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+
+%\input amstex
+%\input amsppt.sty
+
+\input Kohler.sty
+
+ \hsize= 14truecm
+
+\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
+$$\aligned
+{}^{t}\theta ^{*}{}'' & :E\otimes {\Bbb
+C}\llongrightarrow E^{1,0}\enskip \enskip , \\
+{}^{t}\theta ^{*}{}' &: E\otimes {\Bbb
+C}\llongrightarrow E^{0,1}\enskip \enskip .
+\endaligned \Eqno (2.13)
+$$
+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 $\mu ,\nu \in \Gamma ^{\loc}(\Lambda ^{*})$
+$$\aligned
+\partial \overline \partial \left \langle \mu
+^{0,1},\nu ^{1,0}\right \rangle &=\left \langle
+\nabla ^{\overline E}{}'\mu ^{0,1},\nabla ^{\overline
+E}{}''\nu ^{1,0}\right \rangle +\left \langle \mu
+^{0,1},\nabla ^{\overline E}{}'\nabla ^{\overline
+E}{}''\nu ^{1,0}\right \rangle \\
+ &= \left \langle \nabla ^{\overline E}{}'\mu
+,\nabla ^{\overline E}{}''\nu \right \rangle +\left
+\langle \mu ^{0,1},\Omega ^{\overline E}\nu
+^{1,0}\right \rangle \\
+ &= -\left \langle {}^{t}\theta ''{}^{t}\theta
+^{*}{}'\mu ,\nu \right \rangle - \left \langle \Omega
+^{\overline E}\mu ^{0,1},\nu ^{1,0}\right \rangle
+\enskip \enskip ; \endaligned
+\Eqno (2.19)$$
+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
+$$\aligned
+i\partial \overline \partial \left \langle \mu ,J\nu
+\right \rangle &=\partial \overline \partial \left
+\langle \mu ^{1,0},\nu ^{0,1}\right \rangle -\partial
+\overline \partial \left \langle \mu ^{0,1},\nu
+^{1,0}\right \rangle \\
+ & = \left \langle \Omega ^{\overline E}\mu
+,\nu \right \rangle +\left \langle ({}^{t}\theta
+'{}^{t}\theta ^{*}{}''+{}^{t}\theta ''{}^{t}\theta
+^{*}{}'),\mu ,\nu \right \rangle \\
+ & = \left \langle (\Omega ^{\overline
+E}+{}^{t}\theta {}^{t}\theta ^{*})\mu ,\nu \right
+\rangle \enskip \enskip .
+\endaligned \Eqno (2.21)$$
+
+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
+$$\aligned
+\overline \partial ^{Z}(\alpha \otimes e^{i\mu
+^{1,0}})&=0\enskip ,\enskip \enskip \enskip \overline
+\partial ^{Z}(\alpha \otimes e^{i'\mu ^{0,1}})=i\,\mu
+^{0,1}\wedge \alpha \otimes e^{i\mu ^{0,1}}\enskip ,\\
+\overline \partial ^{*\,Z}(\alpha \otimes e^{i\mu
+^{0,1}})&=0\enskip ,\enskip \enskip \enskip
+\overline \partial ^{Z\,*}(\alpha \otimes e^{i\mu
+^{1,0}})=-i\,\iota _{{\frak i}^{-1}\mu ^{1,0}}\alpha
+\otimes e^{i\mu ^{1,0}}\enskip , .
