}}_{X}\,,$

this implies that $f$ is not genuinely ramified.

(2) $\Longrightarrow$ (1): Lemma 2.4 says that the maximal semistable
subbundle ${\mathcal{V}}\,\subset\,f_{*}{\mathcal{O}}_{X}$ is a subalgebra. If
$f$ is not genuinely ramified, then by taking the spectrum of $\mathcal{V}$ we
obtain a separable, possibly ramified, covering map

$g\,:\,{\widetilde{Y}}\,=\,{\rm Spec}\,{\mathcal{V}}\,\longrightarrow\,Y$
(2.11)

whose degree coincides with the rank of $\mathcal{V}$. We have
$g_{*}{\mathcal{O}}_{\widetilde{Y}}\,=\,{\mathcal{V}}$, and the inclusion map
${\mathcal{V}}\,\hookrightarrow\,f_{*}{\mathcal{O}}_{X}$ defines a map

$h\,:\,X\,\longrightarrow\,{\widetilde{Y}}$

such that

$g\circ h\,=\,f.$ (2.12)

Since $f$ is separable, from (2.12) it follows that $g$ is also separable. It
can be shown that $g$ is étale. To prove this, first note that
$g_{*}{\mathcal{O}}_{\widetilde{Y}}$ is semistable, because
$g_{*}{\mathcal{O}}_{\widetilde{Y}}\,=\,{\mathcal{V}}$ and ${\mathcal{V}}$ is
semistable. Next, from (2.7) and the semistability of $\mathcal{V}$ it follows
that $\mu(g_{*}{\mathcal{O}}_{\widetilde{Y}})\,=\,\mu_{\rm
max}(g_{*}{\mathcal{O}}_{\widetilde{Y}})\,=\,0$. Now Lemma 2.3 gives that the
map $g$ in (2.11) is étale.

Since $g$ is étale, and (2.12) holds, we conclude that the statement (2)
fails. Hence the statement (2) implies the statement (1).

The equivalence between the statements (2) and (3) follows from the definition
of the étale fundamental group. ∎

Let $f\,:\,X\,\longrightarrow\,Y$ be a nonconstant separable morphism between
irreducible smooth projective curves. Let

$g\,:\,{\widetilde{Y}}\,:=\,{\rm Spec}\,{\mathcal{V}}\,\longrightarrow\,Y$

be the étale covering corresponding to the maximal semistable subbundle
${\mathcal{V}}\,\subset\,f_{*}{\mathcal{O}}_{X}$ (see (2.11); it was shown
that the map in (2.11) is étale). Let

$h\,:\,X\,\longrightarrow\,{\widetilde{Y}}$ (2.13)

be the morphism given by the inclusion map
${\mathcal{V}}\,\hookrightarrow\,f_{*}{\mathcal{O}}_{X}$.

######  Corollary 2.7.

The map $h$ in (2.13) is genuinely ramified.

###### Proof.

Let $\beta\,:\,Z\,\longrightarrow\,{\widetilde{Y}}$ be an étale covering such
that there is a map

$\gamma\,:\,X\,\longrightarrow\,Z$

satisfying the condition $\beta\circ\gamma\,=\,h$. Since
$(g\circ\beta)\circ\gamma\,=\,f$, we have

$g_{*}{\mathcal{O}}_{\widetilde{Y}}\,\subset\,(g\circ\beta)_{*}{\mathcal{O}}_{Z}\,\subset\,f_{*}{\mathcal{O}}_{X}\,;$
(2.14)

also, we have
$\text{deg}((g\circ\beta)_{*}{\mathcal{O}}_{\widetilde{Y}})\,=\,0$, because
$g\circ\beta$ is étale (see Lemma 2.3). But
${\mathcal{V}}\,=\,g_{*}{\mathcal{O}}_{\widetilde{Y}}$ is the maximal
semistable subsheaf of $f_{*}{\mathcal{O}}_{X}$. Hence from (2.14) it follows
that
$g_{*}{\mathcal{O}}_{\widetilde{Y}}\,=\,(g\circ\beta)_{*}{\mathcal{O}}_{Z}$.
This implies that $\text{deg}(\beta)\,=\,1$. Therefore, from Proposition 2.6
we conclude that the map $h$ in (2.13) is genuinely ramified. ∎

##  3\. Properties of genuinely ramified morphisms

######  Lemma 3.1.

Let $f\,:\,C\,\longrightarrow\,D$ be a nonconstant separable morphism between
irreducible smooth projective curves. Then the following three conditions are
equivalent:

  1. (1)

The map $f$ is genuinely ramified.

  2. (2)

$\dim H^{0}(C,\,f^{*}f_{*}{\mathcal{O}}_{C})\,=\,1$.

  3. (3)

The fiber product $C\times_{D}C$ is connected.

###### Proof.

Let ${\widetilde{C\times_{D}C}}$ be the normalization of the fiber product
$C\times_{D}C$; it is a smooth projective curve, but it is not connected
unless $f$ is an isomorphism. We have the commutative diagram

$\textstyle{\widetilde{C\times_{D}C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\pi}_{1}}$$\scriptstyle{\nu}$$\scriptstyle{\widetilde{\pi}_{2}}$$\textstyle{C\times_{D}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D}$
(3.1)

By flat base change [Ha, p. 255, Proposition 9.3],

$f^{*}(f_{*}{\mathcal{O}}_{C})\,\cong\,{\pi_{1}}_{*}(\pi^{*}_{2}{\mathcal{O}}_{C})\,=\,{\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C}\,.$
(3.2)

(1) $\Longrightarrow$ (2): Since $f$ is separable, $f^{*}F$ is semistable if
$F$ is so (see Remark 2.1), and hence the maximal semistable subbundle of
$f^{*}f_{*}{\mathcal{O}}_{C}$ is $f^{*}\mathcal{V}$, where
${\mathcal{V}}\,\subset\,f_{*}{\mathcal{O}}_{C}$ is the maximal semistable
subbundle. If $f$ is genuinely ramified, then the maximal semistable subbundle
of $f^{*}f_{*}{\mathcal{O}}_{C}$ is
$f^{*}{\mathcal{O}}_{D}\,=\,{\mathcal{O}}_{C}$. On the other hand,

$H^{0}(C,\,(f^{*}f_{*}{\mathcal{O}}_{C})/(f^{*}\mathcal{V}))\,=\,0\,,$

because $\mu_{\rm
max}((f^{*}f_{*}{\mathcal{O}}_{C})/(f^{*}\mathcal{V}))\,<\,0$ (see (2.1)).
These together imply that

$\dim H^{0}(C,\,f^{*}f_{*}{\mathcal{O}}_{C})\,=\,1\,;$

to see this consider the long exact sequence of cohomologies associated to the
short exact sequence

$0\,\longrightarrow\,f^{*}\mathcal{V}\,\longrightarrow\,f^{*}f_{*}{\mathcal{O}}_{C}\,\longrightarrow\,(f^{*}f_{*}{\mathcal{O}}_{C})/(f^{*}\mathcal{V})\,\longrightarrow\,0.$

(2) $\Longleftrightarrow$ (3): From (3.2) it follows that

$H^{0}(C,\,f^{*}f_{*}{\mathcal{O}}_{C})\,=\,H^{0}(C,\,{\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C})\,=\,H^{0}(C\times_{D}C,\,{\mathcal{O}}_{C\times_{D}C})\,.$
(3.3)

Consequently, $C\times_{D}C$ is connected if and only if $\dim
H^{0}(C,\,f^{*}f_{*}{\mathcal{O}}_{C})\,=\,1$.

(3) $\Longrightarrow$ (1): Assume that $f$ is not genuinely ramified. We will
prove that $C\times_{D}C$ is not connected.

Let $g\,:\,\widetilde{D}\,\longrightarrow\,D$ be the étale cover of $D$ given
by ${\rm Spec}\,{\mathcal{W}}$, where
${\mathcal{W}}\,\subset\,f_{*}{\mathcal{O}}_{C}$ is the maximal semistable
subbundle (as in (2.11)). The degree of this covering $g$ is at least two,
because $f$ is not genuinely ramified. To prove that $C\times_{D}C$ is not
connected it suffices to show that ${\widetilde{D}}\times_{D}{\widetilde{D}}$
is not connected.

The projection

$\gamma\,:\,{\widetilde{D}}\times_{D}{\widetilde{D}}\,\longrightarrow\,{\widetilde{D}}$

to the first factor is evidently the base change of
$g\,:\,\widetilde{D}\,\longrightarrow\,D$ to $\widetilde{D}$, and hence the
map $\gamma$ is étale. The diagonal
${\widetilde{D}}\,\hookrightarrow\,{\widetilde{D}}\times_{D}{\widetilde{D}}$
is a connected component of ${\widetilde{D}}\times_{D}{\widetilde{D}}$. This
implies that ${\widetilde{D}}\times_{D}{\widetilde{D}}$ is not connected,
because the degree of $\gamma$ is at least two. ∎

######  Definition 3.2.

A nonconstant morphism $f\,:\,C\,\longrightarrow\,D$ between irreducible
smooth projective curves will be called a separable Galois morphism if $f$ is
separable, and there is a reduced finite subgroup
$\Gamma\,\subset\,\text{Aut}(C)$ such that $D\,=\,C/\Gamma$ and $f$ is the
quotient map $C\,\longrightarrow\,C/\Gamma$. Note that a separable Galois
morphism need not be étale. A separable Galois morphism which is genuinely
ramified will be called a genuinely ramified Galois morphism.

######  Proposition 3.3.

Let $f\,:\,C\,\longrightarrow\,D$ be a separable Galois morphism, of degree
$d$, between irreducible smooth projective curves. Then
$f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})$ is a coherent subsheaf of
${\mathcal{O}}^{\oplus(d-1)}_{C}$.

###### Proof.

The Galois group $\text{Gal}(f)$ of $f$ will be denoted by $\Gamma$. For any
point $x\,\in\,C$, let

$\Gamma_{x}\,\subset\,\Gamma$

be the isotropy subgroup that fixes $x$ for the action of $\Gamma$ on $C$. A
point $(x,\,y)\,\in\,C\times_{D}C$ is singular if and only if $\Gamma_{x}$ is
nontrivial. Note that for any $(x,\,y)\,\in\,C\times_{D}C$ the two isotropy
subgroups $\Gamma_{x}$ and $\Gamma_{y}$ are conjugate, because $y$ lies in the
orbit $\Gamma\cdot x$ of $x$. For any $\sigma\,\in\,\Gamma$, let

$C_{\sigma}\,\subset\,C\times_{D}C$ (3.4)

be the irreducible component given by the image of the map

$\beta_{\sigma}\,:\,C\,\longrightarrow\,C\times C\,,\ \
x\,\longmapsto\,(x,\,\sigma(x))\,;$

clearly we have $\beta_{\sigma}(C)\,\subset\,C\times_{D}C$. In this way, the
irreducible components of $C\times_{D}C$ are parametrized by the elements of
the Galois group $\Gamma$. Note that there is a canonical identification

$C\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,C_{\sigma}$ (3.5)

for every $\sigma\,\in\,\Gamma$.

