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diff --git a/macros/generic/occam/Occam94-old/sample.dir/Kohler.tex b/macros/generic/occam/Occam94-old/sample.dir/Kohler.tex new file mode 100644 index 0000000000..9ba9d3e171 --- /dev/null +++ b/macros/generic/occam/Occam94-old/sample.dir/Kohler.tex @@ -0,0 +1,1169 @@ + +%\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. 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Math. {\bf 131} (1990), +163--203, 205--238. +\Benchmark +\cite{GS2} H. Gillet and C. Soul\'e,{\it Analytic +torsion and the arithmetic Todd genus\/}, with an +appendix by D. Zagier, Topology {\bf 30} (1991), +21--54. +\Benchmark +\cite{H} F. Hirzebruch, Tpological Methods in +Algebraic Geometry, 3. ed. 1978. +\Benchmark +\cite{MQ} V. Mathai and D. Quillen, {\it +Superconnections, Thom classes and equivariant +differential forms\/}, Topology {\bf 25} (1986), +85--110. +\Benchmark +\cite{RS} D.B. Ray and I.M. Singer, {\it Analytic +torsion for complex manifolds\/}, Ann. of Math. {\bf +98} (1973), 154--177. +\Benchmark +\cite{S} D. Sullivan, {\it La classe d'Euler r\'eelle +d'un fibr\'e vectoriel \`a groupe structural +$SL_{n}(x)$ est nulle\/}, C.R. Acad. Sci. Paris, {\bf +281}, S\'erie A, (1975), 17--18. +\Benchmark +\cite{V} I. Vaisman, Symplectic Geometry and Secondary +Characteristic Classes, Birkh{\"a}user +1987.\endReferences + + + +\end |