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\Title  
Analytic torsion forms on torus fibrations
\endTitle 
\bigskip
\bigskip

\Author 
Kai K{\"O}HLER
\endAuthor 
\vskip 20mm

{
{\noindent\smc abstract} : \eightpoint     We construct 
analytic torsion forms on holomorphic Torus 
fibrations, which are not necessarily K{\"a}hler 
fibrations. This is done by doubly transgressing the 
top Chern class. Also we establish a corresponding 
double transgression formula and an anomaly formula.
}


\Subheading {0. Introduction} The purpose of this 
paper is to construct analytic torsion forms for torus 
fibrations, which are not necessarily K{\"a}hler 
fibrations. These forms are needed to construct direct 
images in the hermitian $K$-theory, which was 
developped by Gillet and Soul\'e \cite{GS1} in the 
context of Arakelov geometry.

Let $\pi :M\rightarrow B$ be a holomorphic submersion 
with compact basis $B$, compact fibres $Z$ and a 
K{\"a}hler metric $g^{TZ}$ on the fibres. Let $\xi $ 
be a holomorphic vector bundle on $M$, equipped with a 
hermitian metric $h^{\xi }$. Then one could try to 
define analytic torsion forms $T$ associated to $\pi 
$, i.e. real forms on $B$, sums of forms of type 
$(p,p)$, defined modulo~$\partial $- and $\overline 
\partial $-coboundaries. They have to satisfy a 
particular double transgression formula and when the 
metrics $g^{TZ}$ and $h^{\xi }$ change, they have to 
change in a special way to make the forms ``natural'' 
in Arakelov geometry. They must not depend on metrics 
on $B$, and their component in degree zero should be 
the logarithm of the ordinary Ray-Singer torsion 
\cite{RS}.

Such forms were first constructed by Bismut, Gillet 
and Soul\'e \cite{BGS2, Th.2.20} for locally 
K{\"a}hler fibrations and $H^{*}(Z_{b},\xi 
\vert_{Z_{b}})=0\enskip \forall b\in B$. Gillet and 
Soul\'e \cite{GS2} and after them Faltings \cite{F} 
suggested definitions for more general cases. Then 
Bismut and the author gave in \cite{BK} an explicit 
construction of torsion forms $T$ for K{\"a}hler 
fibrations with $\dim H^{*}(Z_{b},\xi 
\vert_{Z_{b}})=\text{const. on }B$. $T$ satisfies the 
double transgression formula
$$
{\overline \partial \partial \over 2\pi i}T=
\ch\big(H^{*}(Z,\xi \vert_{Z}),h^{H^{*}(Z,\xi 
\vert_{Z})}\big)
           -\displaystyle \int 
_{Z}\Td(TZ,g^{TZ})\ch(\xi ,h^{\xi })
\Eqno (0.0)$$
and for two pairs of metrics $(g_{0}^{TZ},h_{0}^{\xi 
})$ and $(g_{1}^{TZ},h_{1}^{\xi })$,
 $T$ satisfies the anomaly formula
$$\Multline
T(g_{1}^{TZ},h_{1}^{\xi })-T(g_{0}^{TZ},h_{0}^{\xi })
   =\widetilde  {\ch}(H^{*}(Z,\xi 
\vert_{Z}),h_{0}^{H^{*}(Z,\xi 
\vert_{Z})},h_{1}^{H^{*}(Z,\xi \vert_{Z})}) \\
-\displaystyle \int _{Z}\left(\widetilde  
{\Td}(TZ,g_{0}^{TZ},g_{1}^{TZ})
           \ch(\xi ,h^{\xi 
}_{0})+\Td(TZ,g_{1}^{TZ})\widetilde  {\ch}(\xi 
,h_{0}^{\xi },h_{1}^{\xi })\right)
\endMultline
\Eqno (0.1)$$
modulo $\partial $- and $\overline \partial 
$-coboundaries. Here $\int _{Z}$ denotes the integral 
along the fibres, $\Td$ and $\ch$ are the Chern-Weil 
forms associated to the corresponding holomorphic 
hermitian connections and $\widetilde  {\Td}$ and 
$\widetilde  {\ch}$ denote Bott-Chern forms as 
constructed in \cite{BGS1, {\S }1f}.

In this paper, we shall construct analytic torsion 
forms $T$ in the following situation: consider a 
holomorphic hermitian vector bundle $\pi 
:(E^{1,0},g^{E})\rightarrow B$ on a compact complex 
manifold. Let $\Lambda $ be a lattice, spanning the 
underlying real bundle $E$ of $E^{1,0}$, so that local 
sections of $\Lambda $ are holomorphic sections of 
$E^{1,0}$. Then the fibration $E/\Lambda \rightarrow 
B$ is a holomorphic torus fibration which is not 
necessarily flat as a complex fibration.

In this situation, $H^{*}(Z,{\Cal O}_{Z})=\Lambda 
^{*}E^{*0,1}$. Classically, the formula
$$
\ch(\Lambda ^{*}E^{*0,1})={c_{\max}\over \Td}(E^{0,1}) 
\Eqno (0.2)
$$
holds on the cohomological level (see e.g. \cite{H, 
Th.10.11}). If one assumes supplementary that the 
volume of the fibres $Z$ is equal to 1, (0.2) holds 
also on the level of forms for the associated 
Chern-Weil forms. Thus, (0.1) suggests that $T$ should 
satisfy 
$$
{\overline \partial \partial \over 2\pi i}T(E/\Lambda 
,g^{E})={c_{\max}\over \Td}(E^{0,1},g^{E})\enskip 
\enskip .\Eqno (0.3)
$$

For two hermitian structures $g_{0}^{E}$ and 
$g_{1}^{E}$ on $E$, one should find the following 
anomaly formula
$$
T(E/\Lambda ,g_{1}^{E})-T(E/\Lambda ,g_{0}^{E})
=\widetilde  
{\Td^{-1}}(g_{0}^{E},g_{1}^{E})c_{\max}(g_{0}^{E})+%'
\Td^{-1}(g_{1}^{E})\widetilde  
{\ch}(g_{0}^{E},g_{1}^{E})\enskip . \Eqno (0.4)
$$

In this paper, such a $T$ will be constructed by 
explicitly doubly transgressing the top Chern class of 
$E^{0,1}$, which was proven to be 0 in cohomology by 
Sullivan \cite{S}.

Our method is closely following an article of Bismut 
and Cheeger \cite{BC}, in which they investigate eta 
invariants on real {\Blackbox}{\Blackbox} $(2n,{\Bbb 
Z})$ vector bundles. In this article, they are 
considering a quotient of a Riemannian vector bundle 
by a lattice bundle. Then they found a Fourier 
decomposition of the infinite dimensional bundle of 
sections on the fibres $Z$, which allowed them to 
transgress the Euler class explicitly via an 
Eisenstein series $\gamma $, i.e.
$$
d\gamma =\Pf\left({\Omega ^{E}\over 2\pi 
}\right)\enskip \enskip ,$$
where $\Pf$ denotes the Pfaffian and $\Omega ^{E}$ the 
curvature.