Let $\widetilde{C\times_{D}C}$ be the normalization of $C\times_{D}C$. The
maps $\beta_{\sigma}$, $\sigma\,\in\,\Gamma$, in (3.4) together produce an
isomorphism

$C\times\Gamma\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\widetilde{C\times_{D}C}\,;$
(3.6)

this map sends any $(y,\,\sigma)\,\in\,C\times\Gamma$ to $(y,\,\sigma(y))$ if
$\Gamma_{y}$ is trivial; if $\Gamma_{y}$ is trivial, then $(y,\,\sigma(y))$ is
a smooth point of $C\times_{D}C$ and hence $(y,\,\sigma(y))$ gives a unique
point of $\widetilde{C\times_{D}C}$. Consequently, we have

$\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}\,=\,{\mathcal{O}}_{C}\otimes_{k}k[\Gamma]\,,$
(3.7)

where $\widetilde{\pi}_{1}$ is the projection in (3.1), and $k[\Gamma]$ is the
group ring. The natural inclusion
${\mathcal{O}}_{C\times_{D}C}\,\hookrightarrow\,\nu_{*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}$,
where $\nu$ is the map in (3.1), induces an injective homomorphism

$\varphi\,:\,{\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C}\,\hookrightarrow\,{\pi_{1}}_{*}\nu_{*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}\,=\,\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}\,,$
(3.8)

where $\pi_{1}$ and $\widetilde{\pi}_{1}$ are the maps in (3.1).

Let

$\xi\,:\,{\mathcal{O}}_{C}\,\longrightarrow\,{\mathcal{O}}_{C}\otimes_{k}k[\Gamma]$
(3.9)

be the composition of homomorphisms

${\mathcal{O}}_{C}\,\longrightarrow\,{\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C}\,\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\,\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}\,=\,{\mathcal{O}}_{C}\otimes_{k}k[\Gamma]$

(see (3.8) and (3.7)). Note that the image $\xi({\mathcal{O}}_{C})$ in (3.9)
is a subbundle of ${\mathcal{O}}_{C}\otimes_{k}k[\Gamma]$, because the section

$\xi(1_{C})\,\subset\,H^{0}(C,\,\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}})\,=\,k[\Gamma]$

is nowhere vanishing, where $1_{C}$ is the constant function $1$ on $C$. There
is a trivial subbundle ${\mathcal{E}}$ of the trivial bundle
${\mathcal{O}}_{C}\otimes_{k}k[\Gamma]$

${\mathcal{O}}^{\oplus(d-1)}_{C}\,=\,{\mathcal{E}}\,\subset\,{\mathcal{O}}_{C}\otimes_{k}k[\Gamma]$

such that

${\mathcal{E}}\oplus\xi({\mathcal{O}}_{C})\,=\,{\mathcal{O}}_{C}\otimes_{k}k[\Gamma]\,.$
(3.10)

To see this, take any point $x\,\in\,C$, and choose a subspace

${\mathcal{E}}_{x}\,\subset\,({\mathcal{O}}_{C}\otimes_{k}k[\Gamma])_{x}\,=\,k[\Gamma]$

such that $k[\Gamma]\,=\,{\mathcal{E}}_{x}\oplus\xi({\mathcal{O}}_{C})_{x}$;
then take

${\mathcal{E}}\,:=\,{\mathcal{O}}_{C}\otimes_{k}{\mathcal{E}}_{x}\,\subset\,{\mathcal{O}}_{C}\otimes_{k}k[\Gamma]\,.$

This subbundle $\mathcal{E}$ clearly satisfies the condition in (3.10).

From the decomposition in (3.10) we conclude that
$({\mathcal{O}}_{C}\otimes_{k}k[\Gamma])/\xi({\mathcal{O}}_{C})\,=\,\mathcal{E}$.
Using the isomorphism in (3.7), the homomorphism $\varphi$ in (3.8) gives a
homomorphism

$\varphi^{\prime}\,:\,({\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C})/{\mathcal{O}}_{C}\,\longrightarrow\,({\mathcal{O}}_{C}\otimes_{k}k[\Gamma])/(\varphi({\mathcal{O}}_{C}))\,=\,({\mathcal{O}}_{C}\otimes_{k}k[\Gamma])/(\xi({\mathcal{O}}_{C}))\,=\,{\mathcal{E}}.$
(3.11)

On the other hand, the isomorphism in (3.2) produces an isomorphism

$({\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C})/{\mathcal{O}}_{C}\,\cong\,f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})\,.$

Combining this isomorphism with the homomorphism $\varphi^{\prime}$ in (3.11)
we get a homomorphism

$f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})\,\longrightarrow\,\mathcal{E}\,=\,{\mathcal{O}}^{\oplus(d-1)}_{C}\,.$

This homomorphism is clearly an isomorphism over the nonempty open subset of
$C$ where $f$ is étale. ∎

Note that
$f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})\,=\,(f^{*}f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{C}$;
but we use $f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})$ due to the
relevance of $(f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}$.

Let

$f\,:\,C\,\longrightarrow\,D$

be a genuinely ramified Galois morphism, of degree $d$, between irreducible
smooth projective curves; see Definition 3.2. As before, the Galois group
$\text{Gal}(f)$ will be denoted by $\Gamma$, so we have

$\\#\Gamma\,=\,d\,.$

Assume that $d\,>\,1$.

As in (3.4), the irreducible component of $C\times_{D}C$ corresponding to
$\sigma\,\in\,\Gamma$ will be denoted by $C_{\sigma}$.

The following lemma formulated in the above set-up will be used in proving a
variation of Proposition 3.3.

######  Lemma 3.4.

There is an ordering of the elements of $\Gamma$

$\Gamma\,=\,\\{\gamma_{1},\,\gamma_{2},\,\cdots,\,\gamma_{d}\\}$

and a self-map

$\eta\,:\,\\{1,\,2,\,\cdots,\,d\\}\,\longrightarrow\,\\{1,\,2,\,\cdots,\,d\\}$

such that

  1. (1)

$\gamma_{1}\,=\,e$ (the identity element of $\Gamma$),

  2. (2)

$\eta(1)\,=\,1$,

  3. (3)

$\eta(j)\,<\,j$ for all $j\,\in\,\\{2,\,\cdots,\,d\\}$, and

  4. (4)

$C_{\gamma_{j}}\bigcap C_{\gamma_{\eta(j)}}\,\not=\,\emptyset$ (see (3.4) for
notation).

###### Proof.

Set $\Gamma_{0}\,:=\,\gamma_{1}$ to be the identity element $e\,\in\,\Gamma$;
also, set $\eta(1)\,=\,1$. Set $N_{0}\,=\,1$.

Let

$\Gamma_{1}\,\subset\,\Gamma$

be the subset consisting of all $\gamma\,\not=\,e$ such that the action of
$\gamma$ on $C$ has a fixed point. Therefore, $\Gamma_{1}$ consists of all
$\gamma\,\not=\,e$ such that the irreducible component
$C_{\gamma}\,\subset\,C\times_{D}C$ intersects the component
$C_{e}\,=\,C_{\gamma_{1}}$. We note that $\Gamma_{1}$ is nonempty, because
otherwise $C_{\gamma_{1}}$ would be a connected component of $C\times_{D}C$,
while from Lemma 3.1(3) we know that $C\times_{D}C$ is connected; recall that
$\Gamma\,\not=\,\\{e\\}$ and $f$ is genuinely ramified.

If $\\#\Gamma_{1}\,=\,N_{1}-1\,=\,N_{1}-N_{0}$, set $\gamma_{j}\,\in\,\Gamma$,
$2\,\leq\,j\,\leq\,N_{1}$, to be distinct elements of $\Gamma_{1}$ in an
arbitrary order. Set

$\eta(j)\,=\,1$

for all $2\,\leq\,j\,\leq\,N_{1}$.

If $\Gamma_{1}\bigcup\Gamma_{0}\,\not=\,\Gamma$, let

$\Gamma_{2}\,\subset\,\Gamma\setminus(\Gamma_{1}\cup\Gamma_{0})$

be the subset consisting of all
$\gamma\,\in\,\Gamma\setminus(\Gamma_{1}\bigcup\Gamma_{0})$ such that the
irreducible component

$C_{\gamma}\,\subset\,C\times_{D}C$

intersects the component $C_{\sigma}$ for some $\sigma\,\in\,\Gamma_{1}$. Note
that such a component $C_{\gamma}$ does not intersect $C_{\gamma_{1}}$,
because in that case we would have $\gamma\,\in\,\Gamma_{1}$.

If $\\#\Gamma_{2}\,=\,N_{2}-N_{1}$, set $\gamma_{j}\,\in\,\Gamma$,
$N_{1}+1\,\leq\,j\,\leq\,N_{2}$, to be distinct elements of $\Gamma_{2}$ in an
arbitrary order. For every $N_{1}+1\,\leq\,j\,\leq\,N_{2}$, set

$\eta(j)\,\in\,\,\\{2,\,\cdots,\,N_{1}\\}$

such that the component $C_{\gamma_{j}}\,\subset\,C\times_{D}C$ intersects the
component $C_{\gamma_{\eta(j)}}$; the above definition of $\Gamma_{2}$ ensures
that such a $\eta(j)$ exists. If there are more than one
$m\,\in\,\\{2,\,\cdots,\,N_{1}\\}$ such that
$C_{\gamma_{j}}\,\subset\,C\times_{D}C$ intersects the component
$C_{\gamma_{m}}$, then choose $\eta(j)$ arbitrarily from them.

Now inductively define

$\Gamma_{n}\,\subset\,\Gamma\setminus(\bigcup_{i=0}^{n-1}\Gamma_{i})\,,$

if $\Gamma_{n}\,\not=\,\emptyset$, to be the subset consisting of all
$\gamma\,\in\,\Gamma\setminus(\bigcup_{i=0}^{n-1}\Gamma_{i})$ such that the
irreducible component $C_{\gamma}\,\subset\,C\times_{D}C$ intersects the
component $C_{\sigma}$ for some $\sigma\,\in\,\Gamma_{n-1}$. Note that such a
component $C_{\gamma}$ does not intersect $\bigcup_{i=0}^{n-2}\Gamma_{i}$,
because in that case $\gamma\,\in\,\bigcup_{i=0}^{n-1}\Gamma_{i}$.

