##
**Lectures on the arithmetic Riemann-Roch theorem. Notes taken by Shouwu Zhang.**
*(English)*
Zbl 0744.14016

Annals of Mathematics Studies. 127. Princeton, NJ: Princeton University Press,. x, 100 p. (1992).

In the course of the recent ten years, a rapid progress in applying methods from algebraic geometry to arithmetic problems has been achieved. In the early 1970’s the Russian mathematician S. Yu. Arakelov established an intersection theory for divisors on arithmetic surfaces, essentially by compactifying those surfaces with respect to the archimedean places and then introducing suitable hermitean structures on arithmetic line bundles. A decisive break through was made around 1982, when the author of the present book proved a Riemann-Roch theorem for arithmetic surfaces by amplifying Arakelov’s theory. Faltings’ innovation was based upon the construction of volume forms on the cohomology of hermitean bundles, and this allowed the extension of various techniques and theorems for complex surfaces to arithmetic surfaces. The main application of this discovery was the author’s spectacular proof of the Mordell conjecture [cf. the author, Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005)]. At about the same time, D. Quillen gave a similar construction, using Cauchy-Riemann operators on Riemann surfaces and the Ray-Singer theory of analytic torsion [cf. D. Quillen, Funct. Anal. Appl. 19, 31-34 (1985); translation from Funkts. Anal. Prilozh. 19, No. 1, 37-41 (1985; Zbl 0603.32016)].

These achievements lead to a growing interest in this area, in particular with regard to significant applications in mathematical physics (string theory), and in the course of the recent five years the theory of arithmetic varieties underwent an enormous development. After P. Deligne’s generalization of Faltings’ volume forms to a wider class of arithmetic varieties [cf. P. Deligne in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 93-117 (1987; Zbl 0629.14008)], H. Gillet and C. Soulé succeeded in establishing both an arithmetic intersection theory for general arithmetic varieties and a hermitean \(K\)-theory for them [cf. H. Gillet and C. Soulé, Publ. Math., Inst Hautes Étud. Sci. 72, 93-174 (1990)]. Together with J. M.Bismut they constructed, in the sequel, the determinant of cohomology of an arbitrary arithmetic variety. Finally, J. M. Bismut and G. Lebeau [cf. C. R. Acad. Sci., Paris, Sér. I 309, No. 7, 487-491 (1989; Zbl 0681.53034)], using probability-theoretic methods (stochastic integration), proved a Riemann-Roch theorem for closed immersions of arithmetic varieties, i.e., a Riemann-Roch-type result for the determinant of cohomology with respect to closed immersions.

In the spring of 1990 the author gave a course at Princeton University on these very recent results by Gillet-Soulé and Bismut-Lebeau. The present book grew out of these lectures, which were originally intended to explain the various techniques involved. However, the present text appears as an overworked and subsequently improved version of this lecture course. The aim of the book is now to present the arithmetic Riemann-Roch theory in a more coherent and technically simplified way. The author has managed, in an ingenious manner, to widely replace all the probability-theoretic arguments in the original approach — which are barely familiar to algebraic geometers and number theorists — by algebraic, complex-analytic and geometric considerations. This might help to make the topic more accessible to a wider class of interested readers, although the material still remains highly demanding and complicated.

Chapter I provides an introduction to the classical Riemann-Roch theorem for smooth morphisms of regular schemes over. The author explains \(K\)- groups and Chow groups, Chern classes, deformations of regular embeddings to normal cones, the reduction of the Riemann-roch problem to the case of projective bundles, and proves then the Riemann-Roch theorem (via flag schemes) in this situation. Precisely this strategy is used, later on, to derive the general Riemann-Roch theorem for arithmetic schemes.

In Chapter II the author develops the theory of arithmetic Chern classes (due to Gillet-Soulé) by a new method which is closer to the classical approach of Grothendieck. After a brief survey on Kähler manifolds and their Hodge theory, the construction of arithmetic \(K\)-groups, arithmetic Chow groups, and arithmetic Chern classes is carried out in a remarkably simplified way.

Chapter III contains the analytical framework that the author developed for the purpose of replacing the stochastic arguments by Bismut-Lebeau in proving the arithmetic Riemann-Roch theorem. The new ingredient consists in computing the asymptotic behavior of the diagonal values of the heat kernel of the Laplacian on certain Riemann manifolds. The presentation is based on a simplified ad-hoc procedure which avoids fancy technicalities, and suffices for all relevant cases later on.

This is used, in chapter IV, to prove a local index theorem for Dirac operators on compact Kähler manifolds. This chapter begins with a survey on Clifford algebras and Dirac operators, and closes, after the proof of the local index theorem for them, with the construction of super-Dirac operators as limits of ordinary Dirac operators in the sense of Clifford algebras.

Chapter V is devoted to the construction of direct images in arithmetic \(K\)-theory, i.e. of direct images with respect to smooth proper maps of arithmetic varieties. This is done by using new so-called number operators and various estimates for Laplacians, and provides an approach different from the original ones by Bismut-Gillet-Soulé and Quillen.

Chapter VI puts then everything together and culminates in presenting the author’s proof of the arithmetical Riemann-Roch theorem. Along the line which was followed in proving the classical Riemann-Roch theorem in chapter I, the problem is reduced to some deformation argument with respect to normal cones for closed embeddings, and to the Riemann-Roch problem for projective bundles. Both reduced problems are proved, again in a new way, by constructiong a modified Todd class and using subtle estimates for heat kernels of Laplacians.

The concluding chapter VII is of completing nature. It discusses the very recent theorem of Bismut-Vasserot [cf. J.-M. Bismut and E. Vasserot, Commun. Math. Phys. 125, No. 2, 355-367 (1989; Zbl 0687.32023)] on the behavior of the analytic torsion of ample line bundles on complex algebraic manifolds, and that from the point of view of the author’s approach developed in the present book. It turns out that this methods are nicely applicable to this kind of problems, too.

