Kodaira Dimension and the Yamabe Problem

LaTeX file. With minor typographical errors corrected

Abstract:
  The Yamabe invariant Y(M) of a smooth compact manifold is roughly the
supremum of the scalar curvatures of unit-volume constant-scalar curvature
Riemannian metrics g on M. (To be absolutely precise, one only considers
constant-scalar-curvature metrics which are Yamabe minimizers, but this does
not affect the sign of the answer.) If M is the underlying smooth 4-manifold of
a complex algebraic surface (M,J), it is shown that the sign of Y(M) is
completely determined by the Kodaira dimension Kod (M,J). More precisely, Y(M)
< 0 iff Kod (M,J)=2; Y(M) = 0 iff Kod (M,J)=0 or 1; and Y(M) > 0 iff Kod (M,J)=
-infinity.

\section{Introduction}
 One may define an interesting and natural diffeomorphism invariant
of a compact smooth $n$-manifold $M$  as the supremum
of the scalar curvatures of unit-volume
 constant-scalar-curvature
metrics on $M$. A minor refinement of this definition,
which does not change the sign of the invariant but guarantees
that it is finite, restricts the class of allowed
constant-scalar-curvature metrics to the so-called
Yamabe minimizers.  We will refer to the resulting
invariant as the {\em Yamabe invariant} $Y(M)$ of
our manifold. For a more precise definition,
see \S \ref{yam} below.

For 2-manifolds, this invariant is easy to compute;
 the classical Gauss-Bonnet theorem tells us
that the Yamabe invariant of  a smooth compact surface is
just $4\pi \chi$, where $\chi$ is the
Euler characteristic.
Many important complex-analytic  invariants of
compact complex curves are  thus determined by the Yamabe invariant
of the underlying smooth 2-manifold.

One of the most important invariants of
a compact complex manifold $(M,J)$ is its
{\em Kodaira dimension} $\mbox{Kod} (M)$.
If $M$ has complex dimension $m$, recall that
the {\em canonical line bundle} $K \to M$
is   $K=\wedge^{m,0}$, so that the
holomorphic  sections of $K$ are exactly the
holomorphic $m$-forms on $(M,J)$.
One now defines the Kodaira dimension as
$$\mbox{Kod} (M) = \limsup \frac{\log h^0 (M,K^{\otimes \ell})}{\log \ell}.$$
This can be shown to coincide with the
maximal complex dimension of the image of $M$
under   pluri-canonical maps to complex projective space,
so that $\mbox{Kod} (M)\in \{ -\infty , 0, 1, \ldots, m\}$.
A compact complex $m$-manifold is said to be of {\em general type}
if $\mbox{Kod} M =m$.

For  Riemann surfaces,
the trichotomy $\mbox{Kod} (M) = -\infty , 0, 1$, exactly
coincides with that given by $Y(M) > 0, = 0, < 0$.
The purpose of this paper is to show that
much the same thing happens for compact complex surfaces
$(M^4,J)$. Our main result is

\begin{main} \label{main}
Let $M$ be the underlying 4-manifold of a
 compact complex  surface $(M^4,J)$ with $b_1(M)$ even.
Then
$$
Y(M) = \left\{
\begin{array}{ll} Y(M) > 0 &\mbox{iff } \mbox{Kod} (M,J) = -\infty \\
Y(M) = 0 &\mbox{iff } \mbox{Kod} (M,J) = 0 \mbox{ or } 1 \\
  Y(M)<  0 &\mbox{iff } \mbox{Kod} (M,J) = 2.
\end{array}\right.
$$
\end{main}

The hypothesis that $b_1(M)\equiv 0\bmod 2$
is equivalent to requiring that $(M,J)$ admit
a K\"ahler metric. It is also equivalent to
requiring that $(M,J)$ be a deformation of a
complex algebraic surface. Many of the results
contained in this article will also apply in the
non-K\"ahler case, but the overall picture
remains considerably less clear when $b_1$ is odd.

One of the main tools in the proof of the above
result is Seiberg-Witten theory. One of the
 most striking consequences of this
theory is that any two diffeomorphic complex
algebraic surfaces must have the same Kodaira
dimension \cite{FM,FQ}.  Since the Yamabe invariant is
obviously a diffeomorphism invariant,
Theorem  \ref{main}  casts
this result in an interesting new light.
On the other hand, the Yamabe invariant
does not by itself distinguish Kodaira
dimension $0$ from Kodaira dimension $1$.
The distinction between these cases instead only
emerges  when
one asks a finer question:  when is the Yamabe invariant
actually the scalar curvature of some unit-volume
metric?

 Finally, it should be emphasized that,
despite the cited analogy between complex
dimensions $1$ and $2$,  the phenomenon
 explored here does not persist in higher dimensions.
For example, consider a non-singular complex hypersurface of
 degree $m+3$ in ${\Bbb CP}_{m+1}$, where $m\geq 3$.
This complex $m$-manifold has ample
 canonical bundle, so its Kodaira dimension is $m$.
However, the underlying smooth manifold is
simply connected, non-spin, and has real dimension
$> 4$,   so a surgical  construction of
Gromov and Lawson \cite{GL} shows that it
admits metrics of positive scalar curvature,
and thus has $Y >0$.
Thus  the Yamabe invariant and Kodaira dimension
are correlated only in complex dimensions $1$ and $2$.

\section{Yamabe Invariants}
\label{yam}

This section is expository in nature,
and is intended as a convenient summary of
the Yamabe folklore which will be needed in the
rest of the paper.

Let $M$ be a smooth compact manifold.
A {\em conformal class} on $M$
is by definition a collection of
smooth Riemannian metrics on $M$ of the form
$$[g]=\{ vg ~|~v: M\to {\bf R}^+\},$$
where $g$ is some fixed Riemannain metric.
To each such conformal class, one can
associate a number $Y_{[g]}$, called the
{\em Yamabe constant} of the class, by
$$Y_{[g]} = \inf_{{\hat{g}}\in [g] } \frac{\int_M
s_{\hat{g}}~d\mu_{\hat{g}}}{\left(\int_M
d\mu_{\hat{g}}\right)^{\frac{n-2}{n}}},$$
where $s_{\hat{g}}$ and $d\mu_{\hat{g}}$ respectively
 denote the scalar curvature and volume measure of
${\hat{g}}$.  If $g$ is a Riemannian metric
for which the scalar curvature is everywhere positive,
or everywhere zero, or everywhere negative,
one can show that $Y_{[g]}$ is correspondingly
positive, zero, or negative.

While $Y_{[g]}$ has been defined as
an infimum, we could have instead defined it
as a {\sl minimum}, for a remarkable theorem
\cite{aubin,lp,rick}
of Yamabe, Trudinger, Aubin, and Schoen
asserts that any conformal class
$[g]$ contains a
metric
which actually minimizes the relevant functional.
Such a metric will be called a {\em Yamabe minimizer}.
Any Yamabe minimizer has constant scalar curvature;
conversely, a constant-scalar-curvature metric $g$
with $s_g \leq 0$ is
automatically a Yamabe minimizer.
 Aubin's piece of
the proof
hinges on the observation
that any conformal class on any
$n$-manifold automatically
satisfies
$Y_{[g]}\leq n(n-1) (V_n)^{2/n}$, where
$V_n$ is the  volume of
the  unit $n$-sphere
$S^n\subset {\bf R}^{n+1}$.

Given any smooth compact $n$-manifold, we can therefore
\cite{okob,sch}
define an associated real number by
$$Y(M) = \sup_{[g]} Y_{[g]} =
\sup_{[g]}\inf_{{\hat{g}}\in [g] } \frac{\int_M
s_{\hat{g}}~d\mu_{\hat{g}}}{\left(\int_M
d\mu_{\hat{g}}\right)^{\frac{n-2}{n}}}.$$
By construction, this is a diffeomorphism
invariant of $M$, and it will be called the
  {\em Yamabe invariant} of $M$ in this paper.
Notice that $Y(M) > 0$ iff $M$ admits a
metric of positive scalar curvature.
If $Y(M) \leq 0$, the invariant
is simply the   supremum of
the scalar curvatures of unit-volume
constant-scalar-curvature metrics on $M$,
since any constant-scalar-curvature metric
of non-positive scalar curvature is automatically a
Yamabe minimizer.

The minimax definition of $Y(M)$ is rather unwieldy for
many purposes.   Fortunately,
it can often be calculated in a simpler manner.

\begin{lem}
Let $[g]$ be any conformal class on a smooth compact
$n$-manifold $M$. Then
$$\inf_{{\hat{g}}\in [g]} \int_M |s_{\hat{g}}|^{n/2} d\mu_{\hat{g}}
= |Y_{[g]}|^{n/2}.$$
\label{yobo}
\end{lem}
\begin{proof} If $n=2$, the Gauss-Bonnet theorem
tells us that $\int |s|d\mu \geq
|\int s~d\mu |= |4\pi \chi (M)| =
|Y_{[g]}|$, with equality iff $s$ does not
change sign. We  may therefore assume henceforth that
$n \geq 3$. We will break this into two cases, depending
on the sign of $Y_{[g]}$.

First suppose that $Y_{[g]} \geq 0$ and $n \geq 3$. The H\"older
inequality tells us that  each metric ${\hat{g}}$ satisfies
$$\left(\int_M |s_{\hat{g}}|^{n/2} d\mu_{\hat{g}} \right)^{2/n}\geq
\frac{\int_M s_{\hat{g}}d\mu_{\hat{g}}}{\left(\int_M
 d\mu_{\hat{g}} \right)^{1-\frac{2}{n}}},$$
with equality iff $s_{\hat{g}}$ is a non-negative constant. Since the
infimum of the right-hand side over ${\hat{g}}\in [g]$ is $Y_{[g]}$
and is achieved by a metric of constant scalar curvature $s\geq 0$,
the claim follows.

Finally, we come to the   case in which  $Y_{[g]} < 0$
and $n \geq 3$.
Let $g\in [g]$ be a metric of constant negative scalar
curvature, and express any other metric in $[g]$
as  $\hat{g}=
u^\ell g$,
where $\ell = 4/(n-2)$ and $u$ is some smooth
positive function. The scalar curvatures $s=s_g$ and
$\hat{s}=s_{\hat{g}}$ are related by
$$\hat{s}u^{\ell + 1}= su + (n-1)\ell \triangle u$$
where $\triangle=d^\ast d$ is the positive Laplacian of $g$.
Thus
\begin{eqnarray*}
\int_M \hat{s}u^\ell d\mu  &=&
\int_M  \left(s+(n-1)\ell \frac{1}{u}d^\ast du\right) d\mu\\
&=& \int_M  \left(s-(n-1)\ell
\frac{|d u |^2}{u^2}\right) d\mu\\
&\leq& \int_Ms~d\mu ,
\end{eqnarray*}
where $d\mu=d\mu_g$.
The H\"older inequality therefore implies
 \begin{eqnarray*}
\left(\int_M |\hat{s}u^\ell |^{n/2}d\mu\right)^{2/n}
\left(\int_Md\mu\right)^{\frac{n-2}{n}}
 &\geq&  \int (-\hat{s}u^\ell ) d\mu\\
&\geq & -\int s ~d\mu\\
&=&|Y_{[g]}|
\left(\int_Md\mu\right)^{(n-2)/n}.
\end{eqnarray*}
Hence
$$
\int_M |s_{\hat{g}}|^{n/2}d\mu_{\hat{g}}=
\int_M |\hat{s}u^\ell |^{n/2}d\mu \geq |Y_{[g]}|^{n/2},
$$
with equality iff $u$ is constant.
The result follows.
\end{proof}

This leads to a very useful reinterpretation of $Y(M)$;
cf. \cite{and,bcg,okob,sch}.

\begin{prop}\label{func}
Let $M$ be a smooth compact $n$-manifold, $n \geq 3$.
Then
$$
\inf_g \int_M |s_g|^{n/2}d\mu_g = \left\{
\begin{array}{ll} 0&\mbox{if } Y(M)\geq 0\\
|Y(M)|^{n/2} &\mbox{if } Y(M)\leq 0.
\end{array}\right.
$$
Here the infimum on the left-hand side
is taken over all smooth Riemannian metrics
$g$
on $M$.
\end{prop}
\begin{proof}
By Lemma \ref{yobo},
$$
\inf_g \int_M |s_g|^{n/2}d\mu_g = \inf_{[g]} |Y_{[g]}|^{n/2}.$$
If $Y(M) \leq 0$, we have $Y_{[g]} \leq 0$ for all $[g]$,
so the right-hand-side may be rewritten as
$(-\sup Y_{[g]})^{n/2} = |Y(M)|^{n/2}$, and we are done.
On the other hand, any
  smooth manifold of dimension $\geq 3$ admits metrics
of negative scalar curvature \cite{bes}, and
$Y_{[g]}$  depends continuously
on $[g]$. Since    the space of metrics is
connected,  $Y(M) > 0$ implies there is a conformal
class $[g]$ on $M$ with $Y_{[g]}=0$, and the infimum
therefore vanishes.
\end{proof}

\section{Results from Seiberg-Witten Theory}

It is easy to see that
some
complex surfaces of K\"ahler type
carry metrics with positive scalar curvature.

\begin{prop} \label{rule}
Let $M$ be the underlying $4$-manifold
of a K\"ahler-type complex surface with
Kodaira dimension $-\infty$. Then
$Y(M) > 0$.
\end{prop}

Indeed, the Kodaira-Enriques classification \cite{bpv}
tells us that any such complex surface is either ${\Bbb CP}_2$ or else
is obtained from a ${\Bbb CP}_1$-bundle over a complex
curve by blowing up points. Now ${\Bbb CP}_2$ carries the
Fubini-Study metric, which has positive scalar curvature.
Any $S^2$-bundle over another manifold also carries metrics
of positive scalar curvature; namely, consider any Riemannian
submersion metric with round, totally geodesic fibers,
and then rescale the fibers to have very small diameter while
keeping
the metric on the base fixed. Finally, the blowing-up operation
amounts differentiably to taking connect sums with copies
of $\overline{{\Bbb CP}}_2$; all blow-ups of surfaces with
positive-scalar-curvature metrics therefore
themselves admit
positive-scalar-curvature metrics because a result of
Gromov and Lawson \cite{GL} tells us that the
class of manifolds with positive-scalar-curvature metrics
is closed under surgeries in codimension $\geq 3$.