If $\\#\Gamma_{n}\,=\,N_{n}-\sum_{i=0}^{n-1}\\#\Gamma_{i}\,=\,N_{n}-N_{n-1}$,
set $\gamma_{j}\,\in\,\Gamma$, $N_{n-1}+1\,\leq\,j\,\leq\,N_{n}$, to be
distinct elements of $\Gamma_{n}$ in an arbitrary order. For
$N_{n-1}+1\,\leq\,j\,\leq\,N_{n}$, set

$\eta(j)\,\in\,\,[N_{n-2}+1,\,N_{n-1}]\,=\,\\{1+\sum_{i=0}^{n-2}\\#\Gamma_{i},\,\cdots,\,\sum_{i=0}^{n-1}\\#\Gamma_{i}\\}$

such that the component $C_{\gamma_{j}}\,\subset\,C\times_{D}C$ intersects the
component $C_{\gamma_{\eta(j)}}$. If $C_{\gamma_{j}}$ intersects more than one
such component, choose $\eta(j)$ to be any one from them, as before.

Since $\Gamma$ is a finite group, we have $\Gamma_{n}\,=\,\emptyset$ for all
$n$ sufficiently large. Set

${\mathbb{S}}\,=\,\sum_{i=0}^{\infty}\\#\Gamma_{i}\,=\,\text{Max}_{i\geq
0}\\{N_{i}\\}\,.$

Note that

$\bigcup_{i=1}^{\mathbb{S}}C_{\gamma_{i}}\,\subset\,C\times_{D}C$

is the connected component of $C\times_{D}C$ containing $C_{\gamma_{1}}$.
Hence from Lemma 3.1(3) we know that

$\bigcup_{i=1}^{\mathbb{S}}C_{\gamma_{i}}\,=\,C\times_{D}C.$

In other words, we have ${\mathbb{S}}\ =\,d\,:=\,\\#\Gamma$. This completes
the proof of the lemma. ∎

######  Proposition 3.5.

Let $f\,:\,C\,\longrightarrow\,D$ be a genuinely ramified Galois morphism, of
degree $d$, between irreducible smooth projective curves. Then

$f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})\,\subset\,\bigoplus_{i=1}^{d-1}{\mathcal{L}}_{i}\,,$

where each ${\mathcal{L}}_{i}$ is a line bundle on $D$ of negative degree.

###### Proof.

As in Lemma 3.4, the Galois group $\text{Gal}(f)$ is denoted by $\Gamma$. The
ordering in Lemma 3.4 of the elements of $\Gamma$ produces an isomorphism of
$k[\Gamma]$ with $k^{\oplus d}$. Consequently, from (3.7) we have

$\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}\,=\,{\mathcal{O}}_{C}\otimes_{k}k[\Gamma]\,=\,{\mathcal{O}}^{\oplus
d}_{C}\,.$ (3.12)

Let

$\Phi\,:\,\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}\,=\,{\mathcal{O}}^{\oplus
d}_{C}\,\longrightarrow\,{\mathcal{O}}^{\oplus
d}_{C}\,=\,\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}$

be the homomorphism defined by

$(f_{1},\,f_{2},\,\cdots,\,f_{d})\,\longmapsto\,(f_{1}-f_{\eta(1)},\,f_{2}-f_{\eta(2)},\,\cdots,\,f_{d}-f_{\eta(d)})\,,$
(3.13)

where $\eta$ is the map in Lemma 3.4; more precisely, the $i$-th component of
$\Phi(f_{1},\,f_{2},\,\cdots,\,f_{d})$ is $f_{i}-f_{\eta(i)}$. It is
straightforward to check that

${\mathcal{F}}\,:=\,\Phi({\mathcal{O}}^{\oplus
d}_{C})\,\subset\,{\mathcal{O}}^{\oplus
d}_{C}\,=\,\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}$

is a trivial subbundle of rank $d-1$; the first component of
$\Phi(f_{1},\,f_{2},\,\cdots,\,f_{d})$ vanishes identically, because
$\eta(1)\,=\,1$. More precisely, we have

${\mathcal{F}}\,=\,{\mathcal{O}}^{\oplus(d-1)}_{C}\,\subset\,{\mathcal{O}}^{\oplus
d}_{C}\,=\,\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}\,,$
(3.14)

where ${\mathcal{O}}^{\oplus(d-1)}_{C}$ is the subbundle of
${\mathcal{O}}^{\oplus d}_{C}$ spanned by all
$(f_{1},\,f_{2},\,\cdots,\,f_{d})$ such that $f_{1}\,=\,0$.

From (3.14) it follows immediately that

$\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}\,=\,{\mathcal{O}}^{\oplus
d}_{C}\,=\,{\mathcal{F}}\oplus\xi({\mathcal{O}}_{C})\,,$ (3.15)

where $\xi({\mathcal{O}}_{C})$ is the subbundle of ${\mathcal{O}}^{\oplus
d}_{C}\,=\,{\mathcal{O}}_{C}\otimes_{k}k[\Gamma]$ in (3.9) (see (3.12)).

In (3.1) we have $\widetilde{\pi}_{1}\,=\,\pi_{1}\circ\nu$, and hence, as in
(3.8), there is a natural homomorphism

$\varphi\,:\,{\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C}\,\hookrightarrow\,\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}$
(3.16)

which is an isomorphism over the open subset of $C$ where $f$ is étale.
Therefore, from (3.2) and (3.15) we get an injective homomorphism of coherent
sheaves

$\Psi\,:\,f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})\,\longrightarrow\,{\mathcal{F}}\,=\,{\mathcal{O}}^{\oplus(d-1)}_{C}\,;$
(3.17)

it is similar to (3.11), except that now the direct summand ${\mathcal{F}}$ is
chosen carefully (it was $\mathcal{E}$ in (3.11)). Note that since
$\text{rk}(f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,=\,d-1\,=\,\text{rk}({\mathcal{O}}^{\oplus(d-1)}_{C})$,
the homomorphism $\Psi$ in (3.17) is generically an isomorphism, because it is
an injective homomorphism of coherent sheaves. More precisely, $\Psi$ is an
isomorphism over the open subset of $C$ where the map $f$ is étale.

Consider the map $\eta$ in Lemma 3.4. For every $1\,\leq\,i\,\leq\,d-1$,
choose a point

$z_{i}\,\in\,C_{\gamma_{i+1}}\bigcap C_{\gamma_{\eta(i+1)}}\,;$ (3.18)

this is possible because the fourth property in Lemma 3.4 says that the
intersection $C_{\gamma_{i+1}}\bigcap C_{\gamma_{\eta(i+1)}}$ is nonempty.
Recall from (3.5) that $C$ is identified with $C_{\gamma_{i+1}}$. The point
$z_{i}\,\in\,C_{\gamma_{i+1}}$ in (3.18) will be considered as a point of $C$
using this identification. Let

${\mathcal{L}}_{i}\,:=\,{\mathcal{O}}_{C}(-z_{i})$

be the line bundle corresponding to the point $z_{i}\,\in\,C$.

For every $1\,\leq\,i\,\leq\,d-1$, let

$P_{i}\,:\,{\mathcal{O}}^{\oplus(d-1)}_{C}\,\longrightarrow\,{\mathcal{O}}_{C}$
(3.19)

be the natural projection to the $i$-th factor.

Consider the composition of homomorphisms $P_{i}\circ\Psi$, where $P_{i}$ and
$\Psi$ are constructed in (3.19) and (3.17) respectively. It can be shown that
$P_{i}\circ\Psi$ vanishes when restricted to the point $z_{i}$ in (3.18). To
see this, for any $1\,\leq\,j\,\leq\,d$, let

$\widehat{P}_{j}\,:\,{\mathcal{O}}^{\oplus
d}_{C}\,\longrightarrow\,{\mathcal{O}}_{C}$

be the natural projection to the $j$-th factor. Recall the homomorphism $\Phi$
constructed in (3.13). If $(f_{1},\,f_{2},\,\cdots,\,f_{d})$ in (3.13)
actually lies in the image of ${\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C}$ by
the inclusion map $\varphi$ in (3.16), then from (3.18) we have

$(\widehat{P}_{i+1}\circ\Phi)(f_{1},\,f_{2},\,\cdots,\,f_{d})(z_{i},\,\gamma_{i+1})\,=\,f_{i+1}(z_{i},\,\gamma)-f_{\eta(i+1)}(z_{i},\,\gamma_{\eta(i+1)})\,=\,0\,,$
(3.20)

where
$(z_{i},\,\gamma_{i+1})\,\in\,C\times\Gamma\,=\,\widetilde{C\times_{D}C}$ (see
(3.6)) and the same for $(z_{i},\,\gamma_{\eta(i+1)})$; note that from (3.18)
it follows that the point in $C$ corresponding to
$z_{i}\,\in\,C_{\gamma_{\eta(i+1)}}$ (see (3.18)) by the identification
$C\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,C_{\gamma_{\eta(i+1)}}$
in (3.5) coincides with the point corresponding to
$z_{i}\,\in\,C_{\gamma_{i+1}}$ (the element
$\gamma^{-1}_{i+1}\gamma_{\eta(i+1)}\,\in\,\Gamma$ fixes this point of $C$).
To clarify, there is a slight abuse of notation in (3.8) in the following
sense: sections of
$\widetilde{\pi}_{1*}{\mathcal{O}}_{\widetilde{C\times_{D}C}}$ over an open
subset $U\,\subset\,C$ are identified with function on
$\widetilde{\pi}^{-1}(U)$. So $(f_{1},\,f_{2},\,\cdots,\,f_{d})$ in (3.20) is
considered as a function on $\widetilde{\pi}^{-1}(U)$; the above condition
that $(f_{1},\,f_{2},\,\cdots,\,f_{d})$ in (3.20) lies in the image of
${\pi_{1}}_{*}{\mathcal{O}}_{C\times_{D}C}$ by the inclusion $\varphi$ map in
(3.16) means that $(f_{1},\,f_{2},\,\cdots,\,f_{d})$ coincides with
$\widehat{f}\circ\nu$ for some function $\widehat{f}$ on $\pi^{-1}_{1}(U)$,
where $\nu$ is the map in (3.1). Now from (3.20) it follows that
$P_{i}\circ\Psi$ vanishes when restricted to the point $z_{i}\,\in\,C$.

