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**Hermitian structures on the derived category of coherent sheaves.**
*(English.
French summary)*
Zbl 1248.18011

Given a proper holomorphic map of complex manifolds \(X\rightarrow Y\) and a Hermitian holomorphic vector bundle on \(X\), it is an interesting question how to define the direct image of this bundle. The aim of the paper under review is to set up the theoretical basis and language needed to deal with this problem. Tackling it, one is led to work in the bounded derived category of coherent sheaves \(D^b(X)\) and to consider Hermitian structures on objects of this category. In fact, it is reasonable to work in the algebraic category, that is, with smooth complex algebraic varieties.

In Section 2 the authors define and characterize meager complexes, which are roughly speaking bounded acyclic complexes of Hermitian vector bundles whose Bott-Chern classes vanish. They then introduce the tight equivalence relation (whose definition involves meager complexes) between bounded complexes of Hermitian vector bundles and study the set of classes of tightly related complexes. This set is a monoid and the submonoid of acyclic complexes modulo meager complexes has the structure of an abelian group \(\overline{\text{KA}}(X)\), which is universal for additive Chern-Bott classes of acyclic complexes of Hermitian vector bundles.

Section 3 continues the story by defining Hermitian structures on objects of \(D^b(X)\). Namely, a Hermitian metric on an object \(F\in D^b(X)\) consists of an isomorphism \(E\cong F\) where \(E\) is a bounded complex of algebraic vector bundles and \(h\) is a smooth Hermitian metric on the complex of associated complex vector bundles. There is an equivalence relation on Hermitian metrics on a fixed object \(F\) and the authors define a category \(\overline{D}^b(X)\) as follows. Its objects are pairs \((F,h)\) of an object in \(D^b(X)\) and an equivalence class of metrics \(h\), and the morphisms between two pairs \((F,h)\) and \((F',h')\) are just the morphisms between \(F\) and \(F'\) in \(D^b(X)\). By definition, there exists a fully faithful forgetful functor \(\overline{D}^b(X)\rightarrow D^b(X)\). Roughly speaking, \(\overline{D}^b(X)\) turns out to be a principal fibered category over \(D^b(X)\) with structural group \(\overline{\text{KA}}(X)\) provided with a flat connection. After this result, some elementary constructions in \(\overline{D}^b(X)\) are studied, for example, derived tensor products and distinguished triangles. Another important construction is that of a Hermitian cone of a morphism in \(\overline{D}^b(X)\) which is an object defined only up to something called tight isomorphism.

In the next section the authors define Bott-Chern classes for isomorphisms and distinguished triangles in \(\overline{D}^b(X)\), for every additive genus. In the last section a similar construction is done for multiplicative genera. Furthermore, the authors define a category whose objects are smooth complex algebraic varieties and morphisms are pairs formed by a projective morphism \(f\) of smooth complex varieties together with a Hermitian structure on the relative tangent complex \(T_f\). It is the Hermitian cone construction which enables the authors to define a composition rule for these morphisms.

In Section 2 the authors define and characterize meager complexes, which are roughly speaking bounded acyclic complexes of Hermitian vector bundles whose Bott-Chern classes vanish. They then introduce the tight equivalence relation (whose definition involves meager complexes) between bounded complexes of Hermitian vector bundles and study the set of classes of tightly related complexes. This set is a monoid and the submonoid of acyclic complexes modulo meager complexes has the structure of an abelian group \(\overline{\text{KA}}(X)\), which is universal for additive Chern-Bott classes of acyclic complexes of Hermitian vector bundles.

Section 3 continues the story by defining Hermitian structures on objects of \(D^b(X)\). Namely, a Hermitian metric on an object \(F\in D^b(X)\) consists of an isomorphism \(E\cong F\) where \(E\) is a bounded complex of algebraic vector bundles and \(h\) is a smooth Hermitian metric on the complex of associated complex vector bundles. There is an equivalence relation on Hermitian metrics on a fixed object \(F\) and the authors define a category \(\overline{D}^b(X)\) as follows. Its objects are pairs \((F,h)\) of an object in \(D^b(X)\) and an equivalence class of metrics \(h\), and the morphisms between two pairs \((F,h)\) and \((F',h')\) are just the morphisms between \(F\) and \(F'\) in \(D^b(X)\). By definition, there exists a fully faithful forgetful functor \(\overline{D}^b(X)\rightarrow D^b(X)\). Roughly speaking, \(\overline{D}^b(X)\) turns out to be a principal fibered category over \(D^b(X)\) with structural group \(\overline{\text{KA}}(X)\) provided with a flat connection. After this result, some elementary constructions in \(\overline{D}^b(X)\) are studied, for example, derived tensor products and distinguished triangles. Another important construction is that of a Hermitian cone of a morphism in \(\overline{D}^b(X)\) which is an object defined only up to something called tight isomorphism.

In the next section the authors define Bott-Chern classes for isomorphisms and distinguished triangles in \(\overline{D}^b(X)\), for every additive genus. In the last section a similar construction is done for multiplicative genera. Furthermore, the authors define a category whose objects are smooth complex algebraic varieties and morphisms are pairs formed by a projective morphism \(f\) of smooth complex varieties together with a Hermitian structure on the relative tangent complex \(T_f\). It is the Hermitian cone construction which enables the authors to define a composition rule for these morphisms.

Reviewer: Pawel Sosna (Hamburg)

### MSC:

18E30 | Derived categories, triangulated categories (MSC2010) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

### Keywords:

derived category; coherent sheaves; meager complexes; Hermitian vector bundles; Bott-Chern classes; multiplicative genera
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\textit{J. I. Burgos Gil} et al., J. Math. Pures Appl. (9) 97, No. 5, 424--459 (2012; Zbl 1248.18011)

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