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**Singular Bott-Chern classes and the arithmetic Grothendieck Riemann Roch theorem for closed immersions.**
*(English)*
Zbl 1192.14019

To every hermitian vector bundle on a complex manifold, the Chern-Weil theory associates a family of closed characteristic forms which represent the characteristic classes of the vector bundle. Unfortunately, this construction is not compatible with exact sequences of hermitian vector bundles. For instance, let \(\overline{\varepsilon}: 0\to \overline{E}'\to \overline{E}\to \overline{E}''\to 0\) be a short exact sequence of hermitian holomorphic vector bundles, then the alternating sum of Chern characteristic forms \(\text{ch}(\overline{E}')-\text{ch}(\overline{E})+\text{ch}(\overline{E}'')\) is not equal to zero in general. It is well known that the Bott-Chern secondary characteristic class \(\widetilde{\text{ch}}(\overline{\varepsilon})\) measures the difference; precisely speaking, it satisfies the differential equation

\[ \frac{\overline{\partial}\partial}{2\pi i}\widetilde{\text{ch}}(\overline{\varepsilon})=\text{ch}(\overline{E}')-\text{ch}(\overline{E})+\text{ch}(\overline{E}''). \tag{\(*\)} \]

There are three different ways of defining the Bott-Chern secondary characteristic classes. The first one is the original definition of R. Bott and S. S. Chern [Matematika, Moskva 14, No.2, 117–154 (1970; Zbl 0208.35101)]. The second one was introduced by H. Gillet and C. Soulé [Bull. Am. Math. Soc., New Ser. 15, 209–212 (1986; Zbl 0618.58035)]. This second definition was used by J.-M. Bismut, H. Gillet and C. Soulé [Commun. Math. Phys. 115, No. 1, 49–78; 79–126; No. 2, 301–351 (1988; Zbl 0651.32017)] to prove that the Bott-Chern secondary characteristic classes have a family of axiomatic characterizations:

Also, in J.-M. Bismut, H. Gillet and C. Soulé [Commun. Math. Phys. 115, No.1, 49-78; 79-126; No.2, 301-351 (1988; Zbl 0651.32017)], a third definition of Bott-Chern classes was given based on the theory of superconnections. To solve a similar differential equation as in (*) with respect to the resolution of hermitian vector bundle associated to a closed immersion of complex manifolds, the singular Bott-Chern current was defined by J.-M. Bismut, H. Gillet and C. Soulé [Duke Math. J. 60, No. 1, 255–284 (1990; Zbl 0697.58005)]. Precisely, let \(i: Y\to X\) be a closed immersion of complex manifolds with hermitian normal bundle \(\overline{N}\). Suppose that \(\overline{\eta}\) is a hermitian vector bundle on \(Y\) and that \(\overline{\xi}.\) is a complex of hermitian vector bundles providing a resolution of \(i_*\overline{\eta}\) on \(X\) whose metrics satisfy Bismut assumption (A). Then there exists a singular current \(T(\overline{\xi}.)\) which is a sum of \((p,p)\)-type currents satisfying the differential equation

\[ \frac{\overline{\partial}\partial}{2\pi i}T(\overline{\xi}.) = i_*(\text{ch}(\overline{\eta})\text{Td}^{-1}(\overline{N}))-\sum_k(-1)^k\text{ch}(\overline{\xi}_k). \]

Its construction is also based on the theory of superconnections and therefore it can be viewed as a generalization of the third definition of Bott-Chern secondary characteristic classes. This singular Bott-Chern current plays a crucial role in the proof of Gillet and Soulé’s arithmetic Riemann-Roch theorem for closed immersions. Later, Zha gave another definition of singular Bott-Chern currents in his Ph.D thesis and used it to give a proof of a different version of the arithmetic Riemann-Roch theorem. Zha’s definition is analogous to Bott and Chern’s original definition of Bott-Chern secondary characteristic classes.

The main results of the article under review are the following. Firstly, the authors give an axiomatic definition of a theory of singular Bott-Chern classes which can be viewed as a generalization of the second definition of the Bott-Chern secondary characteristic classes. At the same time, they also study all possible different ways of defining such theories of singular Bott-Chern classes. The main tool used by the authors was the construction of the deformation to the normal cone. Secondly, the authors prove that by adding an additional condition of homogeneity to the usual axiomatic characterizations (i), (ii) and (iii) of Bott-Chern classes, the theory of singular Bott-Chern classes can be determined uniquely. This fact can be used to show that the theories of singular Bott-Chern classes introduced by Bismut, Gillet and Soulé, and by Zha are actually agree with each other in certain sense. At last, the authors give a proof of the arithmetic Riemann-Roch theorem for closed immersions which unifies Bismut, Gillet and Soulé’s proof and Zha’s proof in the same picture.