The case considered here is a bit more sophisticated 
because not only the metric but also the complex 
structure has not to have any direct relation with the 
flat structure. It turns out that the right choice for 
the holomorphic structure on $E^{0,1}$ is not, as in 
\cite{BK}, the by the metric induced structure, but an 
exotic holomorphic structure canonically induced by 
the flat structure on $E$ and the holomorphic 
structure on $E^{1,0}$.

We want to emphasize that here, as in \cite{BC}, the 
use of certain formulas in the Mathai-Quillen calculus 
\cite{MQ} is crucial. The formulas which we are using 
were established by Bismut, Gillet and Soul\'e in 
\cite{BGS5}.

\Subheading {I. Definitions} Let  $\pi 
:E^{1,0}\rightarrow B$ be a $n$-dimensional 
holomorphic vector bundle on a compact complex 
manifold $B$, with underlying real bundle $E$. Assume 
a lattice bundle $\Lambda \subset E$, spanning the 
realisation of $E^{1,0}$, so that a local section of 
$\Lambda $ induces a holomorphic section of $E^{1,0}$. 
Let $M$ be the total space of the fibration $E/\Lambda 
$, where the fibre $Z_{x}$ over a point $x\in B$ is 
given by the torus $E_{x}/\Lambda _{x}$. We call $J$ 
the different complex structures acting on $E$, $TM$ 
or $TB$ with $J\circ J=-1$.

Let $E^{*}$ be the dual bundle to $E$, equipped with 
the complex structure
$$
(J\mu )(\lambda ):=\mu (J\lambda )\enskip \enskip 
\enskip \forall \mu \in E^{*}\enskip ,\enskip \enskip 
\lambda \in E\enskip \enskip . \Eqno (1.0)
$$

In the same way, one defines $T^{*}B$ and $T^{*}M$. We 
get
$$E^{1,0}=\lbrace \lambda \in E\otimes {\Bbb C}\vert 
J\lambda =i\lambda \rbrace \enskip \enskip , \Eqno 
(1.1)
$$
$$
E^{0,1}=\lbrace \lambda \in E\otimes {\Bbb C}\vert 
J\lambda =-i\lambda \rbrace \enskip \enskip , \Eqno 
(1.2)
$$
and similar equations for $E^{*\,1,0}$,  $E^{*\,0,1}$, 
$T^{1,0}M$, $T^{0,1}M$, etc.


For $\lambda \in E$, we define
$$
	\lambda ^{1,0}:={\textstyle {1\over 2}}(\lambda 
-iJ\lambda )\enskip \enskip \enskip \text{and}\enskip 
\enskip \enskip \lambda ^{0,1}:={\textstyle {1\over 
2}}(\lambda +iJ\lambda )\enskip \enskip , \Eqno (1.3)
$$
and in the same manner maps $E^{*}\rightarrow 
E^{*\,1,0}$, $TB\rightarrow T^{1,0}B$, etc.  Let 
$\Lambda ^{*}\in E^{*}$ be the dual lattice bundle
$$
	\Lambda ^{*}:=\lbrace \mu \in E^{*}\vert \mu (\lambda 
)\in 2\pi {\Bbb Z}\enskip \forall \lambda \in \Lambda 
\rbrace \enskip \enskip . \Eqno (1.4)
$$
We set $\Lambda ^{1,0}:=\lbrace \lambda ^{1,0}\vert 
\lambda \in \Lambda \rbrace $, similar for $\Lambda 
^{0,1}$, $\Lambda ^{*\,1,0}$ and $\Lambda ^{*\,0,1}$. 
Also we fix a Hermitian metric $g^{E}=\left \langle 
\enskip ,\enskip \right \rangle $ on $E$, i.e. a 
Riemannian metric with the property
$$
	\left \langle J\lambda ,J\eta \right \rangle =\left 
\langle \lambda ,\eta \right \rangle \enskip \enskip 
\forall \lambda ,\eta \in E\enskip \enskip . \Eqno 
(1.5)
$$
This induces a Hermitian metric canonically on 
$E^{*}$. We assume the volumes of the fibres $Z$ of 
$M$ to be equal to $1$.

\Subheading {II. Some connections} Now one finds 
several canonical connections on $E$. First, the 
lattices $\Lambda $ 	and $\Lambda ^{*}$ induce 
(compatible) flat connections $\nabla  $ on $E$ and 
$E^{*}$ by $\nabla \lambda :=0$ for all local sections 
 $\lambda $ of $\Lambda $ (resp. $\nabla \mu :=0$ for 
$\mu \in \Gamma ^{\text{loc}}(\Lambda )$). We shall 
always use the same symbol for a connection on 
$E^{1,0}$, its conjugate on $E^{0,1}$, its realisation 
on $E$ and by duality induced connections on 
$E^{*\,1,0}$, $E^{*\,0,1}$ and $E^{*}$.

Generally, the connection $\nabla $ is not compatible 
with the complex structure $J$ (i.e. $\nabla 
J\mathbin{\not =}0$), so it does not extend to 
$E^{1,0}$. $\nabla $ induces a splitting
$$
	TM=\pi ^{*}E\oplus T^{H}M \Eqno (2.0)
$$
of the tangent space of $M$.

\Theorem {Proposition}   $T^{H}M$ is a complex 
subbundle of $TM$.\endTheorem 

\Proof {Proof} At a point $(x,\Sigma \alpha 
_{i}\lambda _{i})\in M$, $x\in B$, $\alpha _{i}\in 
{\Bbb R}$, $\lambda _{i}\in \Lambda _{x}$, $T^{H}M$ is 
equal to the image of the homomorphism
$$
	\Sigma \alpha _{i}\,T_{x}\lambda 
_{i}\,:\,TB\llongrightarrow        TM\enskip \enskip .
$$
The latter commutes with $J$ by the holomorphy 
condition on $\Lambda $. Thus, $T^{H}M$ is invariant 
by $J$.\qed 

The horizontal lift of $Y\in TB$ to $T^{H}M$ will be 
denoted by $Y^{H}$. Let $\overline \partial ^{E}$ be 
the Dolbeault operator on $E^{1,0}$. Now one can use 
$\nabla $ to construct a canonical holomorphic 
connection $\nabla ^{h}$ on $E^{1,0}$, not depending 
on the metric; furthermore, we will see that $\nabla 
^{h}$ induces a canonical holomorphic structure 
$\overline \partial ^{\overline E}$ on $E^{*\,0,1}$ 
with the property
$$
\overline \partial ^{\overline E}\mu ^{0,1}=0\enskip 
\enskip \enskip \forall \mu \in \Lambda ^{*}\enskip 
\enskip . \Eqno (2.1)
$$

Let us denote by $\nabla '\lambda $, $\nabla ''\lambda 
$ the restrictions of $\nabla .\lambda :TB\otimes 
{\Bbb C}\llongrightarrow        E\otimes {\Bbb C}$ to 
$T^{1,0}B$ and $T^{0,1}B$ (we will use the same 
convention for all connections and for $\End(E\otimes 
{\Bbb C})$-valued one forms on $B$).

