Holomorphic families of immersions and higher analytic torsion forms. (English) Zbl 0899.32013

Bismut, Jean-Michel

Astérisque. 244. Paris: Société Mathématique de France, vii, 275 p. (1997).

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Let \(i:W\rightarrow V\) be an embedding of smooth complex manifolds. Let \(S\) be a complex manifold. Let \(\pi _V:V\rightarrow S\) be a holomorphic submersion with compact fibre X, which restricts to a holomorphic submersion \(\pi _W:W\rightarrow S\), with compact fibre \(Y\). Then we have the diagram of holomorphic maps \[ \begin{tikzcd} Y \ar[r] \ar[d,"i"'] & W \ar[d,"i"] \ar[dr, "\pi_W"] & {}\\ X \ar[r] & V \ar[r,"\pi_V"'] & S \quad . \end{tikzcd}\tag {\(\ast\)} \] Let \(\eta\) be a holomorphic vector bundle on \(W\). Let \((\xi, v)\) be a holomorphic complex of vector bundles on \(V\), which together with a holomorphic restriction maps \(r:\xi _{0| W}\rightarrow \eta\), provides a resolution of the sheaf \(i_*\eta\). Let \(\omega ^V\), \(\omega ^W\) be real \((1,1)\) forms on \(V\), \(W\) which are closed, and which, when restricted to the relative tangent bundles \(TX\), \(TY\), are the Kähler forms of Hermitian metrics \(g^{TX}\), \(g^{TY}\) on \(TX\), \(TY\). Let \(g^{\xi _0},\ldots ,g^{\xi _m},g^{\eta}\) be Hermitian metrics on \(\xi _0,\ldots , \xi _m, \eta\). Let \(P^S\) be the vector space of smooth real differential forms on \(S\) which are sums of forms of type \((p,p)\). Let \(P^{S, 0}\) be the subspace of the \(\alpha \in P^S\), such that there exist smooth forms \(\beta\) and \(\gamma\) on \(S\), with \(\alpha =\overline\partial\beta + \partial \gamma\). Let \(T(\omega ^W,g^\eta)\) be the form in \(P^S\) constructed by J.-M. Bismut, M. Gillet and C. Soulé [Commun. Math. Phys. 115, No. 1, 79-126 (1988; Zbl 0651.32017)] and by J.-M. Bismut and K. K. Köhler [J. Algebr. Geom. 1, No. 4, 647-684, (1992; Zbl 0784.32023)]. Such forms are called higher analytic torsion forms. By the same procedure as in the above cited papers he constructs analytic torsion forms \(T(\omega^V,g^\eta)\in P^S\). The purpose of this paper is to prove an extension of a result of J.-M. Bismut and G. Lebeau [Theorem 0.1 in Publ. Math., Inst. Hautes Étud. Sci. 74, 1-297 (1991; Zbl 0784.32010)], which corresponds to their main result when \(S\) is a point. This extension is stated in the two theorems which give expressions containing Todd classes , analytic torsion forms and the suitable Bott-Chern classes.
The paper is organized as follows. In Chapter 1 the author establishes various results on the differential geometry of families of smooth embeddings, in the \(C^{\infty}\) category. In Chapter 2 the results of the first two above cited papers on higher analytic torsion forms are recalled. In Chapter 3 he describes the geometric setting of \((*)\), and also the objects which appear in the main theorems. In Chapter 4 the author gives some basic formulas on two parameter differential forms. In Chapter 5 the author’s results [J. Am. Math. Soc. 3, No. 1, 159-256 (1990; Zbl 0702.58071)] on the higher analytic torsion forms associated to a short exact sequence are recalled. Chapter 6 contains the proof of the first main theorem. In Chapter 7 a new horizontal bundle on \(V\) and the conjugate superconnections are introduced. Chapters 8-13 are devoted to the proof of the intermediate results. In Chapter 14 the second main theorem is proved. Finally, in Chapter 15 he shows that the objects appearing [J. Am. Math. Soc. 3, 159-256 (1990;l loc. cit.)] in the construction of the higher analytic torsion forms of a short exact sequence are a model for many of the arguments used in the paper.
The results contained in this paper have been announced in [the author, C. R. Acad. Sci., Paris, Sér. I 320, No. 8, 969-974 (1995; Zbl 0844.32010)].

Reviewer: W.Mozgawa (Lublin)

MSC:

32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
57R20	Characteristic classes and numbers in differential topology
32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
58J10	Differential complexes

Keywords:

analytic torsion form; Quillen metrics; Bott-Chern classes; index theory; Quillen superconnection; Ray-Singer torsion; anomaly; Bott-Chern current; zeta function; adiabatic limit; Dolbeault cohomology; hypercohomology; Todd form; Chern-Weil forms

Citations:

Zbl 0784.32023; Zbl 0784.32010; Zbl 0844.32010; Zbl 0651.32017; Zbl 0702.55071; Zbl 0702.58071