On the other hand, it is a truly remarkable consequence of
Seiberg-Witten theory that the converse of the
above statement is also true:

\begin{thm} \label{sw}
Let $M$ be the underlying 4-manifold of a
complex surface of K\"ahler type with
Kodaira dimension $\geq 0$. Then $Y(M) \leq 0$.
\end{thm}

Remember,
this is just a fancy way of saying that
$M$ does not admit metrics of positive scalar curvature.
When $b^+(M) > 1$, this was first proved by
Witten \cite{witten}. In the $b^+=1$ case, this was
  proved for minimal surfaces in \cite{leb1}, and
then in the
non-minimal case by Friedman and Morgan \cite{FM}.

The  proof proceeds by first
showing that for every metric
$g$ on $M$, there must be a solution of the Seiberg-Witten
equations
\begin{eqnarray*} D^{\theta}\Phi &=&0 \\
 F_{\theta}^+&=&i \sigma(\Phi) \end{eqnarray*}
and an unknown unitary connection $\theta$
on some line bundle $L\to M$  and
 an unknown twisted spinor $\Phi\in \Gamma (S_+\otimes L^{1/2})$.
Here $D^\theta$ is the Dirac operator coupled to
$\theta$, $ F_{\theta}^+$ is the projection
of the curvature of $\theta$ to the
bundle $\Lambda^+$ of $g$-self-dual 2-forms,
and $\sigma : S_+\otimes L^{1/2}\to \Lambda^+$
is the real-quadratic map induced by
the natural
isomorphism $\odot^2{\bf S}_+=\Lambda^+\otimes {\bf C}$.
Moreover, $L$ can either be taken to be the anti-canonical line bundle
of $(M,J)$ or its pull-back via some
diffeomorphism $M\to M$.  However, these equations imply
the Weitzenb\"ock formula
$$0=4\nabla^{\theta\ast}\nabla^\theta + s\Phi + |\Phi|^2\Phi,
$$
where $s$ is the scalar curvature of $g$. Taking the
inner product with $\Phi$ and integrating, this tells
us
that
$$ 0 = \int_M \left[
|2\nabla^\theta\Phi|^2+ s |\Phi |^2 + |\Phi |^4
\right]
d\mu ,$$
which is a contradiction if $s > 0$.

Pushing this argument further yields a stronger
result \cite{leb3}
for surfaces of general type:

\begin{thm}\label{gtype}
Let $M$ be the underlying 4-manifold of a
complex surface with Kodaira dimension $2$. Then $Y(M) < 0$.
Moreover, if
$X$ is the minimal model of $M$, then
$Y(M)=Y(X)= -4\pi \sqrt{2c_1^2(X)}.$
\end{thm}

(Recall \cite{bpv}
that the minimal model $X$ is characterized by the
fact that $M$ can be  blown down to $X$, but that $X$ cannot be
blown down any further.
The minimal model $X$ of a surface $M$ of general type
 is unique, and  $c_1^2  (X) > 0$.)

Indeed, applying the
 Schwarz inequality to the integrated Weitzenb\"ock formula
  tells us that
$$(\int_M  s^2~d\mu )^{1/2} (\int_M
  |\Phi|^4 ~d\mu)^{1/2} \geq
 \int_M  (-s)|\Phi|^2~d\mu
\geq  \int_M  |\Phi|^4 ~d\mu .$$
We must therefore have
$$\int_Ms^2d\mu \geq \int_M  |\Phi|^4 ~d\mu =
8\int_M|F_{\theta}^+|^2d\mu$$
for any metric. However \cite{leb2},
$$ \int_M|F_{\theta}^+|^2d\mu \geq 4\pi^2([c_1(L)]^+)^2$$
where $[c_1(L)]^+$ is self-dual part of the
$g$-harmonic deRham representative of $c_1(L)$.
Moreover, for each $g$ one can show \cite{leb3} that
 $L$ can be chosen so that $([c_1(L)]^+)^2\geq c_1^2(X)$.
This shows that
$$ \int_M s^2 d\mu \geq 32\pi^2 c_1^2(X)$$
for all metrics on $M$. But  by gluing
together K\"ahler-Einstein orbifolds, gravitional instantons, and
standard metrics  on the blow-up of ${\bf C}^2$ at the
origin, one can also
construct \cite{leb3}
 sequences of metrics
which show that this bound is actually sharp:
$$\inf_g \int_M s^2 d\mu = 32\pi^2 c_1^2(X) >0.$$
Thus $Y(M) = -4\pi \sqrt{2c_1^2(X)} < 0$ by
Proposition \ref{func}.

\section{Rational Elliptic Surfaces Revisited}

A complex surface $(M^4,J)$ is said to be {\em elliptic} if it admits
a holomorphic map $M\to \Sigma$ to a complex curve such that
the regular fibers are elliptic curves $T^2$. While this notion
is of greatest importance in the study of surfaces of
Kodaira dimension $0$ and $1$, there are also elliptic
surfaces of Kodaira dimension $-\infty$. One family of
examples of the latter type are the rational elliptic surfaces
treated this section.

The smooth oriented 4-manifold ${\Bbb CP}_2\# 9 \overline{{\Bbb CP}}_2$
may be concretely realized as the blow-up of ${\Bbb CP}_2$
at $9$ points. One  choice of these $9$ points is
to take them to be the points of intersection of a generic pair of
elliptic (i.e. cubic) curves in the projective plane.
There is then exactly  a ${\Bbb CP}_1$'s worth of cubics passing through the
$9$ points, corresponding to linear combinations of the
 defining equations of the original pair of curves.
When  ${\Bbb CP}_2$ is blown up at these  $9$   points,
this pencil of cubics  gives rise to a holomorphic projection
 ${\Bbb CP}_2\# 9 \overline{{\Bbb CP}}_2\to {\Bbb CP}_1$, the generic fiber
of which is a smooth elliptic curve $E\approx T^2$.
The term {\em rational elliptic surface} is  thus
used to describe
${\Bbb CP}_2\# 9 \overline{{\Bbb CP}}_2$ equipped
 with  the deformation class of this projection to
${\Bbb CP}_1$.

There is a  useful way of constructing special
rational elliptic surfaces by analogy to
the Kummer construction of K3's.
Let $\Lambda \cong {\bf Z}\oplus {\bf Z}$
be any lattice in $\bf C$, and let
$E={\bf C}/\Lambda$ be the corresponding elliptic curve.
Recall that
each such curve can be realized as a branched cover over
${\Bbb CP}_1$ because reflection $z\mapsto -z$
through the origin in $\bf C$ induces in an involution
of $E$ with $4$ fixed-points\footnote{corresponding the
half-lattice $\frac{1}{2}\Lambda$}; the orbifold quotient
$E/{\bf Z}_2$ can then be identified with
a Riemann surface of genus $0$, and this
realizes $E$ as a branched cover
$\wp : E\to \cp_1$. The involution of $E$ that
occurs in this argument is called the {\em Weierstrass
involution} of $E$, and will be denoted here by
$\Phi : E\to E$. Notice that $\Phi$ is an isometry
of the obvious flat metric on $E$.

Now
let $\Psi : {\Bbb CP}_1 \to {\Bbb CP}_1$ be the involution
$[z_0:z_1]\mapsto [-z_0: z_1]$, which has two fixed
points and amounts to a $180^\circ$
rotation of $S^2$ around an axis. The product
involution
$$\Phi \times \Psi : E\times {\Bbb CP}_1 \to E\times {\Bbb CP}_1$$
then
has 8 fixed points, and
the quotient $E\times {\Bbb CP}_1/{\bf Z}_2$ is
a complex orbifold with 8 singularities
modeled on ${\bf C}^2/\pm 1$. Let $\tilde{M}$ be
the blow-up of
of $E\times {\Bbb CP}_1$ at the 8 fixed points of
$\Phi \times \Psi$, and let $M = \tilde{M}/{\bf Z}_2$.
The non-singular complex surface $M$ is thus
obtained  from $E\times {\Bbb CP}_1/{\bf Z}_2$
by replacing each of the $8$ singular points with a
$(-2)$-curve.

Now consider the  holomorphic map
$\varpi : M \to {\Bbb CP}_1$
induced by following the
second-factor projection $E\times {\Bbb CP}_1\to {\Bbb CP}_1$
with $[z_0: z_1] \mapsto [z_0^2 : z_1^2]$. All
but two fibers of $\varpi$ are copies of $E$;
the only two exceptions are the fibers over the
north and south poles $[0:1]$ and $[1:0]$. Each of these
two exceptional fibers  consists of five $(-2)$-curves,
linked together according to the Dynkin diagram
$\tilde{D}_4$:

\begin{center}
\begin{picture}(240,60)(0,3)
\put(100,30){\circle*{3}}
\put(80,50){\circle*{3}}
\put(80,10){\circle*{3}}
\put(120,50){\circle*{3}}
\put(120,10){\circle*{3}}
 \put(80,10){\line(1,1){40}}
\put(80,50){\line(1,-1){40}}
\end{picture}
\end{center}
These  two exceptional
fibers are of thus of Kodaira type $I_0^*(\tilde{D}_4)$.

Now
$\varpi : M \to \cp_1$ is deformation
equivalent to the above-described standard model of the
rational elliptic surface.
In fact, $M$ can be blown down to ${\Bbb CP}_1\times {\Bbb CP}_1$
in such a way that the fibers of $\varpi$ are sent to
the elliptic curves of the pencil
$$t_0x_0(x_0-x_1)y_0^2+ t_1x_1(x_0-ax_1)y_1^2=0,$$
where the number $a$ is given in terms of the lattice
$\Lambda$ by the Weierstrass  $\jmath$-function.
Notice that for each $[t]\in {\Bbb CP}_1$ this locus is
 an element
of the anti-canonical linear system $|{\cal O}(2,2)|$ on
${\Bbb CP}_1\times{\Bbb CP}_1$, and that each
elliptic curve of the pencil passes through the
points    $([0:1],[0:1],)$,
$([1:1],[0:1])$,
$([a:1],[1:0])$, and
$([1:0],[1:0])$, and all are
 tangent to the
  second-factor ${\Bbb CP}_1$ at these points.