\Theorem {Lemma 1}   $\nabla '$ maps $\Gamma 
(E^{1,0})$ into $\Gamma (T^{1,0}B\otimes E^{1,0})$. 
The connection on $E^{1,0}$
$$
	\nabla ^{h}:=\nabla '+\overline \partial ^{E}  \Eqno 
(2.2)
$$
is a holomorphic connection. Its curvature $(\nabla 
^{h})^{2}$ is a $(1,1)$-form. 

The dual connection on $E^{*}$ satisfies
$$
	\nabla ^{h''}\mu ^{0,1}=0\enskip \enskip \forall 
\lambda \in \Lambda ^{*}\enskip \enskip ; \Eqno (2.3)
$$
hence it induces a canonical holomorphic structure 
$\overline \partial ^{\overline E}$ on $E^{*\,0,1}$, 
depending only on the flat structure on $E$ and the 
holomorphic structure on $E^{1,0}$.\endTheorem 

\Proof {Proof} The lift of $\nabla $ to $M$ is given 
by
$$
	(\pi ^{*}\nabla )_{Y^{H}}Z=[Y^{H},Z]\enskip \enskip 
\forall Z\in \Gamma (TZ)\cong \Gamma (TE),Y\in \Gamma 
(TB)\enskip \enskip , \Eqno (2.4)
$$
in particular
$$
(\pi ^{*}\nabla )_{Y^{H\,1,0}}(\pi ^{*}\lambda 
^{1,0})=[Y^{H^{1,0}},\pi ^{*}\lambda ^{1,0}]\enskip 
\enskip \forall \lambda \in \Gamma (E)\enskip \enskip 
. \Eqno (2.5)
$$
The r.h.s. of (2.5) takes values in $T^{1,0}Z$, hence 
$\nabla '$ maps in fact $E^{1,0}$ to $E^{1,0}$ (this 
is equivalent to the equation
$$
	\nabla _{JY}J=J\nabla _{Y}J\enskip \enskip \forall 
Y\in TB\enskip \enskip )\enskip . \Eqno (2.6)
$$
This proves the first part of the Lemma. Now one 
computes for $\mu \in \Gamma ^{\text{loc}}(\Lambda 
^{*})$, $\lambda \in \Gamma ^{\text{loc}}(\Lambda )$
$$\Multline
   0=\overline \partial (\mu (\lambda ))=(\nabla 
^{h''}\mu ^{0,1})(\lambda ^{0,1})+(\nabla ^{h''}\mu 
^{1,0})(\lambda ^{1,0})\\
     +\mu ^{0,1}(\nabla ^{h''}\lambda ^{0,1})+\mu 
^{1,0}(\nabla ^{h''}\lambda ^{1,0})\enskip 
.\endMultline
\Eqno (2.7)$$
By condition, $\nabla ^{h''}\lambda ^{1,0}=0$; also 
$0=\nabla ''\mu =\nabla ''\mu ^{1,0}+\nabla ''\mu 
^{0,1}$, so
$$\Multline
  0=-\overline \partial (\mu ^{0,1}(\lambda ^{1,0})) 
   =(-\nabla ''\mu ^{0,1})(\lambda ^{1,0})+\mu 
^{0,1}(-\nabla ''\lambda ^{1,0})\\
   =(\nabla ^{h''}\mu ^{1,0})(\lambda ^{1,0})+\mu 
^{0,1}(\nabla ^{h''}\lambda ^{0,1})\enskip .
\endMultline
 \Eqno (2.8)$$

This proves the second part of the Lemma.\qed 

In fact, one could simply verify that $\nabla ^{h}$ is 
just the ``complexification'' of $\nabla $
$$
	\nabla ^{h}=\nabla -{\textstyle {1\over 2}}J\nabla J  
\Eqno (2.9)
$$
both on $E$ and $E^{*}$.

The metric $\left \langle \cdot ,\cdot \right \rangle 
$ induces an isomorphism of real vector bundles  
\hbox{${\frak i}:E\rightarrow E^{*}$,} so that ${\frak 
i}\circ J=-J\circ {\frak i}$.

\Definition {Definition}  Let $\nabla ^{\overline E}$ 
be the hermitian holomorphic connection on 
$E^{*\,0,1}$ associated to the canonical holomorphic 
structure in Lemma~1. We denote by ${}^{t}\theta 
^{*}:TB\otimes {\Bbb C}\rightarrow \End(E^{*}\otimes 
{\Bbb C})$ the one-form given by
$$
	{}^{t}\theta ^{*}:=\nabla -\nabla ^{\overline E}  
\Eqno (2.10)
$$
and by $\vartheta $ the one-form on $B$ with 
coefficients in $\End(E^{*})$
$$
	\vartheta  _{Y}:={\frak i}^{-1}\nabla {\frak 
i}\enskip \enskip \forall Y\in TB\enskip \enskip .  
\Eqno (2.11)
$$\endDefinition 
 $\nabla ^{\overline E}$ should not be confused with 
the hermitian holomorphic connection on $E^{1,0}$ 
associated to its original holomorphic structure, 
which we shall not use in this article.

The transposed of  ${}^{t}\theta ^{*}$ with respect to 
the natural pairing $E\otimes E^{*}\rightarrow {\Bbb 
R}$ will be denoted by $\theta ^{*}$, thus
$$
({}^{t}\theta ^{*}\mu )(\lambda )=\mu (\theta 
^{*}\lambda )\enskip \enskip \forall \mu \in 
E^{*}\enskip ,\enskip \enskip \lambda \in E\enskip 
\enskip . \Eqno (2.12) 
$$
The duals of ${}^{t}\theta ^{*}$ and $\theta ^{*}$ 
will be denoted by ${}^{t}\theta $ and $\theta $. This 
notation is chosen to be compatible with the notation 
in \cite{BC}. By definition, ${}^{t}\theta ^{*}$ 
satisfies
$$\aligned
{}^{t}\theta ^{*}{}'' &amp; :E\otimes {\Bbb 
C}\llongrightarrow         E^{1,0}\enskip \enskip , \\
{}^{t}\theta ^{*}{}' &amp;: E\otimes {\Bbb 
C}\llongrightarrow         E^{0,1}\enskip \enskip . 
\endaligned \Eqno (2.13)
$$
Notice that the connection $\nabla +{\Cal V}$ on 
$E^{*}$ is just the pullback of $\nabla $ by the 
isomorphism ${\frak i}^{-1}$. 