Since $P_{i}\circ\Psi$ vanishes when restricted to the point $z_{i}$, we have

$P_{i}\circ\Psi(f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,\subset\,{\mathcal{L}}_{i}\,=\,{\mathcal{O}}_{C}(-z_{i})\,\subset\,{\mathcal{O}}_{C}\,.$
(3.21)

From (3.17) and (3.21) it follows immediately that

$f^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})\,\hookrightarrow\,\bigoplus_{i=1}^{d-1}{\mathcal{L}}_{i}\,.$

Since $\text{deg}({\mathcal{L}}_{i})\,=\,-1$, the proof of the proposition is
complete. ∎

##  4\. Pullback of stable bundles and genuinely ramified maps

######  Lemma 4.1.

Let $f\,:\,C\,\longrightarrow\,D$ be a genuinely ramified morphism between
irreducible smooth projective curves. Let $V$ be a semistable vector bundle on
$D$. Then

$\mu_{\rm
max}(V\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,<\,\mu(V)\,.$

###### Proof.

First assume that the map $f$ is Galois. Take the line bundles
${\mathcal{L}}_{i}$, $1\,\leq\,i\,\leq\,d-1$, in Proposition 3.5, where
$d\,=\,\text{deg}(f)$. Then from Proposition 3.5 we have

$\mu_{\rm
max}(V\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,\leq\,\mu_{\rm
max}(V\otimes(\bigoplus_{i=1}^{d-1}{\mathcal{L}}_{i}))\,\leq\,{\rm\max}\\{\mu(V\otimes{\mathcal{L}}_{i})\\}_{i=1}^{d-1}\,,$

because $V\otimes{\mathcal{L}}_{i}$ is semistable. On the other hand,

$\mu(V\otimes{\mathcal{L}}_{i})\,<\,\mu(V)\,,$

because $\text{deg}({\mathcal{L}}_{i})\,<\,0$. Combining these, we have

$\mu_{\rm
max}(V\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,<\,\mu(V)\,,$

giving the statement of the proposition.

If the map $f$ is not Galois, consider the smallest Galois extension

$F\,:\,\widehat{C}\,\longrightarrow\,D$ (4.1)

such that there is a morphism
$\widehat{f}\,:\,\widehat{C}\,\longrightarrow\,C$ for which

$f\circ\widehat{f}\,=\,F\,.$ (4.2)

Note that $\widehat{C}$ is irreducible and smooth, and $F$ is separable. From
(4.2) it follows that

$f_{*}{\mathcal{O}}_{C}\,\subset\,F_{*}{\mathcal{O}}_{\widehat{C}}\,.$ (4.3)

First assume that the map $F$ in (4.1) is genuinely ramified. From (4.3) it
follows that

$(f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}\,\subset\,({F}_{*}{\mathcal{O}}_{\widehat{C}})/{\mathcal{O}}_{D}\,.$
(4.4)

Since $F$ is Galois, from Proposition 3.5 we know that
$({F}_{*}{\mathcal{O}}_{\widehat{C}})/{\mathcal{O}}_{D}$ is contained in a
direct sum of line bundles of negative degree. Hence the subsheaf
$(f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}$ in (4.4) is also contained in a
direct sum of line bundles of negative degree. This implies that

$\mu_{\rm
max}(V\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,<\,\mu(V)\,,$

giving the statement of the proposition.

Therefore, we now assume that $F$ is not genuinely ramified. Let

$(F_{*}{\mathcal{O}}_{\widehat{C}})_{1}\,\subset\,F_{*}{\mathcal{O}}_{\widehat{C}}$

be the maximal semistable subbundle. Let

$g\,:\,\widehat{D}\,\longrightarrow\,D$ (4.5)

be the étale cover defined by the spectrum of the bundle
$(F_{*}{\mathcal{O}}_{\widehat{C}})_{1}$ of ${\mathcal{O}}_{D}$–algebras (see
Lemma 2.4); that the map $g$ in (4.5) is étale follows from Lemma 2.3 and
(2.7), because
$g_{*}{\mathcal{O}}_{\widehat{D}}\,=\,(F_{*}{\mathcal{O}}_{\widehat{C}})_{1}$
and $(F_{*}{\mathcal{O}}_{\widehat{C}})_{1}$ is semistable. We note that the
Galois group $\text{Gal}(F)$ for $F$ acts naturally on
$F_{*}{\mathcal{O}}_{\widehat{C}}$, and this action of $\text{Gal}(F)$
preserves the subbundle $(F_{*}{\mathcal{O}}_{\widehat{C}})_{1}$; indeed, this
follows from the uniqueness of the maximal semistable subbundle
$(F_{*}{\mathcal{O}}_{\widehat{C}})_{1}$. Therefore, $\text{Gal}(F)$ acts on
$\widehat{D}$, and the map $g$ in (4.5) is $\text{Gal}(F)$–equivariant for the
trivial action of $\text{Gal}(F)$ on $D$. Consequently, the covering $g$ in
(4.5) is Galois.

Consider the following commutative diagram

$\textstyle{\widehat{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{f}}$$\scriptstyle{h}$$\scriptstyle{\widehat{g}}$$\textstyle{C\times_{D}\widehat{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$$\textstyle{\widehat{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D}$
(4.6)

The existence of the map $h$ in (4.6) is evident. The map $f$ being genuinely
ramified, it follows from Lemma 2.6 that the homomorphism between étale
fundamental groups

$f_{*}\,:\,\pi_{1}^{\rm et}(C)\,\longrightarrow\,\pi_{1}^{\rm et}(D)$

induced by $f$ is surjective. This implies that the fiber product
$C\times_{D}\widehat{D}$ is connected. The diagram in (4.6) should not be
confused with the one in (3.1) — in (4.6), $C\times_{D}\widehat{D}$ is smooth
as $g$ is étale.

We will prove that the map $\widehat{g}$ in (4.6) is genuinely ramified and
Galois. For this, first recall the earlier observation that $\text{Gal}(F)$
acts on $\widehat{D}$. The map $\widehat{g}$ is evidently equivariant for the
actions of $\text{Gal}(F)$ on $\widehat{C}$ and $\widehat{D}$. This
immediately implies that the map $\widehat{g}$ is Galois. From Corollary 2.7
it follows that $\widehat{g}$ is genuinely ramified.

We will next prove that $\widehat{C}\times_{D}\widehat{C}$ is a disjoint union
of curves isomorphic to $\widehat{C}\times_{\widehat{D}}\widehat{C}$. For
this, first note that $\widehat{C}\times_{D}\widehat{C}$ maps to
$\widehat{D}\times_{D}\widehat{D}$, and the curve
$\widehat{D}\times_{D}\widehat{D}$ is a disjoint union of copies of
$\widehat{D}$ as $g$ is étale Galois. The component of
$\widehat{C}\times_{D}\widehat{C}$ lying over any of these copies of
$\widehat{D}$ is isomorphic to $\widehat{C}\times_{\widehat{D}}\widehat{C}$,
and therefore $\widehat{C}\times_{D}\widehat{C}$ is a disjoint union of curves
isomorphic to $\widehat{C}\times_{\widehat{D}}\widehat{C}$.

From (4.3) we have

$F^{*}f_{*}{\mathcal{O}}_{C}\,\subset\,F^{*}F_{*}{\mathcal{O}}_{\widehat{C}}\,.$
(4.7)

Since $\widehat{C}\times_{D}\widehat{C}$ is a disjoint union of curves
isomorphic to $\widehat{C}\times_{\widehat{D}}\widehat{C}$, from (3.3) it
follows that $F^{*}F_{*}{\mathcal{O}}_{\widehat{C}}$ is a direct sum of copies
of $\widehat{g}^{*}\widehat{g}_{*}{\mathcal{O}}_{\widehat{C}}$. It was shown
above that $\widehat{g}$ is genuinely ramified and Galois. So from Proposition
3.5 we know that
$\widehat{g}^{*}((\widehat{g}_{*}{\mathcal{O}}_{\widehat{C}})/{\mathcal{O}}_{\widehat{D}})$
is contained in a direct sum of line bundles of negative degree.

Since $\widehat{g}$ is genuinely ramified, we know from Lemma 3.1, Remark 2.1
and (2.7) that

${\mathcal{O}}_{\widehat{C}}\,=\,H^{0}({\widehat{C}},\,\widehat{g}^{*}\widehat{g}_{*}{\mathcal{O}}_{D})\otimes{\mathcal{O}}_{\widehat{C}}$

is the maximal semistable subbundle of
$\widehat{g}^{*}\widehat{g}_{*}{\mathcal{O}}_{D}$. Since
$F^{*}F_{*}{\mathcal{O}}_{\widehat{C}}$ is a direct sum of copies of
$\widehat{g}^{*}\widehat{g}_{*}{\mathcal{O}}_{\widehat{C}}$, this implies that

$H^{0}({\widehat{C}},\,F^{*}F_{*}{\mathcal{O}}_{\widehat{C}})\otimes{\mathcal{O}}_{\widehat{C}}\,\subset\,F^{*}F_{*}{\mathcal{O}}_{\widehat{C}}$
(4.8)

is the maximal semistable subbundle. On the other hand, we have
$F^{*}f_{*}{\mathcal{O}}_{C}\,=\,\widehat{f}^{*}f^{*}f_{*}{\mathcal{O}}_{C}$.
So from Lemma 3.1, Remark 2.1 and (2.7) we know that

${\mathcal{O}}_{\widehat{C}}\,=\,H^{0}({\widehat{C}},\,F^{*}f_{*}{\mathcal{O}}_{C})\otimes{\mathcal{O}}_{\widehat{C}}\,\subset\,F^{*}f_{*}{\mathcal{O}}_{C}$
(4.9)

is the maximal semistable subbundle.

Consider the inclusion homomorphism in (4.7). From (4.8) and (4.9) we conclude
that using this homomorphism, the quotient
$F^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})$ is contained in

$F^{*}F_{*}{\mathcal{O}}_{\widehat{C}}/(H^{0}({\widehat{C}},\,F^{*}F_{*}{\mathcal{O}}_{\widehat{C}})\otimes{\mathcal{O}}_{\widehat{C}})\,.$
(4.10)

Since $F^{*}F_{*}{\mathcal{O}}_{\widehat{C}}$ is a direct sum of copies of
$\widehat{g}^{*}\widehat{g}_{*}{\mathcal{O}}_{\widehat{C}}$, the vector bundle
in (4.10) is isomorphic to a direct sum of copies of
$\widehat{g}^{*}((\widehat{g}_{*}{\mathcal{O}}_{\widehat{C}})/{\mathcal{O}}_{\widehat{D}})$.