To see this explicitly, let us first
consider $E\times {\Bbb CP}_1$ as a ruled surface
over $E$, and blow it up at the $8$
relevant points to obtain $\tilde{M}$:
\begin{center}
\begin{picture}(240,110)(0,3)
\put(50,110){\line(1,0){160}}
\put(50,110){\line(0,-1){60}}

\put(20,110){\line(0,-1){60}}
\put(45,80){\vector(-1,0){15}}
\put(-5,80){${\Bbb CP}_1$}

\put(75,110){\line(2,-3){13}}
\put(85,100){\line(0,-1){40}}
\put(75,50){\line(2,3){13}}

\put(105,110){\line(2,-3){13}}
\put(115,100){\line(0,-1){40}}
\put(105,50){\line(2,3){13}}
\put(100,100){\makebox(0,0){$-1$}}
\put(100,60){\makebox(0,0){$-1$}}
\put(123,80){\makebox(0,0){$-2$}}

\put(140,110){\line(2,-3){13}}
\put(150,100){\line(0,-1){40}}
\put(140,50){\line(2,3){13}}

\put(175,110){\line(2,-3){13}}
\put(185,100){\line(0,-1){40}}
\put(175,50){\line(2,3){13}}

\put(122,44){\makebox(0,0){$-4$}}
\put(122,116){\makebox(0,0){$-4$}}
\put(230,80){\makebox(0,0){$\tilde{M}$}}
\put(230,15){\makebox(0,0){$E$}}

\put(50,50){\line(1,0){160}}
\put(210,110){\line(0,-1){60}}
\put(135,40){\vector(0,-1){15}}
\put(50,15){\line(1,0){160}}

\put(75,15){\circle*{3}}
\put(105,15){\circle*{3}}
\put(140,15){\circle*{3}}
\put(175,15){\circle*{3}}
\end{picture}
\end{center}
Dividing by $\Phi\times\Psi$ then gives us a
non-minimal rational ruled surface:
\begin{center}
\begin{picture}(240,110)(0,3)
\put(50,110){\line(1,0){160}}
\put(50,110){\line(0,-1){60}}

\put(20,110){\line(0,-1){60}}
\put(45,80){\vector(-1,0){15}}
\put(35,70){$\varpi$}
\put(-5,80){${\Bbb CP}_1$}

\put(75,110){\line(2,-3){13}}
\put(85,100){\line(0,-1){40}}
\put(75,50){\line(2,3){13}}

\put(105,110){\line(2,-3){13}}
\put(115,100){\line(0,-1){40}}
\put(105,50){\line(2,3){13}}
\put(100,100){\makebox(0,0){$-2$}}
\put(100,60){\makebox(0,0){$-2$}}
\put(123,80){\makebox(0,0){$-1$}}

\put(140,110){\line(2,-3){13}}
\put(150,100){\line(0,-1){40}}
\put(140,50){\line(2,3){13}}

\put(175,110){\line(2,-3){13}}
\put(185,100){\line(0,-1){40}}
\put(175,50){\line(2,3){13}}

\put(122,44){\makebox(0,0){$-2$}}
\put(122,116){\makebox(0,0){$-2$}}
\put(230,80){\makebox(0,0){${M}$}}
\put(230,15){\makebox(0,0){${\Bbb CP}_1$}}

\put(50,50){\line(1,0){160}}
\put(210,110){\line(0,-1){60}}
\put(135,40){\vector(0,-1){15}}
\put(50,15){\line(1,0){160}}

\put(75,15){\circle*{3}}
\put(105,15){\circle*{3}}
\put(140,15){\circle*{3}}
\put(175,15){\circle*{3}}
\end{picture}
\end{center}
(Note that
$\varpi$, which is depicted horizontally in the
above picture, is {\em not}
the ruled-surface projection, which is
here
depicted vertically.)
Contracting  $8$ exceptional  curves in a judicious order,

\begin{center}
\begin{picture}(240,110)(0,3)
\put(50,110){\line(1,0){160}}
\put(50,110){\line(0,-1){60}}

\put(80,110){\line(0,-1){60}}
\put(110,110){\line(0,-1){60}}
\put(145,110){\line(0,-1){60}}
\put(180,110){\line(0,-1){60}}

\put(80,50){\circle*{3}}
\put(145,50){\circle*{3}}

\put(80,54){\circle*{3}}
\put(145,54){\circle*{3}}

\put(110,110){\circle*{3}}
\put(180,110){\circle*{3}}

\put(110,106){\circle*{3}}
\put(180,106){\circle*{3}}

\put(122,44){\makebox(0,0){$0$}}
\put(122,116){\makebox(0,0){$0$}}
\put(240,80){\makebox(0,0){${\Bbb CP}_1 \times {\Bbb CP}_1$}}
\put(230,15){\makebox(0,0){${\Bbb CP}_1$}}

\put(50,50){\line(1,0){160}}
\put(210,110){\line(0,-1){60}}
\put(135,40){\vector(0,-1){15}}
\put(50,15){\line(1,0){160}}

\put(151,80){\makebox(0,0){$0$}}

\put(80,15){\circle*{3}}
\put(110,15){\circle*{3}}
\put(145,15){\circle*{3}}
\put(180,15){\circle*{3}}
\end{picture}
\end{center}
we get a rational ruled surface which can be
recognized as
$\cp_1\times\cp_1$ by inspecting self-intersections
of holomorphic curves. Our   pencil now becomes
the sub-family of the linear system $|{\cal O}(2,2)|$
consisting of curves which are tangent to the second factor at
$4$   points in the claimed special position.
It follows that $\varpi : M \to {\Bbb CP}_1$ is
deformation equivalent to the
standard model of the rational elliptic surface,
since we may first deform our pencil into a generic
sub-pencil of $|{\cal O}(2,2)|$, and then
map ${\Bbb CP}_1\times{\Bbb CP}_1$ birationally
to ${\Bbb CP}_2$ in the usual way, using one
of the
base points of the pencil as our center.

This leads to the following:

\begin{prop} \label{eh}
Let $V$ be the (flat) 4-orbifold obtained from
${\bf R}\times T^3$ by dividing by the
involution induced by
$-1:  {\bf R}^4\to {\bf R}^4$,
let $B=({\bf R}\times S^1)/{\bf Z}_2$ denote the analogous
2-orbifold, and let $\pi : V\to B$ be the
map induced by projection ${\bf R}\times T^3\to {\bf R}\times S^1$
to the first two coordinates. Then
a smooth model of the  rational elliptic
surface ${\Bbb CP}_2\# 9\overline{{\Bbb CP}}_2 \to {\Bbb CP}_1$
may be obtained from $\pi : V\to B$ by
replacing each of the $8$ singular point of $V$ with a
2-sphere  of self-intersection $-2$, forgetting the
  orbifold structure on $B$, and adding
a smooth fiber at infinity in   the
 obvious manner.
\end{prop}
 \begin{proof}
We have already seen that the rational elliptic
surface can be viewed as a desingularization
of $(S^2\times T^2)/{\bf Z}_2$. Now let
$x\in S^2$ be some point which is not fixed by
the involution, and let $x^\prime\neq x$
be its image under the involution.
Identify $S^2-\{x,x^\prime\}$ with
the cylinder ${\bf R}\times S^1$ in such a
manner that the involution becomes
simultaneous reflection in both factors. The result follows.
\end{proof}

The orbifold $V$ comes equipped with a family of
flat orbifold metrics. Replacing the
singularities with gravitational instantons yields
the main result of this section:

\begin{prop} Let $\tilde{V}$ denote the complement of
a generic fiber in the rational elliptic
surface ${\Bbb CP}_2\# 9 \overline{{\Bbb CP}}_2\to {\Bbb CP}_1$,
let $\pi: \tilde{V}\to {\bf C}$ be the induced elliptic
fibration, normalized so that all critical
values are contained in open unit disk    $\Delta\subset {\bf C}$.
 Let $f$ be any given flat  metric on
the 2-torus $T^2$.
 Then there is a family $g_t$, $t\in [1,\infty )$
of Riemannian metrics
 on $\tilde{V}$
such that
\begin{itemize}
\item the Ricci curvature of $g_t$ converges uniformly to $0$ as
$t\to\infty$;
\item $(\pi^{-1} ({\bf C} - \Delta),g_t)$
is isometric to $([0,\infty )\times S^1, dx^2+d\theta^2)
\times (T^2, f/t)$; and
\item $\lim_{j\to \infty} \mbox{ Vol } (\pi^{-1}(\Delta), g_t ) = 0$.
\end{itemize}
\end{prop}
\begin{proof}
The basic idea is to replace the singularities of
of $V$ with  Eguchi-Hanson metrics.
The {\em Eguchi-Hanson metric} \cite{eh,leb0} is a complete Ricci-flat
 metric on the ${\cal O}(-2)$ line bundle over ${\Bbb CP}_1$,
or in other words on the blow-up of ${\bf C}^2/(\pm 1)$
at the origin.
Essentially by introducing polar coordinates on
 $({\bf R}^4-0)/(\pm 1)$, we may think of this as
the completion of the metric on
 $(1,\infty)\times SO(3)$ given by
$$g_{EH}=\frac{dr^2}{1-\frac{1}{r^4}}+ r^2\left( \sigma_1^2+\sigma_2^2+
(1-\frac{1}{r^4})\sigma_3^2\right)$$
where the left-invariant co-frame $\{ \sigma_j\}$ is
orthonormal for the curvature $+1$ bi-invariant metric.
By rescaling and homothety, we can turn this into
a 1-parameter family
$$\frac{dr^2}{1-\frac{A}{r^4}}+ r^2\left( \sigma_1^2+\sigma_2^2+
(1-\frac{A}{r^4})\sigma_3^2\right), ~~ r > \sqrt[4]{A}$$
of Ricci-flat metrics which bubble off to
the flat orbifold metric on ${\bf R}^4/(\pm 1)$
as $A\to 0$.

Let us now modify the Eguchi-Hanson metric. Let
$\phi (t)$ be a smooth monotonely non-increasing function
which is identically $1$ on
$[0,1]$ and  identically $0$ on $[2,\infty ]$.
For small values of the parameter $A$,
the modified metric
$$\frac{dr^2}{1-\phi(r)\frac{A}{r^4}}+ r^2\left( \sigma_1^2+\sigma_2^2+
(1-\phi(r)\frac{A}{r^4})\sigma_3^2\right), ~~r > \sqrt[4]{A}$$
then agrees with the Euclidean metric for
$r > 2$, is Ricci-flat when $r < 1$, and has
Ricci curvature $< CA$ for some constant $C$ independent
of the small parameter $A$. Making
another homothety and rescaling, the metric
$$\frac{dr^2}{1-\phi(r/\varepsilon)\frac{A\varepsilon^4}{r^4}}+
r^2\left( \sigma_1^2+\sigma_2^2+
(1-\phi(r/\varepsilon)\frac{A\varepsilon^4}{r^4})\sigma_3^2\right),
~~r > \varepsilon \sqrt[4]{A}$$
is Euclidean for $r > 2\varepsilon$, Ricci-flat for $r < \varepsilon$,
and has Ricci curvature $< CA/\varepsilon^2$. Thus the metrics
$$g_\varepsilon=\frac{dr^2}{1-\phi(r/\varepsilon)
\frac{\varepsilon^8}{r^4}}+
r^2\left( \sigma_1^2+\sigma_2^2+
(1-\phi(r/\varepsilon)\frac{\varepsilon^8}{r^4})\sigma_3^2\right),
~~ r> \varepsilon^2$$
 have    Ricci curvature uniformly $O(\varepsilon^2)$
as $\varepsilon \searrow 0$.
Notice that the volume form of $g_\varepsilon$ coincides with the
Euclidean volume form
$r^3dr\wedge\sigma_1\wedge\sigma_2\wedge\sigma_3$,
but that the appropriate domain of integration is
the   $r > \varepsilon^2$
rather than $r > 0$. Thus,
  smoothing a flat orbifold   singularity modeled
on ${\bf R}^4/(\pm 1)$ by gluing in the
modified Eguchi-Hanson  metric $g_\varepsilon$
 reduces the  total volume, namely by $\pi^2\varepsilon^8/2$.