\Theorem {Lemma 2}  The hermitian connection $\nabla 
^{\overline E}$ on $E^{*\,0,1}$ is given by
$$
	\nabla ^{\overline E}=(\nabla +\vartheta )'+\overline 
\partial ^{\overline E}=\nabla ^{h}+\vartheta '\enskip 
\enskip . \Eqno (2.14)
$$
Its curvature on $E^{*\,0,1}$ is given by
$$
\Omega ^{\overline E}= \overline \partial ^{\overline 
E}\vartheta '\enskip \enskip , \Eqno (2.15)
$$
and it is characterized by the equation
$$
\left \langle (\Omega ^{\overline E}+\theta \theta 
^{*})\mu ,\nu \right \rangle =i\partial \overline 
\partial \left \langle \mu ,J\nu \right \rangle 
\enskip \enskip \forall \mu ,\nu \in \Gamma 
^{\loc}(\Lambda ^{*})\enskip \enskip . \Eqno (2.16)
$$\endTheorem 

\Proof {Proof} The first part is classical, but we 
shall give a short proof to illustrate our notations. 
For all $\mu \in \Gamma ^{\loc}(\Lambda ^{*})$, $\nu 
\in \Gamma (E^{*})$
$$
\overline \partial \left \langle \mu ^{0,1},\nu 
^{1,0}\right \rangle  
   =\overline \partial (({\frak i}^{-1}\mu )(\nu 
^{1,0}))=({\frak i}^{-1}\mu )((\nabla +\vartheta 
)''\nu ^{1,0})\enskip \enskip ; \Eqno (2.17)
$$
but also
$$
\overline \partial \left \langle \mu ^{0,1},\nu 
^{1,0}\right \rangle  
   =\left \langle \mu ^{0,1},\nabla ^{\overline 
E}{}''\nu ^{1,0}\right \rangle =({\frak i}^{-1}\mu 
)(\nabla ^{\overline E}{}''\nu ^{1,0})\enskip \enskip 
, \Eqno (2.18)
$$
hence $(\nabla +\vartheta )'=\nabla ^{\overline E}{}'$ 
on $E^{*0,1}$. To see the second part, one calculates 
for $\mu ,\nu \in \Gamma ^{\loc}(\Lambda ^{*})$
$$\aligned
\partial \overline \partial \left \langle \mu 
^{0,1},\nu ^{1,0}\right \rangle  &amp;=\left \langle 
\nabla ^{\overline E}{}'\mu ^{0,1},\nabla ^{\overline 
E}{}''\nu ^{1,0}\right \rangle +\left \langle \mu 
^{0,1},\nabla ^{\overline E}{}'\nabla ^{\overline 
E}{}''\nu ^{1,0}\right \rangle  \\
    &amp;= \left \langle \nabla ^{\overline E}{}'\mu 
,\nabla ^{\overline E}{}''\nu \right \rangle +\left 
\langle \mu ^{0,1},\Omega ^{\overline E}\nu 
^{1,0}\right \rangle  \\
    &amp;= -\left \langle {}^{t}\theta ''{}^{t}\theta 
^{*}{}'\mu ,\nu \right \rangle  - \left \langle \Omega 
^{\overline E}\mu ^{0,1},\nu ^{1,0}\right \rangle 
\enskip \enskip ; \endaligned
\Eqno (2.19)$$
but also
$$
\partial \overline \partial \left \langle \mu 
^{1,0},\nu ^{0,1}\right \rangle =\left \langle 
{}^{t}\theta '{}^{t}\theta ^{*}{}''\mu ,\nu \right 
\rangle +\left \langle \Omega ^{\overline E}\mu 
^{0,1},\nu ^{1,0}\right \rangle \enskip \enskip .
\Eqno (2.20)$$
Taking the difference and using (2.13), one finds
$$\aligned
i\partial \overline \partial \left \langle \mu ,J\nu 
\right \rangle &amp;=\partial \overline \partial \left 
\langle \mu ^{1,0},\nu ^{0,1}\right \rangle -\partial 
\overline \partial \left \langle \mu ^{0,1},\nu 
^{1,0}\right \rangle  \\
         &amp; = \left \langle \Omega ^{\overline E}\mu 
,\nu \right \rangle +\left \langle ({}^{t}\theta 
'{}^{t}\theta ^{*}{}''+{}^{t}\theta ''{}^{t}\theta 
^{*}{}'),\mu ,\nu \right \rangle  \\
         &amp; =  \left \langle (\Omega ^{\overline 
E}+{}^{t}\theta {}^{t}\theta ^{*})\mu ,\nu \right 
\rangle \enskip \enskip .
\endaligned \Eqno (2.21)$$

Notice that $i\partial \overline \partial \left 
\langle \mu ,J\nu \right \rangle  = \overline 
{i\partial \overline \partial \left \langle \mu ,J\nu 
\right \rangle }$ is in fact a real form.\qed 

\Subheading {III. Computation of the Levi-Civita 
superconnection} The analytic torsion forms of a 
fibration are defined using a certain superconnection, 
acting on the infinite dimensional bundle of forms on 
the fibres. In this section, this superconnection will 
be investigated for the torus fibration  $\smallmatrix 
M\\\pi \,\downarrow \\B\endsmallmatrix$.

Let $F:=\Gamma (Z,\Lambda T^{*\,0,1}Z)$ be the 
infinite dimensional bundle on $B$ with the 
antiholomorphic forms on $Z$ as fibres. By using the 
holomorphic hermitian connection $\nabla ^{\overline 
E}$ on $E^{*\,0,1}$, one can define a connection 
$\widetilde  \nabla $ on $F$ setting
$$
	\widetilde  \nabla _{Y}h:=(\pi _{*}\nabla ^{\overline 
E})_{Y^{H}}h\enskip \enskip \forall Y\in \Gamma 
(TB)\enskip ,\enskip \enskip h\in \Gamma (B,F)\enskip 
\enskip . \Eqno (3.0)
$$

The metric $\left \langle \enskip ,\enskip \right 
\rangle $ on $E$ induces a metric on $Z$. Then $F$ has 
a natural $TZ\otimes {\Bbb C}$ Clifford module 
structure, given by the actions of
$$
c(Z^{1,0}):=\sqrt 2 {\frak i}(Z^{1,0})\Lambda \enskip 
\enskip \text{and}\enskip \enskip c(Z^{0,1}):=-\sqrt 
2\iota _{Z^{0,1}}\enskip \enskip \enskip \forall z\in 
TZ\enskip \enskip . \Eqno (3.1)
$$
 $\iota _{Z^{0,1}}$ denotes here interior 
multiplication. Clearly $$c(Z)c(Z')+c(Z')c(Z)=-2\left 
\langle Z,Z'\right \rangle \enskip \forall Z,Z'\in 
TZ\otimes {\Bbb C}\enskip \enskip .\Eqno  (3.3)$$

 Let $\overline \partial ^{Z}$, $\overline \partial 
^{Z*}$ be the Dolbeault operator and its dual on $Z$, 
and let
$$
	D:=\overline \partial ^{Z}+\overline \partial ^{Z*}  
\Eqno (3.3)
$$
denote the Dirac operator action on $F$. In fact, for 
an orthonormal basis $(e_{i})$ of $TZ\otimes {\Bbb C}$ 
and the hermitian connection $\nabla ^{Z}$ on $Z$
$$
	D={1\over \sqrt 2} \sum c(e_{i})\nabla 
^{Z}_{e_{i}}\enskip \enskip . \Eqno (3.4)
$$

A form $\mu =\mu ^{1,0}+\mu ^{0,1}\in \Lambda ^{*}$ 
can be identified with a ${\Bbb R}/2\pi {\Bbb 
Z}$-valued function on $Z$. In particular, the ${\Bbb 
C}$-valued function $e^{i\mu }$ is welldefined on $Z$. 
Then one finds the analogue of Theorem~2.7 in 
\cite{BC}.

