# zbMATH — the first resource for mathematics

On the arithmetic Chern character. (English. French summary) Zbl 1326.14064
Summary: We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern class of the sequence and its Chern character with support on the finite fibers.
Next, we compute these classes in the situation encountered by the second author when proving a “Kodaira vanishing theorem” for arithmetic surfaces.
##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
Full Text:
##### References:
 [1] Bott (R.), Chern (S.S.).— Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114, p. 71-112 (1965). · Zbl 0148.31906 [2] Bismut (J.-M.), Gillet (H.), Soulé (C.).— Analytic torsion and holomorphic determinant bundles I, II, III. Comm. Math. Physics 115, p. 49-78, p. 79-126, p. 301-351 (1988). · Zbl 0651.32017 [3] Fulton W..— Intersection Theory. Springer 1984. · Zbl 0885.14002 [4] Gillet (H.), Soulé (C.).— Characteristic classes for algebraic vector bundles with hermitian metrics I, II. Annals of Math. 131, p. 163-203, p. 205-238 (1990). · Zbl 0715.14006 [5] Gillet (H.), Soulé (C.).— An arithmetic Riemann-Roch theorem. Invent. Math. 110, p. 473-543 (1992). · Zbl 0777.14008 [6] Gillet (H.), Soulé (C.).— Direct images in non-archimedean Arakelov theory. Annales de l’institut Fourier, 50, p. 363-399 (2000). · Zbl 0969.14015 [7] Soulé (C.).— A vanishing theorem on arithmetic surfaces. Invent. Math. 116, no. 1-3, p. 577-599 (1994). · Zbl 0834.14013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.