Now let $f$ be any fixed flat metric
on the 2-torus, and let $\imath$ denote the injectivity
radius of $(T^2,f)$, and
set $\hat{\imath}=\min (\imath , \pi)$.
 Endow ${\bf R}\times T^3$ with the sequence
of  flat metrics
$\hat{g_t}=dx^2+d\theta^2+ f/t$, and push these
down as flat orbifold metrics on $V=({\bf R}\times T^3)/{\bf Z}_2$.
For each $t$, let $\varepsilon_t = \hat{\imath}/4\sqrt{t}$,
so that the balls of radius $2\varepsilon_t$ centered at the
orbifold singularities of $V$ are pairwise disjoint.
On the blow-up $\tilde{V}$ of $V$, we can then define a
metric $g_t$ as the $\hat{g_t}$ on the complement of
these balls, and the modified Eguchi-Hanson metrics
$g_{\varepsilon_t}$ on the blown-up interior of these balls .
Since $\varepsilon_t\to 0$, the Ricci curvature of $g_t$
tends uniformly to zero. Moreover,  the volume of
the region $|x| \leq a$ in $(X,g_t)$ is less than
$2\pi a \alpha/t \to 0$, where $\alpha$ is the area of
$(T^2,f)$, so the volume of the pre-image of any
compact set in $\bf C$ tends to $0$ as $t\to\infty$.
\end{proof}

\section{Collapsing  Elliptic Surfaces}

\begin{defn}
Let $M$ be a smooth manifold. We will say that $M$
{\em collapses with bounded scalar curvature}
if there is a sequence $g_j$ of smooth metrics
on $M$ for which the absolute value
of the scalar curvatures is uniformly bounded,
but with total volume tending to zero:
$$|s_{g_j}| < {\cal B}, ~~\lim_{j\to \infty}\mbox{Vol}(M,g_j) =0.$$
Similarly, we say that $M$ collapses with
bounded Ricci (respectively,   sectional)
curvature  if there is  a sequence $g_j$ of smooth metrics
on $M$ with
  uniformly  bounded Ricci (respectively, sectional)
curvature and total volume tending to zero.
\end{defn}

This collection of definitions is loosely inspired by the
work of Cheeger and Gromov \cite{cg}, who studied
sequences of metrics $g_j$ with bounded sectional curvature
and {\em injectivity radius} tending to zero at all points.
A standard comparison argument for the volume of
small balls shows that
 a sequence of metrics with sectional
curvature bounded above can have volume tending with to zero
only if  the point-wise injectivity radii
uniformly tend to zero, too.
Note, however, that a   sequence
of metrics $g_j$ with bounded sectional curvature
and injectivity radius tending to zero may have volume
bounded away from zero --- as  even happens for
flat  metrics on the 2-torus.
Nonetheless, a remarkable result of Cheeger and Gromov \cite{cg2}
asserts
that if a compact manifold admits
 a sequence of metrics $g_j$
with bounded sectional curvature and injectivity radius
uniformly tending to zero,
 there must be other sequences
$\tilde{g}_j$ with bounded sectional curvature for which the
volume tends to zero as well.  Thus the
present definition of collapse with bounded sectional
curvature
is in accord with the definitions used by others.

Collapse with bounded scalar curvature
is directly relevant to the computation
of Yamabe invariants  by virtue of the
following result:

\begin{prop} \label{sam}
Let $M$ be a smooth compact $n$-manifold, $n\geq 3$. Then
the following are equivalent:
\begin{description}
\item{(i)} $M$ collapses with bounded scalar curvature;
\item{(ii)} $\inf_{g}\int_M |s_g|^{n/2}d\mu_g =0$;
\item{(iii)} $Y(M)\geq 0$.
\end{description}
\end{prop}
\begin{proof}
The implication $(i)\Rightarrow (ii)$ is trivial, since
$\int_M |s_{g_j}|^{n/2}d\mu_{g_j}\to 0$ for any sequence of
metrics with bounded scalar curvature and volume tending to
zero. The implication $(ii)\Rightarrow (iii)$ follows
from Proposition \ref{func}. Finally,
the  implication $(iii)\Rightarrow (i)$
follows from the fact \cite{aubin} that any $n$-manifold, $n \geq 3$,
admits of metrics of constant
negative scalar curvature; if $Y(M)\geq 0$,
 there is a thus a sequence of unit-volume metrics with
constant negative scalar curvature tending to $0$ from below,
and rescaling these metrics so they have $s\equiv -1$ then
makes the volumes of the rescaled metrics tend to zero.
\end{proof}

If $M$ is the underlying 4-manifold of a
complex  elliptic surface, these various notions of
collapse can be completely understood.

\begin{thm}\label{scol}
Let $M$ be the underlying 4-manifold of a
complex elliptic surface. Then $M$ collapses with bounded
scalar curvature.
\end{thm}

\begin{thm}\label{rcol}
Let $M$ be the underlying 4-manifold of a
complex elliptic surface. Then $M$ collapses with bounded
Ricci curvature $\Leftrightarrow$
$M$ is minimal.
\end{thm}

\begin{thm}\label{kcol}
Let $M$ be the underlying 4-manifold of a
complex elliptic surface. Then $M$ collapses with bounded
sectional curvature $\Leftrightarrow$
$\chi(M)=0$.
\end{thm}

In order to prove these results, we will
need special diffeomorphic models \cite{FM,mats} of elliptic
surfaces. These are built up in stages, starting with
surfaces we shall call {\em twisted products}.
Let $E$ be an elliptic curve, equipped with a compatible flat
metric, let $o\in E$ be a chosen base-point,
and let $G$ be the finite group of
orientation-preserving isometries $E\to E$ which
fix $o$. (Thus $G$ is a cyclic group of order
$2$, $4$, or $6$.) A twisted product surface
is simply the total space $B$ of a fiber-bundle $B\to \Sigma$ over a
compact complex curve, with fiber $E$ and
structure group $G$.

 Now let $B\to \Sigma$ be such a twisted product,
and let a finite collection of points $x_j\in \Sigma$
be specified, and assign each of these points
 some integer multiplicity
$m_j > 1$.  Give
$\Sigma$ an obifold structure by introducinging orbifold
local coordinates  of the form $\hat{z}={z}^{1/m_j}$
near $x_j$, where
$z$ is a local complex coordinate on $\Sigma$   with $z(x_j)=0$.
A process called  the {\em logarithmic transform}
now allows us to  delete the fiber $E_j\cong E$ over each $x_j$,
and replace
it with $E_j/{\bf Z}_{m_j}$, where the ${\bf Z}_{m_j}$-action
is generated by any specified translation
$\tau_j : E_j\to E_j$ of order $m_j$. Namely, let
$\tau_j(t)$ be a 1-complex-parameter group of translations
of $E_{x_j}$
with $\tau_j (1)=\tau_j$, let $\Delta\subset {\bf C}$ be a  disk centered
at $0$ and contained in the range of the orbifold coordinate
$\hat{z}$, and glue $(E_j\times \Delta)/\langle (\tau_j, e^{2\pi i/m_j})\rangle$
to $B-E_{x_j}$ by
$[(y,\tilde{z})]\leftrightarrow (\tau_j (-m_j\log (\hat{z})/2\pi i )(y),
z:=\hat{z}^{m_j})$.

With these constructions in mind, we can now
prove

\begin{lem} Let $M$ be the underlying 4-manifold of any
 complex elliptic surface  with Euler characteristic
$0$. Then there is a  compact $2$-orbifold   $\Sigma$
and a
smooth submersive map $\pi : M \to \Sigma$,
such that for every orbifold metric $h$ on $\Sigma$
there is
a Riemannian metric $g$ on $M$ for which
$\pi$ becomes a Riemannian submersion
with  flat, totally geodesic
fibers. Moreover, $\pi$ can be chosen so
that its fibers over some open disk in $\Sigma$ are also
regular fibers of the original elliptic fibration of $M$.
\end{lem}
\begin{proof} Any complex elliptic surface
with Euler characteristic zero is
$C^\infty$-deformation equivalent to some logarithmic  transform of
a twisted-product surface  \cite[Proposition 7.2]{FM}.
But this model is an orbi-bundle over the $2$-orbifold $\Sigma$
with structure group equal to the isometry group of
a flat $2$-torus $E$. Starting from any
orbifold metric $h$ on $\Sigma$, now construct a metric
$g$ on $M$ by gluing together local product metrics
with a  partition of unity subordinate to a trivializing
cover of $\Sigma$. Each fiber of $\pi: M\to \Sigma$
is then a flat quotient of $E$, and each fiber has vanishing
second fundamental form because  the orthogonal space of
the fibers is the horizontal subspace of a connection
with structure group equal to the isometry group of the fiber.
\end{proof}

The ideas of Cheeger and Gromov \cite{cg}
now immediately lead to a

\bigskip

\noindent {\bf Proof of Theorem \ref{kcol}.}
Consider the metrics
$$g_t = \frac{1}{t}g+(1-\frac{1}{t})\pi^*h$$
on $M$ as $t\to\infty$. We have
$$\mbox{Vol}(M,g_t)= \frac{1}{t}\mbox{Vol}(M,g)\to 0,$$
so it suffices to show that the components of
the curvature tensor remain bounded in an orthonormal
frame as $t\to\infty$.  But this follows from O'Neill's
 Riemannian submersion formulas \cite{bes}. Indeed, the vertical
sectional curvature is $0$ for all $t$, and the
horizontal sectional curvature is given by
$$K(H)=K(\Sigma) - \frac{3}{4}g_t(v,v)\to K(\Sigma),$$
while the sectional curvature of the 2-plane $P$
spanned by a vertical vector and
a horizontal vector is
$$K(P)= \frac{1}{4}g_t(v^{\|},v^{\|}) \to 0.$$
Here $K(\Sigma)$ is
the Gauss curvature of $(\Sigma,h)$, and
$v$ is the ($t$-independent) vertical component of $[w_1,w_2]$,
where $w_1$ and $w_2$ are horizontal vectors which
project to an oriented orthonormal frame on $\Sigma$,
whereas $v^{\|}$ is the
orthogonal projection of $v$ into $P$.
This shows that the point-wise norm of the curvature tensor,
 and hence all sectional curvatures,
 remain uniformly bounded
as $t\to \infty$. Any elliptic surface $M$
with $\chi (M)=0$ therefore collapses
with bounded sectional curvature.

Conversely, if $M$  collapses with bounded sectional
curvature,
the  Gauss-Bonnet formula tells one that
$\chi (M) =0$.
\hfill \rule{.5em}{1em

The next operation we will need in order to
create smooth models of elliptic surfaces is the
{\em fiber sum}. The fiber sum of two elliptic
surfaces is the smooth manifold obtained by
removing a tubular neighborhood of a regular fiber in
each and then then identifying the boundaries
in a manner compatible with the orientations and
given local trivializations.
Any minimal complex elliptic surface  $M$ is
 diffeomorphic \cite[Corollary 2.17]{FM} to a fiber sum of
an elliptic surface  $\check{M}$
with Euler characteristic $0$
and copies of the rational elliptic surface.

\bigskip

\noindent {\bf Proof of Theorem \ref{rcol}.}
  Let us thus begin with some elliptic surface $\check{M}$
with $\chi (\check{M})=0$, and let $\pi : \check{M}\to \Sigma$
be a smooth submersion of the type used in the previous proof.
Let $x_1, \ldots , x_k\in \Sigma$ be the points we will use as the
centers of the fiber sum construction. Let $U_1, \ldots , U_k$
be disjoint disks around these points, chosen so that
$M\to \Sigma$ may be trivialized over their closures,
and identify
each with the   disk of radius $2$ about   $0\in {\bf C}$. Choose $h$ so
that its restriction to the annulus $|z|\in [ 1, 2]$
is isometric to the cylinder $S^1\times [1,2]$, and now
construct the family of metrics $g_t$ used above, with
the stipulation $g$ is actually taken to be a product metric on
$\pi^{-1}(U_1), \ldots , \pi^{-1} (U_k)$.
Letting $\tilde{U}_1\subset U_1, \ldots , \tilde{U}_k\subset U_k$
each correspond to
the unit sub-disk $|z| < 1$,
the Riemannian
manifold-with-boundary $(M-\cup \pi^{-1}(\tilde{U}_j),g_t)$ has ends
isometric to $([0,1]\times S^1, dt^2+ d\theta^2) \times (E, f/t)$,
where $f$ is a fixed flat metric on an elliptic curve $E$.

However, these cylindrical ends
 precisely match those of the metrics on $\tilde{V}$
constructed in Proposition \ref{eh}, and we can therefore
glue these   objects together to obtain metrics with
bounded Ricci curvature and volume tending to
zero on the fiber sum $M$ of $\check{M}$ and $k$
rational elliptic surfaces.  Since a diffeomorphic
model of any minimal elliptic surface can be
constructed in this way, it follows that  any minimal
elliptic surface collapses
with bounded Ricci curvature, as claimed.