It was shown above that
$\widehat{g}^{*}((\widehat{g}_{*}{\mathcal{O}}_{\widehat{C}})/{\mathcal{O}}_{\widehat{D}})$
is contained in a direct sum of line bundles of negative degree. Therefore,
the vector bundle in (4.10) is contained in a direct sum of line bundles of
negative degree. Consequently, the subsheaf

$F^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})\,\subset\,F^{*}F_{*}{\mathcal{O}}_{\widehat{C}}/(H^{0}({\widehat{C}},\,F^{*}F_{*}{\mathcal{O}}_{\widehat{C}})\otimes{\mathcal{O}}_{\widehat{C}})$

is also contained in a direct sum of line bundles of negative degree.

Since $F^{*}(f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})$ is contained in a
direct sum of line bundles of negative degree, we conclude that

$\mu_{\rm
max}((F^{*}V)\otimes(F^{*}((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})))\,<\,\mu(F^{*}V)\,;$

note that $F^{*}V$ is semistable by Remark 2.1 as $F$ is separable. From this
it follows that

$\mu_{\rm
max}(V\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,=\,\mu_{\rm
max}(F^{*}(V\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})))/\text{deg}(F)$
$<\,\mu(F^{*}V)/\text{deg}(F)\,=\,\mu(V)\,,$

because $F^{*}V$ is semistable. This completes the proof. ∎

######  Remark 4.2.

When the characteristic of the base field $k$ is zero, the tensor product of
two semistable bundles remains semistable [RR, p. 285, Theorem 3.18]. We note
that Lemma 4.1 is a straight-forward consequence of it, provided the
characteristic of $k$ is zero.

######  Lemma 4.3.

Let $f\,:\,C\,\longrightarrow\,D$ be a genuinely ramified morphism between
irreducible smooth projective curves. Let $V$ and $W$ be two semistable vector
bundles on $D$ with

$\mu(V)\,=\,\mu(W)\,.$

Then

$H^{0}(D,\,{\rm Hom}(V,\,W))\,=\,H^{0}(C,\,{\rm Hom}(f^{*}V,\,f^{*}W))\,.$

###### Proof.

Using the projection formula, and the fact that $f$ is a finite map, we have

$H^{0}(C,\,{\rm Hom}(f^{*}V,\,f^{*}W))\,\cong\,H^{0}(D,\,f_{*}{\rm
Hom}(f^{*}V,\,f^{*}W))\,\cong\,H^{0}(D,\,f_{*}f^{*}{\rm Hom}(V,\,W))$
$\,\cong\,H^{0}(D,\,{\rm Hom}(V,\,W)\otimes
f_{*}{\mathcal{O}}_{C})\,\cong\,H^{0}(D,\,{\rm Hom}(V,\,W\otimes
f_{*}{\mathcal{O}}_{C}))\,.$ (4.11)

Let

$0\,=\,B_{0}\,\subset\,B_{1}\,\subset\,\cdots\,\subset\,B_{m-1}\,\subset\,B_{m}\,=\,W\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})$

be the Harder–Narasimhan filtration of
$W\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})$ [HL, p. 16, Theorem
1.3.4]. Since $W$ is semistable, and $f$ is genuinely ramified, from Lemma 4.1
we know that

$\mu(B_{i}/B_{i-1})\,\leq\,\mu(B_{1})\,=\,\mu_{\rm
max}(W\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,<\,\mu(W)$

for all $1\,\leq\,i\,\leq\,m$. In view this and the given condition that
$\mu(V)\,=\,\mu(W)$, from (2.1) we conclude that

$H^{0}(D,\,{\rm Hom}(V,\,B_{i}/B_{i-1}))\,=\,0$

for all $1\,\leq\,i\,\leq\,m$; note that both $V$ and $B_{i}/B_{i-1}$ are
semistable. This implies that

$H^{0}(D,\,{\rm
Hom}(V,\,W\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D})))\,=\ 0\,.$

Consequently, we have

$H^{0}(D,\,{\rm Hom}(V,\,W\otimes f_{*}{\mathcal{O}}_{C}))\,=\,H^{0}(D,\,{\rm
Hom}(V,\,W))$

by examining the exact sequence

$0\,\longrightarrow\,{\rm Hom}(V,\,W)\,\longrightarrow\,{\rm Hom}(V,\,W\otimes
f_{*}{\mathcal{O}}_{C})\,\longrightarrow\,{\rm
Hom}(V,\,W\otimes((f_{*}{\mathcal{O}}_{C})/{\mathcal{O}}_{D}))\,\longrightarrow\,0\,.$

From this and (4.11) it follows that

$H^{0}(C,\,{\rm Hom}(f^{*}V,\,f^{*}W))\,=\,H^{0}(D,\,{\rm Hom}(V,\,W))\,.$

This completes the proof. ∎

######  Theorem 4.4.

Let $f\,:\,C\,\longrightarrow\,D$ be a genuinely ramified morphism between
irreducible smooth projective curves. Let $V$ be a stable vector bundle on
$D$. Then the pulled back vector bundle $f^{*}V$ is also stable.

###### Proof.

Consider the Galois extension $F\,:\,\widehat{C}\,\longrightarrow\,D$ and the
diagram in (4.6). Since $V$ is stable, from Lemma 4.3 it follows that $f^{*}V$
is simple. As $V$ is semistable, it follows that $g^{*}V$ is also semistable,
where $g$ is the map in (4.6). Let

$E\,\subset\,g^{*}V$

be the unique maximal polystable subbundle with $\mu(E)\,=\,\mu(g^{*}V)$ [HL,
p. 23, Lemma 1.5.5]; this subbundle $E$ is called the socle of $g^{*}V$. Since
$g^{*}V$ is preserved by the action of the Galois group $\text{Gal}(g)$ on
$g^{*}V$, there is a unique subbundle $E^{\prime}\,\subset\,V$ such that

$E\,=\,g^{*}E^{\prime}\,\subset\,g^{*}V\,.$

As $V$ is stable, we conclude that $E^{\prime}\,=\,V$, and hence $g^{*}V$ is
polystable. So we have a direct sum decomposition

$g^{*}V\,=\,\bigoplus_{j=1}^{m}V_{j}\,,$ (4.12)

where each $V_{j}$ is stable with $\mu(V_{j})\,=\,\mu(g^{*}V)$.

Take any $1\,\leq\,j\,\leq\,m$, where $m$ is the integer in (4.12). Since
$V_{j}$ is stable, and $\widehat{g}$ in (4.6) is Galois (this was shown in the
proof of Lemma 4.1), repeating the above argument involving the socle we
conclude that $\widehat{g}^{*}V_{j}$ is also polystable. On the other hand, as
$\widehat{g}$ is genuinely ramified (see the proof of Lemma 4.1), from Lemma
4.3 it follows that

$H^{0}(\widehat{C},\,\text{End}(\widehat{g}^{*}V_{j}))\,=\,H^{0}(\widehat{D},\,\text{End}(V_{j}))\,.$
(4.13)

But $H^{0}(\widehat{D},\,\text{End}(V_{j}))\,=\,k$, because $V_{j}$ is stable.
Hence from (4.13) we know that
$H^{0}(\widehat{C},\,\text{End}(\widehat{g}^{*}V_{j}))\,=\,k$. This implies
that $\widehat{g}^{*}V_{j}$ is stable, because it is polystable.

Since $\widehat{g}^{*}V_{j}$ is stable, and $\pi_{2}\circ h\,=\,\widehat{g}$
(see (4.6)), we conclude that $\pi_{2}^{*}V_{j}$ is also stable with

$\mu(\pi_{2}^{*}V_{j})\,=\,\mu(\pi_{2}^{*}g^{*}V)$

for all $1\,\leq\,j\,\leq\,m$. This implies that

$\pi_{1}^{*}f^{*}V\,=\,\pi_{2}^{*}g^{*}V\,=\,\bigoplus_{j=1}^{m}\pi_{2}^{*}V_{j}$
(4.14)

is polystable.

The map $\pi_{2}$ is genuinely ramified because $\widehat{g}$ is genuinely
ramified (see the proof of Lemma 4.1) and ${\widehat{g}}\,=\,h\circ\pi_{2}$.
Indeed, if $\pi_{2}$ factors through an étale covering of $\widehat{D}$, then
the genuinely ramified map ${\widehat{g}}$ factors through that étale covering
of $\widehat{D}$, and hence from Proposition 2.6 it follows that $\pi_{2}$ is
genuinely ramified.

Since $\pi_{2}$ is genuinely ramified, and each $V_{j}$ in (4.14) is stable,
from Lemma 4.3 it follows that

$H^{0}(C\times_{D}\widehat{D},\,\text{Hom}(\pi_{2}^{*}V_{i},\,\pi_{2}^{*}V_{j}))\,=\,H^{0}(\widehat{D},\,\text{Hom}(V_{i},\,V_{j}))$
(4.15)

for all $1\,\leq\,i,\,j\,\leq\,m$. We know that $V_{i}$ and $\pi_{2}^{*}V_{i}$
are stable. So from (4.15) we conclude that $V_{i}$ is isomorphic to $V_{j}$
if and only if $\pi_{2}^{*}V_{i}$ is isomorphic to $\pi_{2}^{*}V_{j}$. From
(4.15) it also follows that

$H^{0}(C\times_{D}\widehat{D},\,\text{End}(\pi^{*}_{2}g^{*}V))\,=\,H^{0}(\widehat{D},\,\text{End}(g^{*}V))\,;$
(4.16)

we note that this also follows from Lemma 4.3.

The vector bundle $f^{*}V$ on $C$ is semistable, because $V$ is semistable and
$f$ is separable. Let

$0\,\not=\,S\,\subset\,f^{*}V$ (4.17)

be a stable subbundle with

$\mu(S)\,=\,\mu(f^{*}V)\,.$ (4.18)

Since $S$ is stable with $\mu(S)\,=\,\mu(f^{*}V)$, and the map $\pi_{1}$ is
Galois, using the earlier argument involving the socle we conclude that

$\widetilde{\mathbb{S}}\,:=\,\pi_{1}^{*}S\,\subset\,\pi_{1}^{*}f^{*}V\,=\,\pi_{2}^{*}g^{*}V\,=\,\bigoplus_{j=1}^{m}\pi_{2}^{*}V_{j}\,=:\,\widetilde{V}$
(4.19)

is a polystable subbundle with
$\mu(\widetilde{\mathbb{S}})\,=\,\mu(\widetilde{V})$.