\Theorem {Lemma 3}  For $x\in B$, $F_{x}$ has the 
orthogonal decomposition in Hilbert spaces
$$
	F_{x} = \bigoplus\limits _{\mu \in \Lambda ^{*}_{x}} 
\Lambda E_{x}^{*\,0,1}\otimes \lbrace e^{i\mu }\rbrace 
\enskip \enskip . \Eqno (3.5)
$$

For $\mu \in \Lambda ^{*}_{x}$, $\alpha \in \Lambda 
\,E_{x}^{*\,0,1}$, $D$ acts on $\Lambda 
\,E_{x}^{*\,0,1}\otimes \lbrace e^{i\mu }\rbrace $ as
$$
	D(\alpha \otimes e^{i\mu })={ic({\frak i}^{-1}\mu 
)\over \sqrt 2}\alpha \otimes e^{i\mu }  \Eqno (3.6)
$$
and
$$
	D^{2}(\alpha \otimes e^{i\mu })={\textstyle {1\over 
2}} \left \vert \mu \right \vert ^{2}\alpha \otimes 
e^{i\mu }\enskip \enskip . \Eqno (3.7) 
$$\endTheorem 

\Proof {Proof} The first part of the lemma is standard 
Fourier analysis, using that $\text{vol}(\Lambda )=1$. 
The second part is obtained by calculating
$$\aligned
\overline \partial ^{Z}(\alpha \otimes e^{i\mu 
^{1,0}})&amp;=0\enskip ,\enskip \enskip \enskip \overline 
\partial ^{Z}(\alpha \otimes e^{i'\mu ^{0,1}})=i\,\mu 
^{0,1}\wedge \alpha \otimes e^{i\mu ^{0,1}}\enskip ,\\
\overline \partial ^{*\,Z}(\alpha \otimes e^{i\mu 
^{0,1}})&amp;=0\enskip ,\enskip \enskip \enskip 
\overline \partial ^{Z\,*}(\alpha \otimes e^{i\mu 
^{1,0}})=-i\,\iota _{{\frak i}^{-1}\mu ^{1,0}}\alpha 
\otimes e^{i\mu ^{1,0}}\enskip , .
\endaligned
\Eqno (3.8)$$
\qed 

Now one can determine the action of $\widetilde  
\nabla $ with respect to this splitting. Define a 
connection on the infinite dimensional bundle 
$C^{\infty }(Z,{\Bbb C})$ by setting
$$
	\nabla ^{\infty }_{Y}f:=Y^{H}.f\enskip \enskip 
\forall Y\in TB\enskip ,\enskip \enskip f\in C^{\infty 
}(Z,{\Bbb C})\enskip \enskip . 
\Eqno (3.9)$$

\Theorem {Lemma 3.10}  The connection  $\widetilde  
\nabla $ acts on $F=\Lambda E^{*\,0,1}\otimes 
C^{\infty }(Z,{\Bbb C})$ as
$$
	\widetilde  \nabla =\nabla ^{\overline E}\otimes 
1+1\otimes \nabla ^{\infty }\enskip \enskip ; \Eqno 
(3.10)
$$
hence it acts on local sections of $\Lambda 
E^{*\,0,1}\otimes \lbrace e^{i\mu }\rbrace $ for $\mu 
\in \Gamma ^{\loc}(\Lambda ^{*})$ as $\nabla 
^{E}\otimes 1$. In particular,
$$
\widetilde  \nabla ^{2}=\Omega ^{\overline E}\otimes 
1\enskip \enskip . \Eqno (3.11)
$$\endTheorem 

\Proof {Proof} This is obvious because $\mu $ is a 
flat local section.

\Definition {Definition}  The superconnection $A_{t}$ 
on $\smallmatrix F\\\downarrow \\B\endsmallmatrix$, 
depending on $t\in {\Bbb R}$, $t\geq 0$, given by
$$
	A_{t}:=\widetilde  \nabla +\sqrt tD  \Eqno (3.12)
$$
is called the Levi-Civita 
superconnection.\endDefinition 

In fact, this definition is the analogue to the 
Definition~2.1 in \cite{BGS2}; the torsion term 
appearing there vanishes in the case mentioned here. 
By Lemma~3 and Lemma~4, it is clear that $A^{2}_{t}$ 
acts on $\Lambda E^{*\,0,1}\otimes \lbrace e^{i\mu 
}\rbrace $, $\mu \in \Gamma ^{\loc}(\Lambda ^{*})$, as
$$
A^{2}_{t}=(\nabla ^{\overline E}+i\sqrt {{t\over 
2}}c({\frak i}^{-1}\mu ))^{2}\otimes 1\enskip \enskip 
. \Eqno (3.13)
$$
\Subheading {IV. A transgression of the top Chern 
class} 

In this section, a form $\vartheta $ on $B$ will be 
constructed using the superconnection $A_{t}$, which 
transgresses the top Chern class  $c_{n}({-\Omega 
^{\overline E}\over 2\pi i})$ of $E^{0,1}$. $\vartheta 
$, divided by the Todd class, will define the torsion 
form in section V. We will use the Mathai-Quillen 
calculus \cite{MQ}, in its version described and used 
by \cite{BGS5}. Mathai and Quillen observed that for 
$A\in \End(E)$ skew and invertible and $\Pf(A)$ its 
Pfaffian, the forms $\Pf(A) (A^{-1})^{k}$ are 
polynomial functions in $A$, so they can be extended 
to arbitrary skew elements of $\End(E)$. An 
endomorphism $A\in \End(E^{0,1})$, i.e. $A\in \End(E)$ 
with $J \circ  A = A \circ  J$, may be turned into a 
skew endomorphism of $E \otimes {\Bbb C}$ by replacing
$$
	A \mapsto  {\textstyle {1\over 2}} (A-A^{*}) + 
{\textstyle {1\over 2}} iJ(A+A^{*})\,\,.\Eqno (4.0) 
$$
That means, $A$ is replaced by the operator which acts 
on $E^{1,0}$ as $-A^{*}$ and on $E^{0,1}$ as $A$. This 
is the convention of \cite{BGS5, p. 288} adapted to 
the fact that we are handling with $E^{0,1}$ and not 
with $E^{1,0}$. The same conventions will be applied 
to $\End(TM)$.