Conversely, suppose
 an oriented 4-manifold $M$ admits
 a sequence of metrics $g_j$ with bounded Ricci
curvature $r$ and volume tending to zero. We then
have
$$\lim_{j\to\infty} \int_M \left[ \frac{s_{g_j}^2}{24}-
\frac{|\stackrel{\circ}{r}|^2_{g_j}}{2}\right]d\mu_{g_j}
=0,$$
where $\stackrel{\circ}{r}$ denotes the
trace-free part of the Ricci curvature.
But
$$(2\chi + 3\tau)(M)= \frac{1}{4\pi^2}\int_M \left[ |W_+|^2+
\frac{s_{g}^2}{24}-
\frac{|\stackrel{\circ}{r}|^2_{g}}{2}\right]d\mu_{g}
\geq \frac{1}{4\pi^2}\int_M \left[ \frac{s_{g}^2}{24}-
\frac{|\stackrel{\circ}{r}|^2_{g}}{2}\right]d\mu_{g}$$
for any metric on $M$. It therefore follows that
$(2\chi + 3\tau )(M) \geq 0$.
On the other hand, $2\chi + 3\tau = c_1^2 \leq 0$ if
$M$ is an elliptic surface, and equality occurs iff
$M$ is minimal \cite{bpv}. Hence an elliptic
surface $M$ must be
minimal if it collapses with bounded Ricci curvature.
\hfill \rule{.5em}{1em

Finally, any elliptic surface is obtained from
a minimal elliptic surface by {\em blowing up}
an appropriate number of times.
Recall that the blow-up of ${\bf C}^2$ at the
origin is by definition the line bundle ${\cal O}(-1)$
over ${\Bbb CP}_1$.  A point in an arbitrary complex surface can
similarly be
obtained by replacing a small ball with a
tubular neighborhood of
the zero section in this bundle. This procedure
is then iterated as needed to produce the desired
surface.
\bigskip

\noindent {\bf Proof of Theorem \ref{scol}.}
The idea is to graft Burns metrics  onto the previous
examples. The {\em Burns metric} \cite{leb0} is a scalar-flat
K\"ahler metric on
the blow-up of ${\bf C}^2$ at the origin.
By introducing  polar coordinates, this can be viewed
as the completion of the metric on $(1,\infty)\times S^3$
given by
$$g_{\mbox{\tiny
Burns}}= \frac{dr^2}{1-\frac{1}{r^2}} + r^2\left(\sigma_1^2+
\sigma_2^2+ (1-\frac{1}{r^2})\sigma_3^2\right) ,$$
where $\{\sigma_j\}$ is an $SU(2)$-invariant orthonormal
frame on $S^3$. By repeating the sequence of homotheties,
rescalings, and cut-offs used in the proof of
Proposition \ref{eh}, the metric
$$g_\epsilon =
\frac{dr^2}{1-\phi(\frac{r}{\epsilon})\frac{\epsilon^6}{r^2}}
+ r^2\left(\sigma_1^2+
\sigma_2^2+ (1-\phi(\frac{r}{\epsilon})\frac{\epsilon^6}{r^2}
)\sigma_3^2\right),~~
r > \epsilon^3
$$
is flat for $r > 2\epsilon$, scalar-flat for $r< \epsilon$,
and has scalar curvature    uniformly $O(\epsilon^2)$
as $\epsilon \searrow 0$. Moreover, blowing up a
flat region by replacing a small ball by a
copy of $g_\epsilon$ reduces its total volume, namely
by $\pi^2\epsilon^{12}/2$.

Any elliptic surface is an iterated blow-up of a minimal
one. Up to diffeomorphism, we can therefore
model any such surface by blowing up one of
our previous models at distinct points. But the
constructed metrics on the previous models
can   be taken to be flat in the neighborhood
of certain fibers
provided the orbifold metric
$h$ on $\Sigma$ is also taken to be flat on certain regions.
So let $\check{g}_j$ be a sequence of metrics with
bounded Ricci curvature   and Vol $\to 0$,
each containing a Euclidean ball of radius $\varrho_j$,
on a minimal elliptic surface $\check{M}$.
On any
blow-up $M=\check{M}\# \ell \overline{{\Bbb CP}}_2$,
 we can then
produce a sequence of metrics $g_j$,
with bounded scalar curvature and Vol $\to 0$
by replacing $\ell$ disjoint, Euclidean $(2{\epsilon_j})$-balls with
copies of $g_{\epsilon_j}$, where we may, for example,
take $\epsilon_j=\varrho_j/2\ell$.
\hfill \rule{.5em}{1em

By Proposition \ref{sam}, this immediately implies

\begin{cor}
Any complex elliptic surface $M$ has $Y(M) \geq 0$.
\end{cor}

The reader may prefer to deduce this corollary
directly from  Theorem
\ref{rcol} and the
fact that $Y(\overline{{\Bbb CP}}_2) > 0$ by
citing a general result of \cite{okob}.
At heart, however, this is much the
same proof as is given above.

Combining this corollary with Theorem \ref{sw} now yields

\begin{thm} \label{y0}
Let $M$ be a K\"ahler-type complex    surface
with $\mbox{Kod} (M)=0$ or  $1$. Then    $Y(M)=0$.
\end{thm}
\begin{proof}
Any surface of Kodaira dimension $0$ or $1$ is
deformation-equivalent to an elliptic surface \cite{bpv,FM},
and so has $Y(M)\geq 0$ by the previous result.
But if $b_1(M)=0\bmod 2$, $M$ is of K\"ahler type,
and   Seiberg-Witten theory tells us that
there is no metric of positive scalar curvature
on $M$, so that $Y(M)\leq 0$. Hence $Y(M)=0$, as claimed.
\end{proof}

Combining
this  with Proposition
\ref{rule} and Theorem \ref{gtype}
now proves Theorem \ref{main}.

\section{Concluding Remarks}

Theorem \ref{main} does not allow one to distinguish
between Kodaira dimensions $0$ and $1$ by
comparing Yamabe
invariants. Such a  distinction arises
immediately, however, if one asks whether
$$Y(M)=\sup_{[g]} Y_{[g]}$$
is actually realized by the Yamabe constant
 of some conformal class.

\begin{prop}
Let $M$ be the underlying 4-manifold of
a complex algebraic surface of Kodaira dimension
$\geq 0$. Then $M$ admits a Riemannian metric
of  scalar curvature zero iff $M$ is minimal
and has Kodaira dimension $0$. In particular,
the Yamabe invariant $Y(M)$ is unachieved
whenever $\mbox{Kod} (M) =1$.
\end{prop}

\begin{proof} Any minimal K\"ahler-type
complex surface with
Kodaira
dimension $0$ has $c_1=0$ in
real cohomology, and so carries a
Ricci-flat K\"ahler metric by
Yau's solution \cite{yau} of the Calabi
conjecture. Such a metric has
$s=0$, and so realizes $Y(M)=0$.

 Conversely \cite{bes}, if a manifold
with $Y(M)=0$ admits a scalar-flat metric,
the metric in question must be Ricci-flat,
since otherwise it could be deformed into
a metric of positive scalar curvature  \cite{bes}.
In dimension 4, this implies that
$$2\chi + 3\tau =\frac{1}{4\pi^2}\int_M\left[2|W_+|^2+\frac{s^2}{24}
-\frac{|\stackrel{\circ}{r}|^2}{2}\right]d\mu\geq 0.$$
Our complex surface must therefore satisfy $c_1^2\geq 0$.
If $c_1$ is not a torsion class, the Seiberg-Witten
invariant of $(M,[J])$ is therefore both  metric-independent
and non-zero, even if $b^+(M)=1$.
But the Seiberg-Witten
estimate \cite{leb2}
$$\int_Ms^2d\mu \geq 32\pi^2 (c_1^+)^2$$
then forces $c_1$ to be anti-self-dual with
respect to any scalar-flat metric; thus
$c_1^2\leq 0$, with equality iff $c_1=0$
in real cohomology. Hence $c_1$ is a torsion class,
and our surface must be a minimal surface of
Kodaira dimension $0$.
\end{proof}

While this makes a satisfactory distinction between
{\em minimal} surfaces of Kodaira dimensions $0$ and $1$,
it is still unsatisfactory for non-minimal surfaces.
However, it is not hard to see that there are
sequences of unit-volume constant-scalar-curvature
metrics on any blow-up of, say,  a K3 surface
which bubble off to any given K\"ahler-Einstein
metric on the minimal model. The metrics in this
sequence have uniformly bounded diameter.
It seems plausible to conjecture that this does
not happen in the case of  Kodaira dimension $1$:

\begin{conj}
Let $M$ be the underlying 4-manifold of a
complex algebraic surface of Kodaira dimension
$\geq 0$. Suppose that there is a sequence
of unit-volume constant-scalar-curvature
metrics on $M$ with uniformly bounded diameter and
$s\nearrow 0$. Then $\mbox{Kod} (M) =0$.
\end{conj}

Of course,
beyond simply knowing the sign of $Y(M)$, one would
like to ask to know its actual value. When $b_1(M)$ is even and
$\mbox{Kod} (M) \geq 0$, this  question is completely answered by
Theorems \ref{gtype} and \ref{y0}. For
algebraic surfaces with  $\mbox{Kod} (M) =-\infty$,
however, the only case in which $Y(M)$ is actually
known  is for $M={\Bbb CP}_2$, where $Y(M)$ is actually realized
by the Fubini-Study metric \cite{leb4}. It is, in particular,
unclear whether blowing up leaves $Y(M)$ unchanged when $\mbox{Kod} (M)
=-\infty$, although this is what happens when the
Kodaira dimension is larger.

There are a number of places in this
paper where we have assumed that the surface in question
has $b_1$ even in order to assure that there is
a non-trivial Seiberg-Witten invariant. However,
this is superfluous in Kodaira dimension $0$, since
Kodaira surfaces are symplectic and so have non-trivial
invariants by the work of Taubes \cite{taubes}.
For Kodaira dimension $1$ the situation is less clear,
but Biquard \cite{biq} has also shown that some non-K\"ahler surfaces
in this class have non-trivial Seiberg-Witten invariants.
Thus it would be surprising if the following
were not true:

\begin{conj}
Let $M$ be the underlying 4-manifold of a compact
complex manifold of Kodaira dimension $0$ or $1$.
Then, even if $b_1(M)$ is odd, one still has $Y(M)=0$.
\end{conj}

Even allowing for the case of $b_1$ odd, the following
is an immediate consequence of the results proven here:

\begin{thm}
Let $(M,J)$ be any compact
complex surface. If $Y(M)  < 0$, then
$M$ is either  of general type or   of type $VII$.
\end{thm}

For all known surfaces of type $VII$,
however, one can check by hand that
$Y \geq 0$. It therefore seems reasonable to make the following

\begin{conj}
Let $(M,J)$ be any compact
complex surface. If $Y(M)  < 0$, then
$M$ is    of general type.
\end{conj}

Finally, we have accidentally computed the infima of some
other Riemannian functionals by showing that minimal elliptic
surfaces collapse with bounded Ricci curvature.

\begin{prop}
Let $M$ be the underlying 4-manifold of any
complex elliptic surface.
Then
$$\inf_{[g]} \int_M |W_+|^2d\mu =0.$$
Here $W_+$ denotes the self-dual Weyl
tensor, and the infimum is over the set of all
conformal classes of Riemannian metrics on $M$.
As a consequence
$$\inf_{[g]} \int_M |W_-|^2d\mu = -12\pi^2\tau (M).$$
\end{prop}

Indeed, if $M$ is minimal, this follows from
Theorem \ref{rcol} and the Gauss-Bonnet formula
for
$(2\chi + 3\tau )(M) = 0$. Since the Burns metric
is anti-self-dual, it is easy to extend this
to the non-minimal case; cf. \cite{taubes2}.

By Taubes existence theorem \cite{taubes2} for anti-self-dual
metrics, any complex surface, after being blown up
sufficiently many times, admits metrics with
$W_+=0$. The last result raises, once again, the
fascinating question of determining
precisely how many times a given complex surface
must be blown up before this happens.

\vfill
\noindent
{\bf Acknowledgement.}
The author  warmly
thanks   Michael Anderson, Peter Kronheimer,
and Heberto del Rio
for their helpful comments and suggestions.


Twisted Baryon Number in N=2 Supersymmetric QCD

Abstract:
  We show that the baryon number of N=2 supersymmetric QCD can be twisted in
order to couple the topological field theory of non-abelian monopoles to
$Spin^c$-structures. To motivate the construction, we also consider some
aspects of the twisting procedure as a gauging of global currents in two and
four dimensions, in particular the role played by anomalies.

\section{Introduction}

One of the most impressive consequences of the work by Seiberg and Witten
\cite{swone, swtwo}  has been  the dual description of Donaldson theory in
terms of monopole equations \cite{mfm} (see  \cite{marcolli, morgan, gptres,
tribu, don} for a review). This has opened the way  to new developments  in
four-dimensional topology and has provided astonishing new links between this
field and  the dynamics of supersymmetric gauge theories in four dimensions.
One of these developments  is the non-abelian generalization of the abelian
Seiberg-Witten equations, known as  non-abelian monopoles, which have been
extensively studied both from the mathematical  \cite{okone,
teleman, pt, bradgp}  and the physical point of view \cite{lmna, lmpol, lmt,
tqcd} in the last two years.