Altogether, this treatise provides a new approach to the arithmetic Riemann-Roch problem, and a widely algebraic-geometric method to solve it. The text combines introductory material with a detailed presentation of new methods and results in arithmetical algebraic geometry and hermitean geometry. Although the subject is rather complicated, the author succeeded in providing a comprehensible, fairly self-contained and methodically coherent account of it. This book will certainly become one of the most important standard references for both specialists and interested non-specialists in arithmetic geometry and its applications.

These achievements lead to a growing interest in this area, in particular with regard to significant applications in mathematical physics (string theory), and in the course of the recent five years the theory of arithmetic varieties underwent an enormous development. After P. Deligne’s generalization of Faltings’ volume forms to a wider class of arithmetic varieties [cf. P. Deligne in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 93-117 (1987; Zbl 0629.14008)], H. Gillet and C. Soulé succeeded in establishing both an arithmetic intersection theory for general arithmetic varieties and a hermitean \(K\)-theory for them [cf. H. Gillet and C. Soulé, Publ. Math., Inst Hautes Étud. Sci. 72, 93-174 (1990)]. Together with J. M.Bismut they constructed, in the sequel, the determinant of cohomology of an arbitrary arithmetic variety. Finally, J. M. Bismut and G. Lebeau [cf. C. R. Acad. Sci., Paris, Sér. I 309, No. 7, 487-491 (1989; Zbl 0681.53034)], using probability-theoretic methods (stochastic integration), proved a Riemann-Roch theorem for closed immersions of arithmetic varieties, i.e., a Riemann-Roch-type result for the determinant of cohomology with respect to closed immersions.

In the spring of 1990 the author gave a course at Princeton University on these very recent results by Gillet-Soulé and Bismut-Lebeau. The present book grew out of these lectures, which were originally intended to explain the various techniques involved. However, the present text appears as an overworked and subsequently improved version of this lecture course. The aim of the book is now to present the arithmetic Riemann-Roch theory in a more coherent and technically simplified way. The author has managed, in an ingenious manner, to widely replace all the probability-theoretic arguments in the original approach — which are barely familiar to algebraic geometers and number theorists — by algebraic, complex-analytic and geometric considerations. This might help to make the topic more accessible to a wider class of interested readers, although the material still remains highly demanding and complicated.

Chapter I provides an introduction to the classical Riemann-Roch theorem for smooth morphisms of regular schemes over. The author explains \(K\)- groups and Chow groups, Chern classes, deformations of regular embeddings to normal cones, the reduction of the Riemann-roch problem to the case of projective bundles, and proves then the Riemann-Roch theorem (via flag schemes) in this situation. Precisely this strategy is used, later on, to derive the general Riemann-Roch theorem for arithmetic schemes.

In Chapter II the author develops the theory of arithmetic Chern classes (due to Gillet-Soulé) by a new method which is closer to the classical approach of Grothendieck. After a brief survey on Kähler manifolds and their Hodge theory, the construction of arithmetic \(K\)-groups, arithmetic Chow groups, and arithmetic Chern classes is carried out in a remarkably simplified way.

Chapter III contains the analytical framework that the author developed for the purpose of replacing the stochastic arguments by Bismut-Lebeau in proving the arithmetic Riemann-Roch theorem. The new ingredient consists in computing the asymptotic behavior of the diagonal values of the heat kernel of the Laplacian on certain Riemann manifolds. The presentation is based on a simplified ad-hoc procedure which avoids fancy technicalities, and suffices for all relevant cases later on.

This is used, in chapter IV, to prove a local index theorem for Dirac operators on compact Kähler manifolds. This chapter begins with a survey on Clifford algebras and Dirac operators, and closes, after the proof of the local index theorem for them, with the construction of super-Dirac operators as limits of ordinary Dirac operators in the sense of Clifford algebras.

Chapter V is devoted to the construction of direct images in arithmetic \(K\)-theory, i.e. of direct images with respect to smooth proper maps of arithmetic varieties. This is done by using new so-called number operators and various estimates for Laplacians, and provides an approach different from the original ones by Bismut-Gillet-Soulé and Quillen.

Chapter VI puts then everything together and culminates in presenting the author’s proof of the arithmetical Riemann-Roch theorem. Along the line which was followed in proving the classical Riemann-Roch theorem in chapter I, the problem is reduced to some deformation argument with respect to normal cones for closed embeddings, and to the Riemann-Roch problem for projective bundles. Both reduced problems are proved, again in a new way, by constructiong a modified Todd class and using subtle estimates for heat kernels of Laplacians.

The concluding chapter VII is of completing nature. It discusses the very recent theorem of Bismut-Vasserot [cf. J.-M. Bismut and E. Vasserot, Commun. Math. Phys. 125, No. 2, 355-367 (1989; Zbl 0687.32023)] on the behavior of the analytic torsion of ample line bundles on complex algebraic manifolds, and that from the point of view of the author’s approach developed in the present book. It turns out that this methods are nicely applicable to this kind of problems, too.

Altogether, this treatise provides a new approach to the arithmetic Riemann-Roch problem, and a widely algebraic-geometric method to solve it. The text combines introductory material with a detailed presentation of new methods and results in arithmetical algebraic geometry and hermitean geometry. Although the subject is rather complicated, the author succeeded in providing a comprehensible, fairly self-contained and methodically coherent account of it. This book will certainly become one of the most important standard references for both specialists and interested non-specialists in arithmetic geometry and its applications.

Reviewer: W.Kleinert (Berlin)

### MSC:

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

14C40 | Riemann-Roch theorems |

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

19E08 | \(K\)-theory of schemes |