+\endaligned
+\Eqno (3.8)$$
+\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
+$$
+ \aligned
+{\text{det}_{E^{0,1}}({-\Omega ^{\overline E}\over
+2\pi i} - bI_{\overline E})\over
+\text{det}^{1/2}_{E}(1+\theta ^{*}(\Omega ^{\overline
+E} -2\pi bJ)^{-1}\theta )}
+&=
+{(-1)^{n} \Pf({\Omega ^{\overline E}\over 2\pi
+}-bJ)\over \text{det}^{1/2}_{E}(1+\theta ^{*}(\Omega
+^{\overline E} -2\pi bJ)^{-1}\theta )}\\
+&=
+(-1)^{n} \Pf \left({-\Omega ^{\overline E}-\theta
+\theta ^{*}\over 2\pi }-bJ\right) \\
+&=
+\text{det}_{E^{0,1}}\left({-\Omega ^{\overline E}-
+\theta \theta ^{*}\over 2\pi i} - bI_{\overline
+E}\right)\,\,.\endaligned\Eqno (4.10)
+$$
+\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 $s >n$
+$$
+ \zeta (s) := - {1\over \Gamma (s)} \displaystyle \int
+^{\infty }_{0} t^{s-1} \left(\widetilde \beta _{t} +
+c_{n-1}\big({-\Omega ^{\overline E}\over 2\pi i}\big)
+\right) dt\,\,.\Eqno (4.14)
+$$
+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 $s > n$
+$$
+ \Multline
+ {\overline \partial \partial \over 2\pi i} \zeta (s)
+= {1\over \Gamma (s)} \displaystyle \int ^{\infty
+}_{0} t^{s} {\partial \beta _{t}\over \partial t} dt\\
+ = {1\over \Gamma (s)} \displaystyle \int ^{1}_{0}
+t^{s} {\partial \over \partial t} (\beta _{t}-t^{-n}
+\beta ^{0})dt - {n\over \Gamma (s)} \displaystyle \int
+^{1}_{0} t^{s-1-n} \beta ^{0} dt + {1\over \Gamma (s)}
+\displaystyle \int ^{\infty }_{1} t^{s} {\partial
+\over \partial t} \beta _{t} dt\\
+ = {1\over \Gamma (s)} \displaystyle \int ^{1}_{0}
+t^{s} {\partial \over \partial t} (\beta _{t}-t^{-n}
+\beta ^{0})dt + {1\over \Gamma (s)} {n\over n-s} \beta
+^{0} + {1\over \Gamma (s)} \displaystyle \int
+^{\infty }_{1} t^{s} {\partial \over \partial t} \beta
+_{t} dt\endMultline \Eqno (4.18)
+$$
+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.
+$$
+ \alignat 3
+ \text{I)} &\quad\nabla \omega ^{E}_{0} = 0\,, \tag
+6.0\\
+ \text{II)} &\quad \omega ^{E}_{0}(JX,JY) = \omega
+^{E}_{0}(X,Y) &\qquad \forall & \,X,Y \in E \,, \tag
+6.1\\
+ \text{III)} &\quad \omega ^{E}_{0}(X,JX) > 0 &\qquad
+\forall & \,X\in E\,\,. \tag 6.2
+\endalignat
+$$
+\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
+$$
+ \alignat 2
+\text{I)} \qquad &\omega ^{E}_{0}(JX,JY) = \omega
+^{E}_{0}(X,Y) & \qquad \forall &X,Y \in E\,, \tag
+6.8\\
+\text{II)} \qquad & \omega ^{E}_{0}(X,JX) >0 \,\,
+\text{ at } x_{0} & \qquad \forall &X \in
+E_{x_{0}}\,.\tag 6.9
+\endalignat
+$$
+\endTheorem
+
+
+\References {
+References
+}
+\Benchmark
+\cite{BC} J.-M. Bismut and J. Cheeger, {\it
+Transgressed Euler Classes of $SL(2n, {\Bbb Z})$
+vector bundles, adiabatic limits if eta invariants and
+special values of $L$-functions\/}, Ann. Scient. Ec.
+Norm. Sup. 4e s\'erie, t. 25 (1992), 335--391.
+\Benchmark
+\cite{BGS1} J.-M. Bismut, H. Gillet and C. Soul\'e,
+{\it Analytic torsion and holomorphic determinant
+bundles I\/}, Comm. Math. Phys. {\bf 115} (1988),
+49--78.
+\Benchmark
+\cite{BGS2} ---, {\it Analytic torsion and holomorphic
+determinant bundles II\/}, Comm. Math. Phys. {\bf 115}
+(1988), 79--126.
+\Benchmark
+\cite{BGS5} ---, Complex immersions and Arakelov
+geometry, The Grothendieck Festschrift vol. 1,
+Birkh{\"a}user 1990.
+\Benchmark
+\cite{BK} J.-M. Bismut and Kai K{\"o}hler, {\it Higher
+analytic torsion forms for direct images and anomaly
+formulas\/}, J. Alg. Geom. {\bf 1} (1992, 647--684.
+\Benchmark
+\cite{E} P. Epstein, {\it Zur Theoric allg{\Blackbox}
+Zetafunktionen\/}, Math. Ann. {\bf 56} (1903),
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+\Benchmark
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