It is well-known that the Seiberg-Witten monopole equations involve
${\hbox{\rm Spin}}^c$-structures on  a four-manifold \cite{sping}, and it was shown in
\cite{abelia} that these  naturally arise when  considering the duality
transformations of the underlying physical theory.  For the non-abelian case,
${\hbox{\rm Spin}}^c$-structures were considered in \cite{tqcd} in order to  define twisted
$N=2$ supersymmetric QCD on a general four-manifold. In this note, we will
propose a  different way to couple matter hypermultiplets to
${\hbox{\rm Spin}}^c$-structures in non-abelian  monopole theories. The idea is to consider
an extended twisting procedure by  gauging an additional global symmetry of
the  physical theory. The global symmetry turns out to be the baryon number
$U(1)_B$. We will  show in detail that the procedure is consistent and that
the quantum numbers associated to  this symmetry are the appropriate ones
from the geometrical point of view. To motivate  the construction, we will
review some aspects of the twisting as a gauging procedure.  A general
conclusion of our analysis is that global symmetries not necessarily
associated to the supersymmetry algebra could be considered in the
construction of  a topological field theory.

The organization of this paper is as follows: In section two we consider the
twisting as a gauging in Donaldson-Witten theory and  in two-dimensional
topological sigma models. In section three we consider the  gauging of the
$U(1)_B$ current in $N=2$ supersymmetric QCD and the coupling to
${\hbox{\rm Spin}}^c$-structures.  Finally, in section four we present our conclusions and
outlook.

\section{The twist as a gauging}
\setcounter{equation}{0}

Usually, the twisting procedure has been understood as a modification of the
rotation group of a supersymmetric $N=2$ theory through an embedding of the
global  symmetry associated to the supersymmetry algebra \cite{tqft, tsm, lla,
altwo}.  An alternative (and equivalent) point of view is obtained when the
twisting  procedure is regarded as a gauging of an internal symmetry group, in
which a  global symmetry of the underlying supersymmetric model is promoted to
a  space-time symmetry. In many cases, the gauging is performed by adding to
the Lagrangian of  the original theory a new term, involving the coupling of
the internal current  to the Spin connection of the underlying manifold
\cite{egu,ceco,cmr}.

We will now discuss this approach to the twisting procedure in some detail in
the  case of $N=2$ supersymmetric Yang-Mills, the model originally considered
by Witten in
\cite{tqft}. The field content of the minimal $N=2$ supersymmetric
Yang-Mills theory with gauge group $G$ on ${\bf R}^4$ is the following: a
gauge field $A_{\mu}$,   two Majorana spinors
$\lambda_{i\alpha}$, $i=1,2$, and their conjugates
$\overline\lambda^{i\dot\alpha }$, a complex scalar $B$, and an auxiliary
field $D_{ij}$ (symmetric in $i$ and $j$). The indices $i$, $j$ denote  the
isospin indices of the  internal symmetry group $SU(2)_I$ of $N=2$
supersymmetry. The two Majorana spinors  $\lambda_1$, $\lambda_2$ form a
doublet of $SU(2)_I$. All these fields are considered in the adjoint
representation of the gauge group $G$. The $SU(2)_I$ current of this model is
given by:
\begin{equation}
j^{\mu}_a={\overline \lambda}{\sigma}^{\mu}{\sigma_a}\lambda,
\label{corriente}
\end{equation}
where ${\sigma}^{\mu}=({\bf 1}, i\sigma_a)$, and $\sigma_a$ are the Pauli
matrices.

If we try to formulate the theory on a general four-manifold $X$ with
Euclidean signature, the Majorana spinors $\lambda_{i \alpha}$ and
their conjugates $\overline\lambda^{i\dot\alpha }$ are taken as
independent Weyl spinors with opposite chirality, and we use the
prescription  of minimal coupling to gravity in the Lagrangian. The structure
group of the  spinor bundle is,
\begin{equation}
{\hbox{\rm Spin}}_4 \simeq SU(2)_{L}\times SU(2)_{R},
\label{estr}
\end{equation}
 so that the covariant derivative
acting on positive chirality spinors is given by,
\begin{equation}
D_{\mu}M_{\alpha}=\nabla_{\mu}
M_{\alpha}-i\omega_{\mu}^a(\sigma_{a})_{\alpha}{}^{\beta}
M_{\beta},
\label{cova}
\end{equation}
where $\omega_{\mu}^a$ is the $SU(2)_{L}$ Spin connection on $X$, and
$\nabla_{\mu}$ is  the covariant derivative on flat space (including the gauge
connection). This construction  can always  be done  locally, but globally we
have of course the usual topological obstruction associated  to the second
Stiefel-Whitney class, $w_2(X)$. If $X$ is not Spin, we cannot consistently
couple the original $N=2$ theory to gravity.

The twist of the theory under consideration involves
the  gauging of the $SU(2)_I$ group as the space-time symmetry group
$SU(2)_{L}$. As the  only fields charged with respect to $SU(2)_I$ are
spinors, the gauging is achieved  after adding to the Lagrangian the coupling
of the $SU(2)_L$ connection to the  $SU(2)_I$ current:
\begin{equation}
-\omega_{\mu}^a j^{\mu}_a=
-{\overline \lambda}^{i}_{\dot \alpha}(\sigma^{\mu})^{\alpha
\dot \alpha}(\sigma_a)_i{}^j\omega_{\mu}^a \lambda_{j \alpha}.
\label{acoplo}
\end{equation}
The only change in the Lagrangian is in the fermion kinetic term, where
the coupling to gravity becomes,
\begin{equation}
-{\overline \lambda}^{i}_{\dot \alpha}(\sigma^{\mu})^{\alpha
\dot \alpha}\{ \delta_{\alpha}{}^{\beta}\delta_{i}{}^{j}-i\omega_{\mu}^a
\big((\sigma_a)_{\alpha}{}^{\beta}\delta_{i}{}^{j} +
(\sigma_a)_i{}^{j}\delta_{\alpha}{}^{\beta}\big)\}\lambda_{j\beta}.
\label{kin}
\end{equation}
The connection appearing here is the tensor product connection on the bundle
$S^{+}\otimes S^{+}$, which is isomorphic to $\Omega^{0}_{\bf C}\oplus
\Omega^{2,+}_{\bf C}$. The scalar part corresponds to the antisymmetric part
of  $\lambda_{w\beta}$ while the self-dual two-form corresponds to the
symmetric one.  We can write (\ref{kin}) in terms of space-time fields,
and to do this we introduce a  scalar $\eta$, a one-form $\psi_{\mu}$, and a
self-dual two-form $\chi_{\mu \nu}$ as:
$$
\eta=-\lambda_{\beta}^{\beta},\,\,\,\,\,\,\
\psi_{\mu}=i(\sigma_{\mu})^{\dot \alpha \alpha}
{\overline \lambda}_{\alpha \dot \alpha},
$$
\begin{equation}
\lambda_{(w\beta)}=-{1 \over {2 {\sqrt 2}}} C^{\dot \alpha \dot \beta}
(\bar \sigma_{\mu})_{w \dot \alpha}(\bar \sigma_{\nu})_{\beta \dot \beta}
\chi_{\mu \nu},
\label{formas}
\end{equation}
being ${\bar\sigma}^{\mu}=({\bf 1}, -i\sigma_a)$ on flat space
(on a curved space, the
usual vierbein is included).
In terms of these fields the fermion kinetic
terms can be written, after a lengthy
computation, as
\begin{equation}
-{i \over 2}\psi^{\mu}D_{\mu} \eta - {\sqrt 2}
\psi_{\mu}D_{\nu}\chi^{\nu \mu},
\label{kindos}
\end{equation}
which are the standard fermion kinetic terms in Donaldson-Witten theory.
Notice that the gauging of the $SU(2)_I$ global symmetry allows
one to define the  theory on an arbitrary smooth four-manifold, as the
spinors become the differential forms in (\ref{formas}). Although these
differential forms are complex, they have of course a
natural underlying real structure.  One must restrict the resulting  fields to
be {\it real} differential forms, in order to have the same number of  degrees
of freedom in the untwisted and twisted theory. This  counting of degrees of
freedom must be taken into account if one is interested in  extracting some
information from the dynamics of the untwisted, physical theory.

In general, the gauging of a global symmetry can generate anomalies in the
resulting  theory. In the case of $N=2$ supersymmetric Yang-Mills theory, the
possible  anomalies are related to  the global $SU(2)$ anomaly discovered by
Witten in \cite{sudosano}, and it is easy  to see \cite{tqft} that they only
appear when the corresponding moduli space is not  orientable. This is not
the case for the moduli space  of ASD connections \cite{dk}, which is the one
described by Donaldson-Witten theory,   and the twisted theory is then
anomaly-free.

In the case of $N=2$ supersymmetric Yang-Mills theory in four dimensions the
twisting  involves the gauging of a $SU(2)$ internal symmetry group. In the
twisting of $N=2$  supersymmetric sigma models in two dimensions, the rotation
group is abelian, and the  twisting involves now the global $SO(2)=U(1)$
rotation group. Because of that,  the two-dimensional  situation has a formal
analogy with the twisting of the $U(1)_B$ current to be discussed  in the next
section, and we will examinate it now in some detail.

From the point of view of the gauging procedure, it is better to start with an
$N=2$  supersymmetric sigma model involving chiral multiplets. The target
manifold  $M$ is then K\"ahler and we have a bosonic field $\phi: \Sigma
\longrightarrow M$  and a Dirac spinor $\psi^{I}_{\pm} \in  \Gamma(\Sigma,
S^{\pm}\otimes \phi^{*}(TM))$, where $TM$ is the holomorphic tangent bundle
to $M$. The kinetic fermion term in the action is:
\begin{equation}
S_{\rm f}=\int_{\Sigma}d^{2}z ig_{{\bar I}J}
(\psi^{\bar I}_{+}D_{\bar z}\psi^{J}_{+}+
\psi^{\bar I}_{-}D_{ z}\psi^{J}_{-}),
\label{fermiondos}
\end{equation}
where the covariant derivatives are given in local coordinates by,
\begin{eqnarray}
D_{\bar z} \psi^{J}_{+}&=&\partial_{\bar z}\psi^{J}_{+}+
{i \over 2}\omega_{\bar z}\psi^{J}_{+}+
\Gamma^{J}_{KL}
\partial_{\bar z} \phi^{K} \psi^{L}_{+},
\nonumber\\
D_{z} \psi^{J}_{-}&=&\partial_{z}\psi^{J}_{-}
-{i \over 2}\omega_{z}\psi^{J}_{-}+
\Gamma^{J}_{KL}
\partial_{z} \phi^{K} \psi^{L}_{-}.
\label{covados}
\end{eqnarray}
This theory has a conserved, non-anomalous vector
current $j^{\mu}_{V}=g_{{\bar I}J}
{\overline \psi}^{\bar I}
{\gamma}^{\mu} \psi^{J}$, with components,
\begin{equation}
j^{z}_{V}=2g_{{\bar I}J}\psi^{\bar I}_{-}
\psi^{J}_{-},\,\,\,\,\,\ j^{\bar z}_{V}=
2g_{{\bar I}J}\psi^{\bar I}_{+} \psi^{J}_{+},
\label{jvector}
\end{equation}
and an anomalous axial current
$j^{\mu}_{5}=g_{{\bar I}J}{\overline \psi}^{\bar I}
{\gamma}^{\mu} \gamma_5 \psi^{J}$ with components,
\begin{equation}
j^{z}_{5}=2g_{{\bar I}J}\psi^{\bar I}_{-}
\psi^{J}_{-},\,\,\,\,\,\ j^{\bar z}_{5}=
-2g_{{\bar I}J}\psi^{\bar I}_{+} \psi^{J}_{+}.
\label{axial}
\end{equation}
The anomaly is given by the index of the Dirac operator and reads:
\begin{equation}
\int_{\Sigma} \phi^{*}(c_1(M)).
\label{anomalia}
\end{equation}
To twist the model we can gauge the $U(1)_V$ or the $U(1)_A$ symmetries.  The
first choice  leads to the A model and the second one to the B model
\cite{tsm,lla, mm}.  As in Donaldson-Witten theory, we promote the abelian
global symmetry to a  worldsheet space-time  symmetry, and in this case this
amounts to add to the Lagrangian the coupling of the  corresponding currents
to the worldsheet Spin connection. For the A model we have,
\begin{eqnarray}
S_{\rm f}&-&{i \over 4} \int_{\Sigma} d^2 z {\sqrt h}\omega_{\mu}j^{\mu}_{V}
\nonumber\\
&=& \int_{\Sigma} d^2 z {\sqrt h} i g_{{\bar I}J}\{ \psi^{\bar I}_{+}
({\overline \partial} \psi^{J}_{+}+\Gamma^{J}_{KL}
{\overline \partial} \phi^{K} \psi^{L}_{+})
\nonumber\\
& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
+\psi^{\bar I}_{-}(\partial \psi^{J}_{-}
-i\omega_{z}\psi^{J}_{-}+ \Gamma^{J}_{KL}
\partial \phi^{K} \psi^{L}_{-}) \},
\label{tipoa}
\end{eqnarray}
and for the B model,
\begin{eqnarray}
 S_{\rm f}&-&{i \over 4} \int_{\Sigma} d^2 z {\sqrt h}\omega_{\mu}j^{\mu}_{A}
\nonumber\\
&=&\int_{\Sigma} d^2 z{\sqrt h}ig_{{\bar I}J}\{ \psi^{\bar I}_{+}
({\overline \partial}\psi^{J}_{+}+i\omega_{\bar z}\psi^{J}_{+}+\Gamma^{J}_{KL}
\partial_{\bar z} \phi^{K} \psi^{L}_{-})
\nonumber\\
& &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
+\psi^{\bar I}_{-}(\partial\psi^{J}_{-}-
i\omega_{ z}\psi^{J}_{-}
+\Gamma^{J}_{KL}
\partial \phi^{K} \psi^{L}_{+}) \}.
\label{tipob}
\end{eqnarray}
We see again that, in the twisted models, the fermion fields
have changed their spin
content.  $\psi^{J} _{-}$ becomes a $(0,1)$-form $\rho_{\bar z}^{J}$, while
$\psi^{J} _{+}$ becomes a scalar  $\chi^{J}$ in the type A model,  and a
$(1,0)$-form $\rho^{J}_{z}$ in the type B model.

