Analytic torsion and holomorphic determinant bundles. I: Bott-Chern forms and analytic torsion. II: Direct images and Bott-Chern forms. III: Quillen metrics on holomorphic determinants.

*(English)*Zbl 0651.32017
Commun. Math. Phys. 115, No. 1, 49-78, 79-126 (1988); No. 2, 301-351 (1988).

Let \(\pi\) : \(M\to B\) be a proper holomorphic map of complex analytic manifolds and let \(\xi\) be a holomorphic vector bundle on M. Following F. Knudsen and D. Mumford [Math. Scand. 39, 19-55 (1976; Zbl 0343.14008)], which is written for schemes, the authors define a holomorphic determinant line bundle \(\lambda^{KM}\) of the direct image of \(\xi\) by \(\pi\). Under the additional assumption of \(\xi\) being Hermitian and of \(\pi\) being a smooth family of Kähler manifolds the line bundle \(\lambda^{KM}\) is compared to Quillen’s line bundle \(\lambda\), which is naturally endowed with a metric deduced from the analytic torsion of Ray-Singer. For every \(y\in B\) the fibres \(\lambda_ y\) and \(\lambda_ y^{KM}\) are canonically isomorphic. However it is not at all clear, that the bundles are holomorphically or even smoothly isomorphic.

The three papers under review constitute a complete proof of this statement: The natural isomorphism between the fibres of \(\lambda\) and \(\lambda^{KM}\) define a smooth isomorphism between the bundles. The isomorphism is holomorphic provided B has an open covering \(\{\) \(U\}\) such that \(\pi^{-1}(U)\) admits a Kähler metric (whose restriction to the fibres may differ from their given Kähler metrics). There are some additional results e.g. a formula for the curvature associated with the Quillen metric in terms of the curvatures of the holomorphic Hermitian connections of \(\xi\) and of the tangent bundles of the fibres.

In the first paper the authors use Quillen’s superconnection in order to construct new representatives of the Bott-Chern classes of an acyclic complex of holomorphic Hermitian vector bundles. They establish many properties including an axiomatic description.

In the second paper the main properties of Kähler fibrations are derived. Analytic torsion forms for holomorphic direct images which generalize the torsion of Ray and Singer are constructed by means of a Levi-Civita superconnection. In case of acyclic complexes such forms are calculated by means of Bott-Chern classes. Severe difficulties arise: Asymptotic expansions become singular. In order to calculate the relevant term complicated manipulations on traces and Brownian motion techniques are necessary.

The third paper begins with the analysis of holomorphic determined bundles with special emphasiz on the Quillen metric. Finally the techniques and results developed so far are used to give two proofs of the main result \(\lambda =\lambda^{KM}\), an analytical one and a sheaf theoretic one.

The three papers under review constitute a complete proof of this statement: The natural isomorphism between the fibres of \(\lambda\) and \(\lambda^{KM}\) define a smooth isomorphism between the bundles. The isomorphism is holomorphic provided B has an open covering \(\{\) \(U\}\) such that \(\pi^{-1}(U)\) admits a Kähler metric (whose restriction to the fibres may differ from their given Kähler metrics). There are some additional results e.g. a formula for the curvature associated with the Quillen metric in terms of the curvatures of the holomorphic Hermitian connections of \(\xi\) and of the tangent bundles of the fibres.

In the first paper the authors use Quillen’s superconnection in order to construct new representatives of the Bott-Chern classes of an acyclic complex of holomorphic Hermitian vector bundles. They establish many properties including an axiomatic description.

In the second paper the main properties of Kähler fibrations are derived. Analytic torsion forms for holomorphic direct images which generalize the torsion of Ray and Singer are constructed by means of a Levi-Civita superconnection. In case of acyclic complexes such forms are calculated by means of Bott-Chern classes. Severe difficulties arise: Asymptotic expansions become singular. In order to calculate the relevant term complicated manipulations on traces and Brownian motion techniques are necessary.

The third paper begins with the analysis of holomorphic determined bundles with special emphasiz on the Quillen metric. Finally the techniques and results developed so far are used to give two proofs of the main result \(\lambda =\lambda^{KM}\), an analytical one and a sheaf theoretic one.

Reviewer: K.Lamotke

##### MSC:

32L10 | Sheaves and cohomology of sections of holomorphic vector bundles, general results |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

57R20 | Characteristic classes and numbers in differential topology |

##### Keywords:

characteristic classes; Kähler metric; Quillen metric; Bott-Chern classes; holomorphic Hermitian vector bundles
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\textit{J. M. Bismut} et al., Commun. Math. Phys. 115, No. 1, 49--78, 79--126 (1988; Zbl 0651.32017)

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