Consider the associative algebra $H^{0}(C\times_{D}\widehat{D},\,{\rm
End}(\widetilde{V}))$, where $\widetilde{V}$ is the vector bundle in (4.19).
Define the right ideal

$\Theta\,:=\,\\{\gamma\,\,\in\,H^{0}(C\times_{D}\widehat{D},\,{\rm
End}(\widetilde{V}))\,\mid\,\gamma(\widetilde{V})\,\subset\,\widetilde{\mathbb{S}}\\}\,\subset\,H^{0}(C\times_{D}\widehat{D},\,{\rm
End}(\widetilde{V}))\,,$ (4.20)

where $\widetilde{\mathbb{S}}$ is the subbundle in (4.19). The subbundle
$\widetilde{\mathbb{S}}\,\subset\,\widetilde{V}$ is a direct summand, because
$\widetilde{V}$ is polystable, and
$\mu(\widetilde{\mathbb{S}})\,=\,\mu(\widetilde{V})$. Consequently,
$\widetilde{\mathbb{S}}$ coincides with the subbundle generated by the images
of endomorphisms lying in the right ideal ${\Theta}$. Since $\widetilde{V}$ is
semistable, the image of any endomorphism of it is a subbundle.

Consider $\widetilde{V}$ in (4.19). The identification

$H^{0}(C\times_{D}\widehat{D},\,{\rm
End}(\widetilde{V}))\,=\,H^{0}(\widehat{D},\,\text{End}(g^{*}V))$

in (4.16) preserves the associative algebra structures of

$H^{0}(C\times_{D}\widehat{D},\,{\rm End}(\widetilde{V}))\ \ \text{ and }\ \
H^{0}(\widehat{D},\,\text{End}(g^{*}V))\,,$

because it sends any $\gamma\,\in\,H^{0}(\widehat{D},\,{\rm End}(g^{*}V))$ to
$\pi^{*}_{2}\gamma$. Let

$\widetilde{\Theta}\,\subset\,H^{0}(\widehat{D},\,\text{End}(g^{*}V))$ (4.21)

be the right ideal that corresponds to $\Theta$ in (4.20) by the
identification in (4.16). Let

$\overline{\mathcal{S}}\,\subset\,g^{*}V$ (4.22)

be the subbundle generated by the images of endomorphisms lying in the right
ideal $\widetilde{\Theta}$ in (4.21). Since $g^{*}V$ is semistable, the image
of any endomorphism of it is a subbundle. From the above construction of
$\overline{\mathcal{S}}$ it follows that

$\widetilde{\mathbb{S}}\,=\,\pi^{*}_{2}\overline{\mathcal{S}}\,,$

where $\widetilde{\mathbb{S}}$ is the subbundle in (4.20).

The isomorphism in (4.16) is equivariant for the actions of the Galois group
$\text{Gal}(\pi_{1})\,=\,\text{Gal}(g)$ on

$H^{0}(C\times_{D}\widehat{D},\,{\rm
End}(\widetilde{V}))\,=\,H^{0}(C\times_{D}\widehat{D},\,{\rm
End}(\pi^{*}_{1}f^{*}V))$

and $H^{0}(\widehat{D},\,{\rm End}(g^{*}V))$, because the isomorphism sends
any $\gamma\,\in\,H^{0}(\widehat{D},\,{\rm End}(g^{*}V))$ to
$\pi^{*}_{2}\gamma$. Since $\widetilde{\mathbb{S}}\,=\,\pi_{1}^{*}S$ in (4.19)
is preserved under the action of $\text{Gal}(\pi_{1})$ on $\pi^{*}_{1}f^{*}V$,
it follows that the action of $\text{Gal}(\pi_{1})$ on
$H^{0}(C\times_{D}\widehat{D},\,{\rm End}(\pi^{*}_{1}f^{*}V))$ preserves the
right ideal $\Theta$ in (4.20). These together imply that the action of
$\text{Gal}(g)$ on $H^{0}(\widehat{D},\,{\rm End}(g^{*}V))$ preserves the
right ideal $\widetilde{\Theta}$ in (4.21). Consequently, the subbundle

$\overline{\mathcal{S}}\,\subset\,g^{*}V$

in (4.22) is preserved under the action of $\text{Gal}(g)$ on $g^{*}V$.

Since $\overline{\mathcal{S}}$ is preserved under the action of
$\text{Gal}(g)$ on $g^{*}V$, there is a unique subbundle

${\mathbb{S}}_{0}\,\subset\,V$

such that $\overline{\mathcal{S}}\,=\,g^{*}{\mathbb{S}}_{0}\,\subset\,g^{*}V$.
Given that $V$ is stable, and $\mu({\mathbb{S}}_{0})\,=\,\mu(V)$ (this follows
from (4.18)), we now conclude that ${\mathbb{S}}_{0}\,=\,V$. Hence the
subbundle $S$ in (4.17) coincides with $f^{*}V$. Therefore, we conclude that
$f^{*}V$ is stable. ∎

##  5\. Characterizations of genuinely ramified maps

Let $D$ be an irreducible smooth projective curve, and let

$\phi\,:\,X\,\longrightarrow\,D$

be a nontrivial étale covering with $X$ irreducible. Let $L$ be a line bundle
on $X$ of degree one.

######  Proposition 5.1.

  1. (1)

The direct image $\phi_{*}L$ is a stable vector bundle on $D$.

  2. (2)

The pulled back bundle $\phi^{*}\phi_{*}L$ is not stable.

###### Proof.

Let $\delta$ be the degree of the map $\phi$; note that $\delta\,>\,1$,
because $\phi$ is nontrivial.

We have $\text{deg}(\phi_{*}L)\,=\,\text{deg}(L)\,=\,1$ [Ha, p. 306, Ch. IV,
Ex. 2.6(a) and 2.6(d)]. This implies that

$\text{deg}(\phi^{*}\phi_{*}L)\,=\,\delta\cdot\text{deg}(L)\,=\,\delta\,.$

We have a natural homomorphism

$H\,:\,\phi^{*}\phi_{*}L\,\longrightarrow\,L\,.$ (5.1)

This $H$ has the following property: For any coherent subsheaf
$W\,\subset\,\phi_{*}L$, the restriction of $H$ to
$\phi^{*}W\,\subset\,\phi^{*}\phi_{*}L$

$H_{W}\,:=\,H|_{\phi^{*}W}\,:\,\phi^{*}W\,\longrightarrow\,L$ (5.2)

is a nonzero homomorphism. Note that for any point $y\,\in\,D$, the fiber
$(\phi^{*}\phi_{*}L)_{y}$ is $H^{0}(\phi^{-1}(y),\,L|_{\phi^{-1}(y)})$, and
hence a nonzero element of $(\phi^{*}\phi_{*}L)_{y}$ must be nonzero at some
point of $\phi^{-1}(y)$.

We will first show that $\phi_{*}L$ is semistable. To prove this by
contradiction, let $V\,\subset\,\phi_{*}L$ be a semistable subbundle with

$\mu(V)\,>\,\mu(\phi_{*}L)\,=\,\frac{1}{\delta}\,.$ (5.3)

Consider the nonzero homomorphism

$H_{V}\,:=\,H|_{\phi^{*}V}\,:\,\phi^{*}V\,\longrightarrow\,L$

in (5.2). We have $\mu(\phi^{*}V)\,=\,\delta\cdot\mu(V)\,>\,1$ (see (5.3)),
and also $\phi^{*}V$ is semistable because $V$ is so. Consequently, $H_{V}$
contradicts (2.1). As $\phi_{*}L$ does not contain any subbundle $V$
satisfying (5.3), we conclude that $\phi_{*}L$ is semistable.

Since $\text{rk}(\phi_{*}L)$ is coprime to $\text{deg}(\phi_{*}L)$, the
semistable vector bundle $\phi_{*}L$ is also stable. This proves statement
(1).

The vector bundle $\phi^{*}\phi_{*}L$ is not stable, because the homomorphism
$H$ in (5.1) is nonzero and $\mu(\phi^{*}\phi_{*}L)\,=\,\mu(L)$. ∎

######  Proposition 5.2.

Let $f\,:\,C\,\longrightarrow\,D$ be a nonconstant separable morphism between
irreducible smooth projective curves such that $f$ is not genuinely ramified.
Then there is a stable vector bundle $E$ on $D$ such that $f^{*}E$ is not
stable.

###### Proof.

Since $f$ is not genuinely ramified, from Proposition 2.6 we know that there
is a nontrivial étale covering

$\phi\,:\,X\,\longrightarrow\,D$

and a map $\beta\,:\,C\,\longrightarrow\,X$ such that $\phi\circ\beta\,=\,f$.
As in Proposition 5.1, take a line bundle $L$ on $X$ of degree one. The vector
bundle $\phi_{*}L$ is stable by Proposition 5.1(1).

The vector bundle $\phi^{*}\phi_{*}L$ is not stable by Proposition 5.1(2).
Therefore,

$f^{*}(\phi_{*}L)\,=\,\beta^{*}\phi^{*}(\phi_{*}L)\,=\,\beta^{*}(\phi^{*}\phi_{*}L)$

is not stable. ∎

Theorem 4.4 and Proposition 5.2 together give the following:

######  Theorem 5.3.

Let $f\,:\,C\,\longrightarrow\,D$ be a nonconstant separable morphism between
irreducible smooth projective curves. The map $f$ is genuinely ramified if and
only if $f^{*}E$ is stable for every stable vector bundle $E$ on $D$.

## Acknowledgements

We thank both the referees for going through the paper very carefully and
making numerous suggestions. The first-named author is partially supported by
a J. C. Bose Fellowship. Both authors were supported by the Department of
Atomic Energy, Government of India, under project no.12-R&D-TFR-5.01-0500.

## References

  * [Ha] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
  * [HL] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997.
  * [NS] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540–567.
  * [PS] A. J. Parameswaran and S. Subramanian, On the spectrum of asymptotic slopes, Teichmüller theory and moduli problem, 519–528, Ramanujan Math. Soc. Lect. Notes Ser., 10, Ramanujan Math. Soc., Mysore, 2010.
  * [RR] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. Jour. 36 (1984), 269–291.
  * [Se] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6 (1956), 1–42.


# Resource-Efficient Gesture Recognition using Low-Resolution Thermal Camera
via Spiking Neural Networks and Sparse Segmentation

Ali Safa1,2, Wout Mommen1,3 and Lars Keuninckx1
1 imec, Leuven, Belgium, 2 ESAT, KU Leuven, Belgium, 3 ETRO, VUB, Brussels,
Belgium This research received funding from the Flemish Government under the
“Onderzoeksprogramma Artificiële Intelligentie (AI) Vlaanderen” programme.