It turns out \cite{tsm} that the type A model can be formulated on any almost
Hermitian  target manifold. However, the type B model was obtained through the
gauging of an  anomalous current, and this can give ill-defined models: the
anomaly in  the global current, given in (\ref{anomalia}), is inherited in the
twisted model as a  global anomaly in the  fermion determinant. This leads to
additional restrictions on the geometry of the  target space, as pointed out
in \cite{mm}, because the fact that the $U(1)_A$ current is  chiral  leads to
a non-linear sigma model anomaly. We will present here a computation of  this
global anomaly using the strategy of \cite{aetio}.  The fermion kinetic term
of the type B model is:
\begin{equation}
S_{\rm B}= \int_{\Sigma} d^2 z{\sqrt h}ig_{{\bar I}J} \{ \chi^{\bar I}D_z
\rho^{J}_{\bar z}
+ \theta^{\bar I}D_{\bar z}\rho^{J}_{ z}\},
\label{kinb}
\end{equation}
where $\theta^{\bar I}=\psi_{+}^{\bar I}$ is an scalar in the twisted theory.
The effective action is then a section of the line bundle,
\begin{equation}
{\cal L}={\cal L}_1\otimes {\cal L}_2=({\rm det}\,\ D_z) \otimes
({\rm det}\,\ D_{\bar z}),
\label{ef}
\end{equation}
and the global, topological anomaly is
measured by $c_1({\cal L})=c_1({\cal L}_1) +
c_1({\cal L}_2)$. This can be computed
using the index theorem for families as in \cite{aetio}. Consider first the
evaluation map:
\begin{eqnarray}
{\hat \phi}: {\rm Map}(\Sigma, M) \times \Sigma &\rightarrow& M \nonumber\\
(\phi, \sigma) &\mapsto& \phi(\sigma).
\label{sigevalu}
\end{eqnarray}
By pulling back  differential forms on $M$ through ${\hat \phi}^*$, we get
differential forms on  ${\rm Map}(\Sigma, M) \times \Sigma$, with the natural
bigrading given by the  product structure:
\begin{equation}
{\hat \phi}^*(\omega)={\cal O}^{(0)}_{\omega}+{\cal O}^{(1)}_{\omega}+
{\cal O}^{(2)}_{\omega},
\label{bigrado}
\end{equation}
where the ${\cal O}^{(i)}_{\omega}$ are of degree $i$ with respect to
$\Sigma$. This is precisely the descent procedure of \cite{tsm}.
The topological obstructions are given by:
\begin{equation}
c_1({\cal L}_{1,2})=\int_{\Sigma}
{\rm ch}({\hat \phi}^*({\overline {TM}})) \{\pm 1 -
{1 \over 2}c_1(\Sigma)\},
\label{cherndet}
\end{equation}
where we only keep the degree two forms on
${\rm Map}(\Sigma, M)$. In $c_1({\cal L})$ only
the first descendant of
${\hat \phi}^*({\rm ch}({\overline {TM}}))$ contributes, and we finally get,
\begin{equation}
c_1({\cal L})=\int_{\Sigma}c_1(\Sigma){\cal O}^{(0)}_{c_1(M)},
\label{anomala}
\end{equation}
where ${\cal O}^{(0)}_{c_1(M)}={\hat \phi}_{\sigma}^{*}(c_1(M))$ is the
descendant of zero degree with respect to $\Sigma$,  and ${\hat
\phi}_{\sigma}$ is the map obtained from ${\hat \phi}$ by fixing a point
$\sigma\in \Sigma$ (the cohomology class of the pulled-back form does not
depend  on the $\sigma$ chosen). This result says then that the anomaly in the
global  current $U(1)_A$ is inherited in the  twisted B model as a sigma model
anomaly. The B model has no topological anomalies  if the target is  a
Calabi-Yau manifold or if the worldsheet is a torus ($c_1(\Sigma)=0$). The
last  possibility is natural,  as in this case  the twist does nothing (the
torus is a hyperk\"ahler manifold) and the original $N=2$  supersymmetric
model is anomaly-free.

\section{Twisting the $U(1)_B$ current in $N=2$ supersymmetric QCD}

To be specific, we will consider $N=2$ supersymmetric  QCD with gauge group
$SU(N)$ and  $N_f=1$ in the fundamental representation (we will follow the
notations  and conventions in \cite{lmpol, tesis}). In this case there is one
$N=2$ matter
 hypermultiplet which contains  a complex scalar isodoublet $q=(q_1,q_2)$,
fermions $\psi_{q \alpha}$, $\psi_{\tilde q \alpha}$, $\overline \psi_{q
\dot\alpha}$, $\overline \psi_{\tilde q  \dot\alpha}$, and a complex scalar
isodoublet auxiliary field $F_i$. The fields $q_i$, $\psi_{q \alpha}$,
$\overline \psi_{\tilde q  \dot\alpha}$, and $F_i$ are in the fundamental
representation of the gauge group, while the fields  $q^{i\dagger}$,
$\psi_{\tilde q \alpha}$, $\overline \psi_{q \dot\alpha}$, and $F^{i\dagger}$
are in the conjugate representation. From the point of view of $N=1$
superspace, this multiplet contains two $N=1$ chiral multiplets and therefore
it can be described by two $N=1$ chiral superfields $Q$ and $\widetilde Q$,
{\it i.e.}, these superfields satisfy the constraints $\overline
D_{\dot\alpha} Q=0$ and $\overline D_{\dot\alpha} \widetilde Q=0$.
While the superfield $Q$ is in the fundamental
representation of the gauge group, the superfield $\widetilde Q$ is in the
corresponding conjugate representation. The component fields of these $N=1$
superfields are:
\begin{eqnarray}
Q, \;\; Q^\dagger
 \;\; & \longrightarrow \;\;
q_1, \;\;   \psi_{ q\alpha}  \;\;  F_2, \;\; q^{1\dagger} \;\;
\overline\psi_{ q \dot\alpha}, \;\;  F^{2\dagger}, \nonumber\\
\widetilde Q , \;\; \widetilde Q^\dagger
\;\;& \longrightarrow \;\;  q^{2\dagger}, \;\; \psi_{\tilde q \alpha},
\;\; F^{1\dagger}, \;\; q_2, \;\;
\overline \psi_{\tilde q\dot\alpha},   \;\;  F_1.
\label{compas}
\end{eqnarray}
The $SU(2)_I$ current includes now a
contribution from the bosonic part of the hypermultiplet:
\begin{equation}
j^{\mu}_a={\overline \lambda}{\sigma}^{\mu}{\sigma_a}
\lambda-iq^{\dagger}\sigma_aD^{\mu}q
+iD^{\mu}q^{\dagger}\sigma_a q.
\label{corredos}
\end{equation}
The twist of the theory can also be understood as a gauging of the $SU(2)_I$
current,  as it happens with the pure $N=2$ supersymmetric
Yang-Mills theory. This  is
achieved by  adding to the original Lagrangian  a term $-\omega_{\mu}^a
j^{\mu}_a + q^{\dagger}\omega_{\mu}^a\omega^{\mu b}{\sigma_a}  {\sigma_b}q$.
The kinetic term for the bosons in the resulting theory is then,
\begin{equation}
(D_{\mu}+ i\omega_{\mu}^a{\sigma_a})q^{\dagger}
(D_{\mu}-i\omega^{\mu a}{\sigma_a})q,
\label{kinbos}
\end{equation}
where $D_{\mu}$ is the covariant derivative acting on scalars in  the
fundamental of $SU(N)$. We then see that, after the twisting, the bosonic
fields  $q$ become positive-chirality spinors. The fields in  the $N=2$
hypermultiplet are redefined after the twisting as follows:
\begin{eqnarray}
q_i & \longrightarrow& M_\alpha,\nonumber\\
\psi_{q \alpha}  & \longrightarrow&
-\mu_\alpha /{\sqrt 2}, \nonumber\\
\overline \psi_{\tilde q}^{ \dot\alpha}
& \longrightarrow&  v^{\dot\alpha}/{\sqrt 2},  \nonumber\\
q^{\dagger i}  & \longrightarrow&
\overline M^\alpha,   \nonumber\\
\overline \psi_{q  \dot\alpha}
& \longrightarrow &  \overline v_{\dot\alpha}/{\sqrt 2}, \nonumber\\
\psi_{\tilde q}^ \alpha & \longrightarrow &
 \overline  \mu^\alpha/{\sqrt 2}.\nonumber\\
\label{paloma}
\end{eqnarray}
The motivation for this redefinition is geometrical and can be understood
using the  Mathai-Quillen formulation of the twisted theory \cite{lmna}: the
fields  $\mu_{\alpha}$, $\overline  \mu^\alpha$ are a basis of differential
forms for the  configuration space of monopole fields $M_\alpha$, $\overline
M^\alpha$, and the  fields $v^{\dot\alpha}$, $\overline v_{\dot\alpha}$ are a
basis of fields for the  fibre where the moduli equations take values.

We have seen that, after the twisting, the bosonic fields  $q$ in the
hypermultiplet become positive-chirality spinors: the gauging of  the
$SU(2)_I$ current makes possible to define $N=2$ supersymmetric
Yang-Mills theory on a
curved manifold, but the  obstruction associated to $w_2(X)$ reappears when
matter  hypermultiplets are introduced. From the point of view of
four-dimensional geometry it would be  desirable to construct twisted $N=2$
supersymmetric QCD on a general four-manifold.  In the case of the
Seiberg-Witten monopole equations, the issue is precisely to consider
${\hbox{\rm Spin}}^c$-structures, and one would like to extend this possibility to the
non-abelian generalization of these equations. The  idea is to construct  an
extended twisting procedure by gauging an additional global symmetry of the
physical theory. As the pure Yang-Mills sector is already well-defined with
the usual gauging of the
$SU(2)_I$ isospin group, this symmetry can only act  on the matter sector.
Morevover, ${\hbox{\rm Spin}}^c$-structures involve a $U(1)$ gauge group  associated
to  a line bundle $L$ over the four-manifold $X$. In fact, in four dimensions
we have that,
\begin{equation}
{\hbox{\rm Spin}}^c_4=\{(A,B) \in U(2)\times U(2): {\rm det}(A)={\rm det}(B) \},
\label{spincfour}
\end{equation}
and therefore the structure groups of the complex spinor bundles
$S^{\pm}\otimes L^{1/2}$ are $SU(2)_{L,R}\times U(1)$, with the same $U(1)$
action in both  sectors. We should then gauge a global, non-anomalous $U(1)$
symmetry in the original $N=2$ theory, acting solely on the matter
hypermultiplets. The  required symmetry is precisely the baryon number. Let us
analyze this in some detail.