With $I_{\overline E} \in  \End(E^{0,1})$ the identity 
map, we consider at $Y\in E$ and $b\in {\Bbb R}$
$$
	\alpha _{t} := \text{det}_{T^{0,1} E}\left({-\pi 
^{*}\Omega ^{\overline E}\over  2\pi i} - b 
I_{\overline E}\right) e^{-t({ \left \vert Y\right 
\vert \over 2} + (\pi ^{*} \Omega ^{\overline E}-2\pi 
b J)^{-1})} \Eqno (4.1) 
$$
by antisymmetrization as a form on the total space of 
$E$.
\Definition {Definition}   Let $\widetilde  \beta _{t} 
\in \Lambda  T^{*}B$ be the form
$$
	\widetilde  \beta _{t}:= \sum _{\mu \in \Lambda ^{*}} 
({\frak i}^{-1}\mu )^{*} {\partial \over \partial 
b}\Big\vert_{b=0} \alpha _{t} \Eqno (4.2)
$$
and $\beta _{t} \in \Lambda  T^{*}B$ be the form
$$
	\beta _{t} := \sum _{\mu \in \Lambda ^{*}} ({\frak 
i}^{-1}\mu )^{*} \alpha _{t}\vert_{b=0}\,\,.\Eqno 
(4.3) 
$$
\endDefinition   
The  geometric meaning of $\beta _{t}$ will become 
clear in the proof of Lemma 8. We recall that $\theta 
^{*} = \nabla ^{\overline E} - \nabla $ on $E$, hence 
for $\mu \in \Gamma ^{\loc}(\Lambda ^{*})$
$$
	\nabla ^{\overline E}({\frak i}^{-1}\mu ) = -\theta  
{\frak i}^{-1}\mu  \Eqno (4.4)
$$
and one obtains
$$
	({\frak i}^{-1}\mu )^{*} (\pi ^{*} \Omega ^{\overline 
E} -  2\pi bJ)^{-1} = {\textstyle {1\over 2}} \left 
\langle {\frak i}^{-1}\mu , \theta ^{*}(\Omega 
^{\overline E} - 2\pi bJ)^{-1} \theta  {\frak 
i}^{-1}\mu \right \rangle \,\,.\Eqno (4.5) 
$$
Hence one obtains
\Theorem {Lemma 5} $\widetilde  \beta _{t}$ is given 
by
$$
	\widetilde  \beta _{t} = {\partial \over \partial 
b}\Big\vert_{b=0} \text{det}_{E^{0,1}} \left({-\Omega 
^{\overline E}\over 2\pi i} - bI_{\overline E}\right) 
\sum _{\mu \in \Lambda ^{*}} e^{-{t\over 2} \left 
\langle {\frak i}^{-1}\mu ,(1+\theta ^{*} (\Omega 
^{\overline E}-2\pi bJ)^{-1}\theta ){\frak i}^{-1}\mu 
\right \rangle } \Eqno (4.6)
$$
and
$$
	\widetilde  \beta _{t} = {\partial \over \partial 
b}\Big\vert_{b=0} {\text{det}_{E^{0,1}}({-\Omega 
^{\overline E}\over 2\pi i} - bI_{\overline E})\over  
\text{det}^{1/2}_{E}(1+\theta ^{*}(\Omega ^{\overline 
E} -2\pi bJ)^{-1}\theta )} \sum _{\lambda \in \Lambda 
} e^{-{1\over 2t}\left \langle \lambda ,(1+\theta 
^{*}(\Omega ^{\overline E} -2\pi bJ)^{-1}\theta 
)\lambda \right \rangle } \,\,.\Eqno (4.7)
$$
It has the asymptotics
$$
	\widetilde  \beta _{t} = - c_{n-1} \left({-\Omega 
^{\overline E}\over 2\pi i}\right) + {\Cal 
O}_{t\nearrow \infty }(e^{-t}) \Eqno (4.8)
$$
for $t\nearrow \infty $ and
$$
	\widetilde  \beta _{t} = -(2\pi t)^{-n} c_{n-1} 
\left({-\Omega ^{\overline E}-\theta \theta ^{*}\over  
2\pi i}\right) + {\Cal O}_{t\searrow 0}(e^{-{1\over 
t}}) \Eqno (4.9)
$$
for $t\searrow 0$.
\endTheorem   
\Proof {Proof}  The second equation follows by the 
Poisson summation formula (recall $\vol(\Lambda ) = 
1$). The first asymptotic (4.8) is clear. The second 
asymptotic (4.9) may be proved by using formula (1.40) 
in \cite{BC}, which is obtained by a nontrivial result 
on Brezinians in \cite{Ma, pp. 166-167}. One finds
$$
	\aligned
{\text{det}_{E^{0,1}}({-\Omega ^{\overline E}\over 
2\pi i} - bI_{\overline E})\over  
\text{det}^{1/2}_{E}(1+\theta ^{*}(\Omega ^{\overline 
E} -2\pi bJ)^{-1}\theta )} 
&amp;= 
{(-1)^{n} \Pf({\Omega ^{\overline E}\over 2\pi 
}-bJ)\over  \text{det}^{1/2}_{E}(1+\theta ^{*}(\Omega 
^{\overline E} -2\pi bJ)^{-1}\theta )}\\
&amp;=
(-1)^{n} \Pf \left({-\Omega ^{\overline E}-\theta 
\theta ^{*}\over  2\pi }-bJ\right) \\
&amp;=
\text{det}_{E^{0,1}}\left({-\Omega ^{\overline E}- 
\theta \theta ^{*}\over 2\pi i} - bI_{\overline 
E}\right)\,\,.\endaligned\Eqno (4.10)
$$
\qed 
In the same manner one obtains
\Theorem {Lemma 6} $\beta _{t}$ is given by
$$
	\beta _{t} = \text{det}_{E^{0,1}} \left({-\Omega 
^{\overline E}\over 2\pi i}\right) \sum _{\mu \in 
\Lambda ^{*}} e^{-{t\over 2} \left \langle {\frak 
i}^{-1} \mu ,(1+\theta ^{*} \Omega ^{\overline 
E-1}\theta ){\frak i}^{-1}\mu \right \rangle } \Eqno 
(4.10)
$$
and
$$
\beta _{t} =(2\pi t)^{-n} 
{\text{det}_{E^{0,1}}({-\Omega ^{\overline E}\over 
2\pi i})\over \text{det}_{E}^{1/2}(1+\theta ^{*}\Omega 
^{\overline E-1}\theta )} \sum _{\lambda \in \Lambda } 
e^{-{1\over 2t}\left \langle \lambda ,(1+\theta 
^{*}\Omega ^{\overline E-1}\theta )\lambda \right 
\rangle }\,\,.\Eqno (4.11)
$$
It has the asymptotics
$$
	\beta _{t} = c_{n} \left({-\Omega ^{\overline E}\over 
2\pi i}\right) + {\Cal O}_{t\nearrow \infty 
}(e^{-t})\Eqno (4.12)
$$
for $t\nearrow \infty $ and for $t\searrow 0$
$$
	\beta _{t} = (2\pi t)^{-n} c_{n} \left({-\Omega 
^{\overline E}-\theta \theta ^{*}\over 2\pi i}\right) 
+ {\Cal O}_{t\searrow 0}(e^{-{1\over t}})\,\,.\Eqno 
(4.13)
$$
\endTheorem   
We define the Epstein $\zeta $-function for $s &gt;n$
$$
	\zeta (s) := - {1\over \Gamma (s)} \displaystyle \int 
^{\infty }_{0} t^{s-1} \left(\widetilde  \beta _{t} + 
c_{n-1}\big({-\Omega ^{\overline E}\over 2\pi i}\big) 
\right) dt\,\,.\Eqno (4.14)
$$
Classically, $\zeta $ has a holomorphic continuation 
to $0[E]$. Hence we may define
\Definition {Definition}   Let $\vartheta $ be the 
form on $B$
$$
	\vartheta  := \zeta '(0)\,\,.\Eqno (4.15) 
$$
\endDefinition   
Then $\vartheta $ transgresses the top Chern class :
\Theorem {Theorem 7}   $\vartheta $ permits the 
double-transgression formula
$$
	{\overline \partial \partial \over 2\pi i} \vartheta  
= c_{n} \left({-\Omega ^{\overline E}\over 2\pi 
i}\right)\,\,.\Eqno (4.16)
$$
\endTheorem   
\Proof {Proof}  By \cite{BGS5, Th. 3.10}, one knows 
that
$$
	- t {\partial \over \partial t} \alpha 
_{t}\big\vert_{b=0} = {\overline \partial \partial 
\over 2\pi i} {\partial \over \partial 
b}\Big\vert_{b=0} \alpha _{t}\,\,.\Eqno (4.17)
$$
The minus sign occuring here contrary to \cite{BGS5} 
is caused by the different sign of $J = -i 
I_{\overline E}$ in our formulas.