As a byproduct of this study, the authors obtain two results of independent interest. The first is a Poincaré lemma for the complex of currents with fixed wave front set, the second is that certain direct images of Bott-Chern classes are closed.

\[ \frac{\overline{\partial}\partial}{2\pi i}\widetilde{\text{ch}}(\overline{\varepsilon})=\text{ch}(\overline{E}')-\text{ch}(\overline{E})+\text{ch}(\overline{E}''). \tag{\(*\)} \]

There are three different ways of defining the Bott-Chern secondary characteristic classes. The first one is the original definition of R. Bott and S. S. Chern [Matematika, Moskva 14, No.2, 117–154 (1970; Zbl 0208.35101)]. The second one was introduced by H. Gillet and C. Soulé [Bull. Am. Math. Soc., New Ser. 15, 209–212 (1986; Zbl 0618.58035)]. This second definition was used by J.-M. Bismut, H. Gillet and C. Soulé [Commun. Math. Phys. 115, No. 1, 49–78; 79–126; No. 2, 301–351 (1988; Zbl 0651.32017)] to prove that the Bott-Chern secondary characteristic classes have a family of axiomatic characterizations:

- (i)
- the differential equation (*);
- (ii)
- functoriality; and
- (iii)
- the vanishing of the Bott-Chern class of an orthogonally split exact sequence.

Also, in J.-M. Bismut, H. Gillet and C. Soulé [Commun. Math. Phys. 115, No.1, 49-78; 79-126; No.2, 301-351 (1988; Zbl 0651.32017)], a third definition of Bott-Chern classes was given based on the theory of superconnections. To solve a similar differential equation as in (*) with respect to the resolution of hermitian vector bundle associated to a closed immersion of complex manifolds, the singular Bott-Chern current was defined by J.-M. Bismut, H. Gillet and C. Soulé [Duke Math. J. 60, No. 1, 255–284 (1990; Zbl 0697.58005)]. Precisely, let \(i: Y\to X\) be a closed immersion of complex manifolds with hermitian normal bundle \(\overline{N}\). Suppose that \(\overline{\eta}\) is a hermitian vector bundle on \(Y\) and that \(\overline{\xi}.\) is a complex of hermitian vector bundles providing a resolution of \(i_*\overline{\eta}\) on \(X\) whose metrics satisfy Bismut assumption (A). Then there exists a singular current \(T(\overline{\xi}.)\) which is a sum of \((p,p)\)-type currents satisfying the differential equation

\[ \frac{\overline{\partial}\partial}{2\pi i}T(\overline{\xi}.) = i_*(\text{ch}(\overline{\eta})\text{Td}^{-1}(\overline{N}))-\sum_k(-1)^k\text{ch}(\overline{\xi}_k). \]

Its construction is also based on the theory of superconnections and therefore it can be viewed as a generalization of the third definition of Bott-Chern secondary characteristic classes. This singular Bott-Chern current plays a crucial role in the proof of Gillet and Soulé’s arithmetic Riemann-Roch theorem for closed immersions. Later, Zha gave another definition of singular Bott-Chern currents in his Ph.D thesis and used it to give a proof of a different version of the arithmetic Riemann-Roch theorem. Zha’s definition is analogous to Bott and Chern’s original definition of Bott-Chern secondary characteristic classes.

The main results of the article under review are the following. Firstly, the authors give an axiomatic definition of a theory of singular Bott-Chern classes which can be viewed as a generalization of the second definition of the Bott-Chern secondary characteristic classes. At the same time, they also study all possible different ways of defining such theories of singular Bott-Chern classes. The main tool used by the authors was the construction of the deformation to the normal cone. Secondly, the authors prove that by adding an additional condition of homogeneity to the usual axiomatic characterizations (i), (ii) and (iii) of Bott-Chern classes, the theory of singular Bott-Chern classes can be determined uniquely. This fact can be used to show that the theories of singular Bott-Chern classes introduced by Bismut, Gillet and Soulé, and by Zha are actually agree with each other in certain sense. At last, the authors give a proof of the arithmetic Riemann-Roch theorem for closed immersions which unifies Bismut, Gillet and Soulé’s proof and Zha’s proof in the same picture.

As a byproduct of this study, the authors obtain two results of independent interest. The first is a Poincaré lemma for the complex of currents with fixed wave front set, the second is that certain direct images of Bott-Chern classes are closed.

Reviewer: Shun Tang (Orsay)