The global anomaly-free symmetry of $N=2$ supersymmetric QCD with $N_f$
hypermultiplets is:
\begin{equation}
SU(N_f)\times U(1)_{B}\times SU(2)_I.
\label{fsime}
\end{equation}
For $N_f=1$, the baryon number $U(1)_B$ acts on the hypermultiplet as,
\begin{eqnarray}
Q& \rightarrow& {\rm e}^{i\phi}Q, \,\,\,\,\,\,\,\,\ {\widetilde Q}
\rightarrow {\rm e}^{-i\phi}{\widetilde Q},\nonumber\\
Q^{\dagger} &\rightarrow &{\rm e}^{-i\phi}Q,
\,\,\,\,\,\,\,\,\ {\widetilde Q}^{\dagger}
\rightarrow {\rm e}^{i\phi}{\widetilde Q}^{\dagger}.
\label{barion}
\end{eqnarray}
As it is a vector symmetry, it is non-anomalous. In components it reads:
\begin{eqnarray}
q& \rightarrow& {\rm e}^{i\phi}q, \,\,\,\,\,\,\,\,\ q^{\dagger}
\rightarrow {\rm e}^{-i\phi}q^{\dagger}, \nonumber\\
\psi_{q \alpha}&\rightarrow &{\rm e}^{i\phi}\psi_{q \alpha},
\,\,\,\,\,\,\,\,\
{\overline \psi}_{{\tilde q} \dot  \alpha}
\rightarrow {\rm e}^{i\phi}
{\overline \psi}_{{\tilde q} \dot  \alpha},\nonumber\\
{\overline \psi}_{ q \dot \alpha}&\rightarrow &{\rm e}^{-i\phi}
{\overline \psi}_{ q \dot \alpha},
\,\,\,\,\,\,\,\,\
\psi_{ {\tilde q} \alpha}\rightarrow
{\rm e}^{-i\phi}\psi_{{\tilde q} \alpha}.
\label{barioncom}
\end{eqnarray}
The $U(1)_B$ current associated to this symmetry is:
\begin{equation}
j^{\mu}_{B}=-iD_{\mu}q^{\dagger}q+iq^{\dagger}D^{\mu}q+
{\overline  \psi}_{{ q} \dot  \alpha}
(\sigma^{\mu})^{\dot \alpha \alpha}{\psi}_{{ q} \dot  \alpha}-
{\overline  \psi}_{{\tilde q} \dot  \alpha}
(\sigma^{\mu})^{\dot \alpha \alpha}
{\psi}_{{\tilde q} \dot  \alpha}.
\label{barcor}
\end{equation}
To gauge this $U(1)$ symmetry, consider the determinant line bundle $L$
associated to a ${\hbox{\rm Spin}}^c$-structure on $X$, endowed with a connection
$b_{\mu}$, and add to the  Lagrangian the term:
\begin{equation}
{1 \over 2}j^{\mu}_B b_{\mu} -{1 \over 4}q^{\dagger}b_{\mu}b^{\mu}q.
\label{copla}
\end{equation}
If we gauge both the $SU(2)_I$ and the $U(1)_B$ symmetries, the covariant
derivatives  acting on the components  of the matter hypermultiplet in the
resulting Lagrangian are the appropriate ones  for complex spinors taking
values  in $S^{\pm}\otimes L^{1/2}\otimes E $. To further analyze the
consistency of the  procedure, it is  useful to consider the correspondence
between the fields in the original $N=2$ theory  and the fields appearing in
the Mathai-Quillen formulation of the moduli problem.  This correspondence is
given in (\ref{paloma}), and the structure of (\ref{copla}) implies
that the components of the matter hypermultiplet will be sections of
$L^{\pm {1\over 2}}$ if their baryon number is $\pm 1$.
 Taking into account the baryon number  assignment in
(\ref{barioncom}), we see that the  fields $M_{\alpha}$, $\mu_{\alpha}$ are
sections of $S^{+}\otimes L^{1/2} \otimes E$,  $v^{\dot \alpha}$ is a section
of $S^{-}\otimes L^{1/2}\otimes E$, ${\overline M}^{\alpha}$,  ${\bar
\mu}^{\alpha}$ are sections of $S^{+}\otimes L^{-1/2}\otimes {\overline E} $,
and  ${\bar v}_{\dot \alpha}$ is a section of $S^{-} \otimes L^{-1/2}\otimes
{\overline E}$.  This is then  consistent with the expected structure of the
complex spinor bundles, with the  same determinant line bundle for both
$S^{\pm}$. The  kinetic terms in the twisted theory are then,
\begin{equation}
{\cal L}_{\rm k}=g_{\mu \nu}D^{\mu}_{L}{\overline
M}^{\alpha}D^{\nu}_{L}M_{\alpha}-  { i \over 2}\Big({\bar v}_{\dot
\alpha}D_{L}^{\dot \alpha \alpha}\mu_{\alpha}+ {\bar \mu}^{\alpha}D_{L \alpha
\dot \alpha}v^{\dot \alpha} \Big),
\label{kinbardos}
\end{equation}
where,
\begin{equation}
D^{\mu}_{L}=D_{\mu}+{i\over 2}b_{\mu},
\label{spincov}
\end{equation}
is the covariant derivative associated to the tensor product connection on
$S^{+} \otimes L^{1/2}\otimes E $, $D_{\mu}$ is given in (\ref{cova}),
and $D_L^{\dot \alpha \alpha}$ is the
corresponding Dirac operator. The remaining terms in the Lagrangian are the
same as in the non-abelian monopole theory constructed in \cite{lmna}.

As in the case of the usual twisting involving matter  hypermultiplets
\cite{lmt, tesis}, the Lagrangian obtained after  the gauging of the
$SU(2)_I$ and the $U(1)_B$ symmetries is not $Q$-closed. This is a geneal
feature of  the twisting procedure: in order to guarantee the existence of a
scalar topological  symmetry on a curved manifold it can be necessary to add to
the Lagrangian  terms involving the non-trivial geometry of this manifold.
 If we compute $[Q,{\cal L}]$,  we must use the Weitzenb\"ock formula  for the
${\hbox{\rm Spin}}^c$-case \cite{sping}:
\begin{equation}
D^{\dagger}_{L\alpha\dot\alpha}
D^{\dot\alpha \beta}_L = (D_{L\mu} D^\mu_L  +
{1\over 4} R)\delta_{\alpha}{}^{\beta}
+i F_{\alpha}{}^{\beta}+ {i \over 2}\Omega_{\alpha}{}^{\beta},
\label{miwei}
\end{equation}
where $R$ is the scalar curvature of the manifold, and
$F_{\alpha}{}^{\beta}$  and $\Omega_{\alpha}{}^{\beta}$ are the self-dual
part  of the gauge field strength on $E$ and the curvature of the
line bundle $L$, respectively. The kinetic terms in  (\ref{kinbardos}),
${\cal L}_{\rm k}$,  give a non-zero contribution:
\begin{equation}
[ Q,{\cal L}_{\rm k} ]=-{R \over 4}({\overline M}^{\alpha}
\mu_{\alpha}+{\bar \mu}^{\alpha}
M_{\alpha})-
{i \over 2}\big( {\overline M}^{\alpha}
\Omega_{\alpha}{}^{\beta}\mu_{\beta}+
{\bar \mu}^{\alpha}\Omega_{\alpha}{}^{\beta} M_{\alpha}\big).
\label{noexdos}
\end{equation}
It then follows that the modified Lagrangian,
\begin{equation}
{\cal L}_{\rm top}={\cal L}+{R \over 4}
{\overline M}^{\alpha}M_{\alpha}+{i \over 2}
{\overline M}^{\alpha}\Omega_{\alpha}{}^{\beta}M_{\alpha},
\label{totexdos}
\end{equation}
where ${\cal L}$ contains the kinetic term
${\cal L}_{\rm k}$ in (\ref{kinbardos}) plus the rest of the terms
of the theory of non-abelian monopoles \cite{lmna},
is $Q$-closed on a general
four-manifold. This Lagrangian was obtained in
\cite{tqcd}, and a standard analysis shows that the resulting
topological field theory corresponds
to the moduli problem encoded in the equations
\begin{eqnarray}
F_{\alpha\beta}^{+a}+{i}{\overline
M}_{(\alpha} (T^{a}) M_{\beta )}&=&0,\nonumber\\
D_{L}^{\alpha\dot\alpha}M_\alpha &=&0.
\label{compactoc}
\end{eqnarray}
These are equations for a pair $(A,M)$ consisting of a connection  $A$ on $E$
and a section $M$ of the complex spinor bundle $S^+\otimes L^{1/2}  \otimes
E$, where the connection on the determinant line bundle $L$ is fixed. The
operator  $D_{L}$ is just the Dirac operator for this twisted bundle. Similar
equations  have been considered in the mathematical literature, see
\cite{okone,bradgp,gptres,pt}. Notice that the twist of the $U(1)_B$
current  gives precisely the geometrical content of these moduli equations.
As it was expected, the resulting topological field theory is anomaly free,
because the determinat line bundle of the twisted Dirac operator $D_L$
is always orientabledue to its underlying complex structure.

Usually, the fact that the theory is topological means that correlation
functions  do not depend on the Riemannian metric of the four-manifold. In the
same way one can  easily check that the theory is topological with respect to
the ${\hbox{\rm Spin}}^c$-connection:  the correlation functions do not depend on the
choice of the connection $b_{\mu}$  on the line  bundle $L$, but only on the
topological class of the ${\hbox{\rm Spin}}^c$-structure. To see this,  notice that
the Mathai-Quillen formulation of the model coupled to a ${\hbox{\rm Spin}}^c$-structure is
almost identical to the one presented in \cite{lmna}, the only difference
being that we must consider instead the  expression for the Dirac operator
$D_L$ (including the connection $b_{\mu}$ on the  determinant line bundle
$L$). The $Q$-transformations of  the fields in this off-shell formulation are
identical to the ones in the ${\hbox{\rm Spin}}$ case
 and do not depend on  the connection on $L$. As the full Lagrangian
(\ref{totexdos}) is  $Q$-exact, we have, using the results in \cite{lmna}:
\begin{equation}
{\delta {\cal L}_{\rm top} \over \delta b_{\mu}}=
\{Q, {1 \over 2}({\bar v}_{\dot \alpha}
(\sigma^{\mu})^{\dot \alpha \alpha}M_{\alpha}-{\overline M}^{\alpha}
(\sigma^{\mu})_{\dot \alpha \alpha}v^{\dot \alpha}) \}.
\label{topscon}
\end{equation}
This in turn guarantees that the twisted theory is independent of the
choice of $b_{\mu}$.

Notice that the metric (or Spin connection) and the ${\hbox{\rm Spin}}^c$-connection
enter  the construction on the same footing. In this formulation of
non-abelian monopoles  coupled to ${\hbox{\rm Spin}}^c$-structures, the connection on the
determinant line bundle $L$ is a  background gauge field just like the ${\hbox{\rm Spin}}$
connection. The analysis of the moduli problem  associated to the equations
(\ref{compactoc}) is very  similar to the one presented in \cite{lmna} in the
${\hbox{\rm Spin}}$ case, but one should not  divide by the group of gauge transformations
associated to the $U(1)$ gauge field,  because this is a {\it background}
field. This is in contrast with the corresponding  situation in the abelian
theory. However, the topological correlation functions  depend on the
topological  class of the ${\hbox{\rm Spin}}^c$-structure chosen to gauge the  $U(1)_B$
symmetry. In particular, the virtual dimension of the  moduli space depends
now on the first Chern class of $L$, and other features (like the
orientability, the analysis of reducibles and the structure of the
observables) are almost  identical to the ${\hbox{\rm Spin}}$ case.

\section{Conclusions and outlook}
We have shown that $N=2$ supersymmetric QCD has the possibility of coupling the
matter fields to  ${\hbox{\rm Spin}}^c$-structures once the adequate symmetry has been
identified. The gauging of the  $U(1)_B$ current can be generalized to
theories with more than one hypermultiplet: in this  case there are $N_f$
$U(1)$ symmetries that can be gauged, and this makes possible to  consider
$N_f$ different ${\hbox{\rm Spin}}^c$-structures, as it has been already noticed in
\cite{tqcd}.  An obvious extension of this work would be the computation,
using physical methods, of  the topological correlation functions of this
extended model, generalizing in this way  the results in \cite{lmpol}. One
should be careful with the subtleties involving  ${\hbox{\rm Spin}}^c$-structures that have
been discussed in \cite{abelia}. But the moral of our
 procedure  is perhaps that in the construction of  topological quantum field
theories one could consider not only
the  symmetries associated to the supersymmetry algebra, but {\it any} global
symmetry of the theory which might have a geometrical meaning through an
appropriate gauging.

\begin{center}
\large{ACKNOWLEDGEMENTS}
\end{center}
We would like to thank L. \'Alvarez-Gaum\'e and O. Garc\'\i a-Prada
for useful discussions.
This work was supported in part by DGCIYT under grant PB93-0344.