We define $\beta ^{0}$ by $\beta _{t} = t^{-n} \beta 
^{0} + {\Cal O}_{t\searrow 0}(e^{-1/t})$ as in Lemma 
6. Then one obtains for $s &gt; n$
$$
	\Multline
	{\overline \partial \partial \over 2\pi i} \zeta (s) 
= {1\over \Gamma (s)} \displaystyle \int ^{\infty 
}_{0} t^{s} {\partial \beta _{t}\over \partial t} dt\\
	= {1\over \Gamma (s)} \displaystyle \int ^{1}_{0} 
t^{s} {\partial \over \partial t} (\beta _{t}-t^{-n} 
\beta ^{0})dt - {n\over \Gamma (s)} \displaystyle \int 
^{1}_{0} t^{s-1-n} \beta ^{0} dt + {1\over \Gamma (s)} 
\displaystyle \int ^{\infty }_{1} t^{s} {\partial 
\over \partial t} \beta _{t} dt\\
	=  {1\over \Gamma (s)} \displaystyle \int ^{1}_{0} 
t^{s} {\partial \over \partial t} (\beta _{t}-t^{-n} 
\beta ^{0})dt + {1\over \Gamma (s)} {n\over n-s} \beta 
^{0} + {1\over \Gamma (s)}  \displaystyle \int 
^{\infty }_{1} t^{s} {\partial \over \partial t} \beta 
_{t} dt\endMultline \Eqno (4.18)
$$
and hence for the holomorphic continuation of $\zeta $ 
to 0
$$
	 {\overline \partial \partial \over 2\pi i}  \zeta 
'(0) = \lim_{t\nearrow \infty } \beta _{t} = c_{n}  
\left({-\Omega ^{\overline E}\over 2\pi 
i}\right)\,\,.\Eqno (4.19)
$$
\qed 



\Subheading { V. The analytic torsion form} 

Let $N_{H}$ be the number operator on $B$ acting on 
$\Lambda ^{p} T^{*}B\otimes  F$ by multiplication with 
$p$ $\Tr_{s}\bullet $ will denote the supertrace 
$\Tr(-1)^{N_{H}}\bullet $. Let $\varphi $ be the map 
acting on $\Lambda ^{2p}T^{*}B$ by multiplication with 
$(2\pi i)^{-p}$.
\Theorem {Lemma 8}   Up to a cboundary,
$$
	\varphi  \Tr_{s} N_{H} e^{-A^{2}_{t}} = 
\Td^{-1}\left({-\Omega ^{\overline E}\over 2\pi 
i}\right) \widetilde  \beta _{t}\,\,,\Eqno (5.0) 
$$
where $\Td^{-1}$ denotes the inverse of the Todd 
genus.
\endTheorem   
\Proof {Proof}  Define a form $\widehat \alpha _{t}$ 
on the total space of $E$ with value
$$
	\widehat \alpha _{t} := \varphi  \Tr_{s} N_{H} 
\exp\left(-(\nabla ^{\overline E} + i \sqrt {{t\over 
2}} c(\lambda ))^{2}\right)\Eqno (5.1) 
$$
at $\lambda  \in  E$. Then one observes
$$
	\varphi  \Tr_{s} N_{H} e^{-A^{2}_{t}} = \sum _{\mu 
\in \Lambda ^{*}}  ({\frak i}^{-1}\mu )^{*} \widehat 
\alpha _{t}\,\,.\Eqno (5.2) 
$$
But one knows that
$$
	\widehat \alpha _{t} = {\partial \over \partial 
b}\Big\vert_{b=0} \Td^{-1} \left({-\pi ^{*} \Omega 
^{\overline E}\over 2\pi i} - b I_{E}\right) \alpha 
_{t}\Eqno (5.3) 
$$
by \cite{BGS5, Proof of Th. 3.3}. The result follows.
\qed 

Now we define the analytic torsion form $T(M, \left 
\langle i\right \rangle )$ in \cite{BK} via the $\zeta 
$-function to $\varphi  \Tr_{s} N_{H} e^{-A^{2}_{t}}$, 
modulo $\partial -$ and $\overline \partial 
-$coboundaries.
\Definition {Definition}  The analytic torsion form 
$T(M, g^{E})$ is defined by
$$
	T(M, g^{E}) := \Td^{-1}\left({-\Omega ^{\overline 
E}\over 2\pi i}\right) \vartheta \,\,.\Eqno (5.4) 
$$
\endDefinition   
In particular, we deduce from Theorem 7
$$
	{\overline \partial \partial \over 2\pi i} T(M, 
g^{E}) = \left({c_{n}\over \Td}\right) \left({-\Omega 
^{\overline E}\over 2\pi i}\right) \,\,\,\,.\Eqno 
(5.5)
$$
Now we shall investigate the dependence of $T$ on the 
metric $g^{E}$. For a charactersitic class $\phi  $, 
we shall denote by $\phi  (g^{E})$ its evaluation for 
the hermitian holomorphic connection $ \nabla ^{E}$ on 
$E^{0,1}$ with respect to $\overline \partial $. For 
two Hermitian metrics  $g^{E}_{0}, g^{E}_{1}$ on $E$, 
let $\widetilde  \phi  (g^{E}_{0}, g^{E}_{1})$ denote 
the axiomatically defined Bott-Chern classes of 
\cite{BGS1, Sect. 1f)}. $\widetilde  \phi  $ is living 
in the space of sums of $(p,p)$-forms modulo $\partial 
-$ and $\overline \partial -$coboundaries. It has the 
following property
$$
	{\overline \partial \partial \over 2\pi i}\widetilde  
\phi  (g^{E}_{0}, g^{E}_{1}) = \phi  (g^{E}_{1}) - 
\phi  (g^{E}_{0})\,\,.\Eqno (5.6)
$$
\Theorem {Theorem 9}   Let $g^{E}_{0}, g^{E}_{1}$ be 
two Hermitian metrics on $E$. Then the associated 
analytic torsion forms change by
$$
	T(M, g^{E}_{1})-T(M,g^{E}_{0}) = 
\widetilde{\Td^{-1}}(g^{E}_{0}, g^{E}_{1}) 
c_{n}(g^{E}_{0}) + \Td^{-1}(g^{E}_{1}) 
\widetilde{c_{n}}(g^{E}_{0}, g^{E}_{1})\Eqno (5.7)
$$
modulo $\partial -$ and $\overline \partial 
-$coboundaries.
\endTheorem   
\Proof {Proof}  This follow by the uniqueness of the 
Bott-Chern classes. Using (5.5) and the 
characterization of Bott-Chern classes in \cite{BGS1, 
Th. 1.29}, it is clear that
$$
	T(M,g^{E}_{0}) - T(M, g^{E}_{1}) = 
\left({\widetilde{c_{n}}\over \Td}\right) (g^{E}_{0}, 
g^{E}_{1})\,\,.\Eqno (5.8)
$$
The result follows.\qed 

