%\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. 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