###### Abstract

This work proposes a novel approach for hand gesture recognition using an
inexpensive, low-resolution ($24\times 32$) thermal sensor processed by a
Spiking Neural Network (SNN) followed by Sparse Segmentation and feature-based
gesture classification via Robust Principal Component Analysis (R-PCA).
Compared to the use of standard RGB cameras, the proposed system is
insensitive to lighting variations while being significantly less expensive
compared to high-frequency radars, time-of-flight cameras and high-resolution
thermal sensors previously used in literature. Crucially, this paper shows
that the innovative use of the recently proposed Monostable Multivibrator
(MMV) neural networks as a new class of SNN achieves more than one order of
magnitude smaller memory and compute complexity compared to deep learning
approaches, while reaching a top gesture recognition accuracy of $93.9\%$
using a 5-class thermal camera dataset acquired in a car cabin, within an
automotive context. Our dataset is released for helping future research.

## Supplementary Material

Our dataset can be downloaded at:

http://tinyurl.com/4purd2e3

##  I Introduction

Hand gesture recognition is a popular domain of research for human-machine
interfacing and has been studied in many different contexts, from touchless
interaction with smartphones and computers [10, 11, 8] to in-car driver
monitoring applications [4, 9, 17].

In addition, different sensor modalities have been investigated in the
literature, from RGB and depth cameras [10] to high-frequency radars and high-
resolution thermal camera sensors [20]. Since standard RGB cameras suffer from
high sensitivity to lighting conditions [14, 15], the use of more expensive
sensors such as high-frequency radars [2, 3], as well as high-resolution
thermal [4] and time-of-flight (ToF) cameras [10] has been investigated in
scenarios where the gesture detection system must be robust to lighting
conditions caused by day-night variations, such as for in-car driver gesture
recognition, which is the scenario considered in this paper.

Since radars and ToF cameras can be expensive [18], this work investigates the
use of an inexpensive low-resolution thermal sensors for in-cabin driver
gesture recognition (see Fig. 1), as a sensing modality not affected by day-
night lighting variation.

Crucially, we investigate a complementary approach to previously-proposed deep
learning systems such as [18] (which uses deep Temporal Convolution Networks),
by proposing a first-of-its-kind thermal gesture recognition system using an
event-based or spiking neural networks (SNN) [19]. The SNN acts as a highly
memory- and compute-efficient wake-up system for a downstream sparse
segmentation and feature-based classification pipeline using Robust Principal
Component Analysis (R-PCA) [5]. As SNN model, we use the recently-proposed
Monostable Multivibrator (MMV) Neural Networks as a fully-binarized and highly
compute- and area-efficient SNN design approach [6, 7].

The contributions of this paper are the following:

  1. 1.

We acquire a novel dataset for the study of in-car gesture recognition via
low-resolution thermal camera data (our dataset is released as supplementary
material).

  2. 2.

Using our dataset, we present a novel memory- and compute-efficient approach
for gesture recognition using the recently proposed MMV event-based neural
network followed by R-PCA segmentation and feature-based gesture
classification.

  3. 3.

We show that our proposed approach significantly outperforms by more than one
order of magnitude the use of deep learning methods in terms of memory and
computational costs, while achieving comparable accuracy.

This paper is organized as follows. Section II provides background theory
about MMVs and R-PCA segmentation. Section III presents our dataset
acquisition and gesture classification methods. Section IV provides our
experimental results. Finally, conclusions are provided in Section V.

Figure 1: Thermal gesture sensing setup. A $24\times 32$ pixel MLX90640
thermal camera is mounted in a car cabin and used to acquire data for the
study of gesture recognition.

##  II Background

###  II-A Network of Monostable Multivibrator Neurons (MMVs)

Recently, a new approach to SNNs has emerged through the use of event-driven
binary neurons behaving as Monostable Multivibrator (MMV) timers [6, 7]. Fig.
2 a) shows the working principle of the MMV neurons. The neuron has two types
of inputs: an excitatory (EXC) and an inhibitory (INH) one. As soon as an
input spike is received in the EXC connection, a state counter is triggered
which counts up to $T$ time-steps. As soon as the counter reached $T$, the MMV
counter is reset to zero and an output spike (OUT) is emitted. On the other
hand, if a spike is received in the INH connection while the timer is in the
triggered state, the timer is reset to zero and the MMV does not output any
spikes.

Figure 2: Monostable Multivibrator Neuron. a) MMV neuron behavior. b) Single
MMV neuron connected to an input spiking vector $\bar{\sigma}$ via the OR-ing
binary weights.

Using the event-driven MMV neuron of Fig. 2, networks of fully-connected or
recurrent MMV neurons can be built. MMV networks have the additional
particularity that all synapses or weight connections are strictly binary:
given a $d$-dimensional vector of spikes $\bar{\sigma}$, each entry $i$ in
$\bar{\sigma}$ is either connected to the EXC input, either to the INH input
or not connected at all to the downstream MMV neuron. In addition, the weight-
input inner product found in conventional deep neural networks is replaced by
a simple logical OR operation between the incoming spikes that are connected
to either the EXC or the INH input (see Fig. 2 b). Therefore, during training
[7], the MMV network essentially learns what input spike source $\sigma_{i}$
should be connected to which downstream MMV neuron and with which type of
connection (EXC or INH). In addition to learning the binary EXC and INH
connections, the integer timer period $T_{j}$ of each MMV $j$ is learned as
well during training [7].

Similar to deep learning, training is carried out using backprop with two
additional particularities: a) since MMV neurons output spikes and the
derivative of a spike in function of the MMV state counter value is ill-
defined, this derivative is replaced with a Gaussian function (1) acting as a
surrogate gradient (see [13] for an overview of surrogate gradients in SNNs);
b) since the weights of the MMV network are strictly binary, a quantization-
aware training procedure is followed which slowly binarizes the weights to
either EXC, INH or no connections. We refer the reader to [7] for additional
details behind MMV network training.

$\text{OUT}^{{}^{\prime}}(\text{STATE})\approx\frac{1}{\sqrt{2\pi}}e^{-2\text{
STATE}^{2}}$ (1)

Crucially, MMV networks have been shown to be more than one order of magnitude
more hardware-efficient compared to traditional SNNs due to their drastic
simplification in both neuron model and binary weight connections ($\sim 1$pJ
vs. $\sim 200$nJ of energy consumption on MNIST compared to state-of-the-art
SNN implementations) [7]. Therefore, the compute- and memory-efficient aspect
of MMVs justifies their use for our thermal gesture recognition scenario,
where our aim is to drastically reduce the hardware footprint compared to
conventional deep learning models.

###  II-B Robust Principal Component Analysis (R-PCA)

Robust Principal Component Analysis (R-PCA) [5] is concerned with the recovery
of a low-rank signal matrix $L$ affected by a sparse outlier matrix $S$ with
arbitrary amplitude, from the $n_{1}\times n_{2}$ measurement matrix $M$,
with:

$M=L+S$ (2)

By defining the nuclear norm $||A||_{*}=\sum_{i}\sigma_{i}(A)$ as the sum of
the singular values of the matrix $A$ and $||A||_{1}=\sum_{ij}|A_{ij}|$ as the
$l_{1}$ norm of $A$, it can be shown that the recovery of both $L$ and $S$
from $M$ in (2) can be done by solving the following optimization problem [5]:

$L,S=\arg\min_{L,S}||L||_{*}+\lambda||S||_{1}\hskip 3.0pt\text{s.t}\hskip
3.0ptM=L+S$ (3)

where $\lambda$ is the sparsity defining hyper-parameter [5].

In practice, given a measurement matrix $M$ (such as $N_{c}$ successive
thermal camera data frames), (3) can be solved using the algorithm for
Principal Component Pursuit by Alternating Directions provided in [5], with an
open Python implementation provided in [1]. In this paper, we use R-PCA to
segment the thermal camera images in order to isolate the sparse moving hand
in $S$ from the near-static dense background in $L$ (see Fig. 3 c). After this
segmentation, the trajectory of the hand performing the gesture can be tracked
and used to classify the gesture type (see Section III-B2).

Figure 3: MMV - R-PCA System Overview a) Pre-processing Eq. 4-6 b) Gesture
presence detection c) Gesture classification.

##  III Sensor Processing Methods

###  III-A Data Acquisition and Sensing Setup

We mount a MLX90640 $24\times 32$ thermal camera inside a car cabin facing the
driver seat (see Fig. 1) and use it to collect data at 8 frames per second in
both day and night times. The in-cabin changes in temperature caused by the
day to night shift adds a challenge to this use case. We consider a 5-class
recognition problem composed of: i) circular clock-wise, ii) circular counter-
clock-wise, iii) vertical up-down, iv) horizontal right-left and v) no gesture
(background and driver’s movements). Table I lists all data sequences acquired
in this work and Fig. 4 shows gesture acquisition examples from our dataset.

Acquisition Name | Gesture Type | Frames
---|---|---
no-gesture-{m,n} | no gestures | 1853, 1062
all-gesture-{m,n} | all gestures | 2631, 1109
cirCW-gesture-{m,n} | circular clock-wise | 824, 727
cirCCW-gesture-{m,n} | circular counter-clock-wise | 828, 748
vert-gesture-{m,n} | vertical | 853, 766
hor-gesture-{m,n} | horizontal | 790, 769
TABLE I: Gesture data acquired in this work. In the acquisition name, the
suffix -m denotes the morning-noon daytime while the suffix -n denotes night
time. The acquisition length is reported accordingly. Figure 4: Examples of
gestures acquired in our dataset.

###  III-B Proposed Gesture Recognition Pipeline

Our proposed thermal gesture recognition pipeline is a modular system composed
of two sub-systems: a) a wake-up system using a very compute-efficient binary
MMV network, which is responsible for detecting if any gesture is being
performed by the driver; and b) a gesture classification system based on on-
line R-PCA thermal camera segmentation and hand-crafted feature extraction.
Fig. 3 provides a complete overview of the proposed modular system. The MMV-
based wake-up sub-system enables a drastic saving in energy consumption by
monitoring when an actual gesture is performed (binary detection) and waking
up the more compute-expensive R-PCA scene segmentation pipeline only when
needed. Then, the R-PCA segmentation filters the environment-dependent
background clutter slowly changing through time (i.e., the low-rank
contribution $L$ in Eq. 2) and isolates the pixels corresponding to the fast
hand gesture movements as the matrix $S$ in (2). Thus, the R-PCA segmentation
helps making our system robust to changes in the background, greatly helping
model generalization. The trajectory of the segmented hand movement returned
in $S$ can then be directly classified using robust hand-crafted features (see
Section III-B2).