\Subheading {VI. The K{\"a}hler condition} 

The analytic torsion forms were only constructed in 
\cite{BK} for the case were the fibration is 
K{\"a}hler. That means, there had to exist a 
K{\"a}hler metric on the total space $M$, so that the 
decomposition (2.0) is an orthogonal decomposition. 
Hence it is interesting to see when this happens for 
the case investigated here.
\Theorem {Lemma 10}  The fibration $\smallmatrix 
M\\\downarrow \\B\endsmallmatrix$ is K{\"a}hler iff 
the base $B$ is K{\"a}hler and there exists a falt 
symplectic structure $\omega ^{E}_{0}$ on $E$, which 
is a positive $(1,1)$-form with respect to $J$, i.e.
$$
	\alignat 3
	\text{I)} &amp;\quad\nabla \omega ^{E}_{0} = 0\,, \tag 
6.0\\
	\text{II)} &amp;\quad \omega ^{E}_{0}(JX,JY) = \omega 
^{E}_{0}(X,Y) &amp;\qquad \forall  &amp; \,X,Y \in  E \,, \tag 
6.1\\
	\text{III)} &amp;\quad \omega ^{E}_{0}(X,JX) &gt; 0 &amp;\qquad 
\forall  &amp; \,X\in E\,\,. \tag 6.2
\endalignat
$$
\endTheorem   
It follows easily that $\overline \partial ^{\overline 
E}$ is the by the metric and $\overline \partial 
^{\overline E}$ induced holomorphic structure if $M$ 
is K{\"a}hler. Thus, $T$ coincides with the torsion 
form in \cite{BK} in this case. Furthermore, $\Omega 
^{\overline E} + \theta \theta ^{*} = 0$, so the 
asymptotic  terms in (4.9), (4.13) vanish.
\Proof {Proof}  Let $g$ any Hermitian metric on $TM$, 
so that $g(T^{H}M, TZ) = 0$. Let $\omega := g(\bullet 
,J\bullet )$ be the corresponding K{\"a}hler form. By 
$\omega ^{H}$ and $\omega ^{Z}$ we denote the 
horizontal and the vertical part of $\omega $. Using 
the decomposition (2.0), the condition $d\omega =0$ 
splits into four parts :
\Item {{\bf I)}}    For $Y_{1}, Y_{2}, Y_{3} \in  TB$ 
:
$$
	0 = d\omega (Y^{H}_{1}, Y^{H}_{2}, Y^{H}_{3}) = 
d\omega ^{H} (Y^{H}_{1}, Y^{H}_{2}, Y^{H}_{3})\,,\Eqno 
(6.3)
$$
\Item {{\bf II)}}    for $Y_{2}, Y_{2} \in  TB$, $Z\in 
TZ$ :
$$
	0 = d\omega (Y^{H}_{1}, Y^{H}_{2}, Z) = 2 . \omega 
^{H}(Y^{H}_{1}, Y^{H}_{2})\,,\Eqno (6.4)
$$
\Item {{\bf III)}}    for $Y\in TB, Z_{1}, Z_{2} \in  
TZ$ :
$$
	0 = d\omega (Y^{H}, Z_{1}, Z_{2}) = (L_{Y^{H}} \omega 
^{Z})(Z_{1}, Z_{2})\,,\Eqno (6.5) 
$$
\Item {{\bf IV)}}    for $Z_{1}, Z_{2}; Z_{3} \in TZ$ 
:
$$
	0 = d\omega (Z_{1}, Z_{2},Z_{3}) = d\omega 
^{Z}(Z_{1}, Z_{2}, Z_{3})\,.\Eqno (6.5) 
$$
Conditions I) and II) just mean that 
$g\vert_{T^{H}M\times T^{H}M}$ is he horizonal lift of 
a K{\"a}hler metric on $B$. If  there is a form 
$\omega ^{Z}$ satisfying condition III), then its 
restriction to the zero section of $E$ induces a 
K{\"a}hler form $\omega ^{E}$ on $E$, so that the left 
$\pi ^{*} \omega ^{E}$ satisfies conditions III) and 
IV). Only the following necessary condition remains
\Item {{\bf III\,\,\alpha )}}    There exists a 
Hermitian metric $g^{E}$ on $E$, so that for the 
corresponding K{\"a}hler, form $\omega ^{E}$ and all 
$\lambda _{1}, \lambda _{2}\in  \Gamma ^{\loc}(\Lambda 
)$
$$
	\omega ^{E}(\lambda _{1},\lambda _{2}) = 
\text{const}\,.\Eqno (6.7)
$$
On the other hand, $M$ is clearly K{\"a}hler if this 
condition is satisfied. This proves the Lemma.\qed 

\noindent One may also investigate the local 
K{\"a}hler condition as posed in \cite{BGS1}, 
\cite{BGS2}. Because $B$ is always locally K{\"a}hler, 
the same proof as above shows
\Theorem {Lemma 11}   The fibration $\smallmatrix 
M\\\downarrow \\B\endsmallmatrix$ is locally 
K{\"a}hler at $x_{0} \in  B$ iff there exists locally 
on $B$ at $x_{0}$ a flat symplectic structure $\omega 
^{E}_{0}$ on $E$, so that
$$
	\alignat 2
\text{I)} \qquad &amp;\omega ^{E}_{0}(JX,JY) = \omega 
^{E}_{0}(X,Y) &amp; \qquad \forall  &amp;X,Y \in  E\,, \tag 
6.8\\
\text{II)} \qquad &amp; \omega ^{E}_{0}(X,JX) &gt;0 \,\, 
\text{ at } x_{0} &amp; \qquad \forall  &amp;X \in 
E_{x_{0}}\,.\tag 6.9
\endalignat 
$$
\endTheorem   


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