####  III-B1 MMV gesture detection

We use a recurrent MMV network with $C$ neurons followed by a spike counter
(which returns the spike count at the output of each MMV neuron after the $T$
MMV execution time steps) and a 2-class logistic regression layer (composed of
a $C\times 2$ weight matrix $W_{out}$ followed by $2$ sigmoid neurons) which
classifies the spike count result in order to detect if a gesture is being
initiated by the user (see the MMV architecture in Fig. 3 b). As soon as the
MMV network detects the execution of a gesture, the R-PCA classification
pipeline described in Section III-B2 is woken up.

As input data to the MMV network, a sliding window containing the past $N_{c}$
thermal frames (flatten as 1D vectors) at time-step $k$ is aggregated as the
matrix $M_{th}[k]$ with dimensions $N_{c}\times(h_{th}w_{tw})$, where the
sensor height and width are respectively $h_{th}=24$ and $w_{tw}=32$ in Fig.
1. Then, the thermal camera frames are normalized to a range of pixel values
between $[0,1]$ using:

$M_{th}[k]\xleftarrow{}\frac{M_{th}[k]-\min(M_{th}[k])}{\max(M_{th}[k])-\min(M_{th}[k])}$
(4)

After normalization in (4), the thermal frames are differentiated in time in
order to capture the change in each pixel, as conventionally done with event-
driven neural networks [15]:

$\Delta M_{th}[k]=M_{th}[k]-M_{th}[k-1]$ (5)

Due to differentiation (5), the resulting dimension becomes
$(N_{c}-1)\times(h_{th}w_{tw})$. Then, $\Delta M_{th}[k]$ is transformed into
MMV-compatible spiking (binary) data by thresholding the absolute change in
pixel intensity against a threshold $\theta_{s}$:

$E_{ij}[k]=1\hskip 3.0pt\text{if}\hskip 3.0pt|\Delta
M_{th}[k]|\geq\theta_{s}\hskip 3.0pt\text{else}\hskip 3.0ptE_{ij}[k]=0\hskip
3.0pt\forall i,j$ (6)

Finally, the $(N_{c}-1)\times(h_{th}w_{tw})$ spiking matrix $E[k]$ is reshaped
to $(N_{c}-1)h_{th}\times w_{tw}$ and fed row by row into the MMV network,
with input dimension $w_{tw}$ during $T=(N_{c}-1)h_{th}$ MMV execution time
steps. Fig. 3 a) shows how the thermal data is fed into the MMV network for
gesture detection. The MMV network is trained through backprop following the
method described in Section II-A, and used to detect the execution of a
gesture by the user (see Fig. 3 b).

####  III-B2 R-PCA segmentation and feature-based classification

As soon as the MMV network detects the start of a gesture, the R-PCA and
classification is woken up. The collection of $N_{c}$ consecutive thermal
camera frames $M_{th}[k]$ at time-step $k$ previously used by the MMVs to
detect the presence of a gesture in Fig. 3 a) is decomposed by R-PCA into a
low-rank contribution $L_{th}[k]$ containing the quasi-static background, and
a sparse contribution $S_{th}[k]$ containing non-zero pixels corresponding to
the faster gesture movement (see Fig. 3 c). The centroid location of the hand
performing the gesture is derived from the sparse image $S_{th}[k]$ as the
center coordinate $[x_{k},y_{k}]$ of the hand blob [12], and noted
$\bar{p}_{c}[k]=[x_{k},y_{k}]$. Then, the hand is tracked throughout time $k$
by low-pass filtering the evolution of $\bar{p}_{c}[k]$ to alleviate noise and
registering the low-pass-filtered coordinate $\bar{p}_{c}^{l}[k]$ as a new
point in the track history:

$T=\\{\bar{p}_{c}^{l}[k],\forall k=k^{*}-L,...,k^{*}\\}$ (7)

where $k^{*}$ is the current time step, $L$ is the track length and:

$\bar{p}_{c}^{l}[k+1]=\beta\bar{p}_{c}^{l}[k]+(1-\beta)\bar{p}_{c}[k+1]$ (8)

where $\beta\in[0,1]$ is the decay parameter setting the strength of the low-
pass filter.

Finally, the gesture class is derived from the track history $T$ in (7) as
follows:

  1. 1.

If the difference between maximum extend in the $x$ and $y$ directions
$D_{x}=\max(T_{x})-\min(T_{x})$ and $D_{y}=\max(T_{y})-\min(T_{y})$ is smaller
than a threshold $\theta_{c,1}$: $|D_{x}-D_{y}|<\theta_{c,1}$; and the minimum
diameter is larger than a threshold $\theta_{c,2}$:
$\min(D_{x},D_{y})>\theta_{c,2}$, the gesture is classified as circular.

The circular gesture is considered clock-wise if the angular evolution of the
track $T_{\phi}=\arctan\frac{T_{y}}{T_{x}}$ is decreasing and considered
counter-clock-wise in the opposite case.

  2. 2.

Else, we test if the gesture is vertical or horizontal by first measuring the
variances of the track evolution along the x-axis and along the y-axis:
$\sigma_{x}^{2}$ and $\sigma_{y}^{2}$. If the variance along the y-axis is
larger than along the x-axis $\sigma_{y}^{2}>\sigma_{x}^{2}$, the gesture is
registered as vertical.

  3. 3.

Finally, if the inverse is true: $\sigma_{x}^{2}>\sigma_{y}^{2}$, the gesture
is registered as horizontal.

The first case effectively verifies circularity of the track by checking if
diameters along the x- and y- are similar (as it is the case for a circle),
and also checks whether the circle is larger than a certain minimum diameter
$\theta_{c,2}$. If the first case is not verified, the vertical or horizontal
nature of the gesture is identified by checking along which axis changes occur
the most by, measuring the variances of $T_{x}$ and $T_{y}$.

##  IV Experimental Results

We evaluate the accuracy of the proposed system as follows. We first train the
MMV wake-up system using only the no-gesture-m and all-gesture-m acquisitions
in Table I, randomly partitioned into a training and validation set following
a 70%-30% split. The training is done using the Adam optimizer with learning
rate $\eta=5\times 10^{-3}$ and batch size $32$ during 50 epochs. MMV network
binarization starts from epoch 10 and ends at epoch 25. After binarization,
the model keeps on being trained for an addition 25 epochs and MMV accuracy is
assessed on the validation set after each epoch. The MMV network parameters
with maximum validation accuracy is kept as the final model. Training is
carried for three MMV network configurations using respectively 125, 250 and
500 neurons. In addition, Table II lists the various MMV and R-PCA hyper-
parameters used in our experiments (tuned manually for reducing compute
complexity while not sacrificing accuracy).

$N_{c}$ | $h_{tw}$ | $w_{tw}$ | $L$ | $\beta$ | $\theta_{s}$ | $\theta_{c,1}$ | $\theta_{c,2}$ | $\lambda$
---|---|---|---|---|---|---|---|---
$5$ | $24$ | $32$ | $10$ | $0.5$ | $0.2$ | $5$ | $5$ | $0.05$
TABLE II: System hyper-parameters.

Then, the MMV network is integrated into our proposed modular gesture
recognition system described in Section III-B (see Fig. 3) and accuracy is
assessed using all other acquisitions of Table I. This assessment approach
makes the problem challenging since the night-time data is not used during the
model training, but rather only used during testing. These aspects enable a
challenging and fair assessment of model generalization to sensing contexts
outside its training data (i.e., daytime vs. night-time).

The performance of the proposed MMV - R-PCA system is reported in Table III
and compared against the state-of-the-art 9-class deep learning method
proposed in [18] using a Temporal Convolution Network (TCN) with the same low-
resolution sensor used in this work (see Fig. 1). We also compare our proposed
system against a baseline MMV-only classifier (using 500 MMV neurons), without
the R-PCA classification pipeline of Section III-B2.

Model | Accuracy [%] | Parameters (kB) | Avg. FLOPS
---|---|---|---
TCN f128 [18] | 95.9 | 12550 | 622.4M
TCN f64 [18] | 93 | 1090 | 34.01M
Ours (125) | 92.6 | 36.7 | 129k + 1.7M
Ours (250) | 93.56 | 50.5 | 257k + 1.7M
Ours (500) | 93.9 | 101.4 | 513k + 1.7M
MMV only (500) | 85.4 | 76.7 | 1.28M
TABLE III: Accuracy, Parameter Utilization and FLOPS. The average FLOPS is
computed assuming a maximum user gesture execution rate of 1 gesture per
minute. For our proposed systems (with 125, 250 or 500 neurons), the Avg.
FLOPS is reported as ”MMV + R-PCA”. We also train an MMV-only setup (without
R-PCA) as baseline model.

Table III shows that our proposed MMV - R-PCA system significantly outperforms
the use of TCN in terms of parameter utilization (memory) and average Float
Operation Per Seconds (FLOPS)111The FLOPS for the Singular Value Decomposition
(SVD) executed for R-PCA [1] is computed by considering the formula provided
by [16] for the number of operations in SVD:
$n_{svd}=2N_{c}(h_{th}w_{th})^{2}+11(h_{th}w_{th})^{3}$ and considering a
maximum number of $100$ iterations in R-PCA [1].. In addition, Table III
demonstrates that our proposed R-PCA-based classification approach of Section
III-B2 leads to a significant gain of $+8.5\%$ in accuracy compared to the
MMV-only baseline classifier. On the other hand, the TCN in [18] achieves
superior accuracy to our system, with the ability to recognize more classes
than ours (9 vs. 5 classes), while suffering from more than one order of
magnitude more memory and compute complexity.

##  V Conclusion

This paper has presented a memory- and compute-efficient gesture recognition
system using low-resolution thermal camera data. After acquiring a novel
thermal camera dataset for gesture recognition, a first-of-its-kind system
using the recently-proposed MMV approach followed by conventional R-PCA and
feature-based gesture classification has been presented. Thanks to its use of
an ultra-low-complexity MMV wake-up system, it was shown that the proposed
gesture detector achieves close accuracy to state-of-the-art deep learning
systems while consuming more than one order of magnitude less memory and
compute power. Hence, the proposed approach is well-suited for application
cases where accuracy can be slightly traded off for a significant gain in
memory and compute efficiency. Finally, we released our dataset for helping
future research.

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