## Hermitian vector bundles and extension groups on arithmetic schemes. I: Geometry of numbers.(English)Zbl 1187.14027

The goal of Arakelov Theory is to produce (deep) arithmetic results by combining tools from algebraic geometry over Spec $$\mathbb Z$$ with those from complex geometry. Indeed, it was first invented with the hope of proving the Mordell conjecture on the finiteness of the set of rational points on a curve of genus at least two over a number field.
(This dream was eventually realized by G. Faltings [Invent. Math. 73, 349–366 (1983; Zbl 0588.14026)].) When interpreted with hindsight, the simplest example of the Arakelov philosophy is Minkowski’s geometry of numbers, which has been described beautifully by L. Szpiro [Astérisque 127, 11–28 (1985; Zbl 1182.11029)].
In the paper under review, the authors begin a sequence of investigations of arithmetic extension groups. Let $$X$$ be an arithmetic scheme — i.e., a separated scheme that is flat and of finite type over Spec $$\mathbb Z$$, and such that the associated complex variety $$X_{\mathbb C}$$ is smooth. If $$F$$ and $$G$$ are locally free coherent $${\mathcal O}_X$$-modules (vector bundles on $$X$$), an arithmetic extension $$({\mathcal E}, s)$$ of $$F$$ by $$G$$ is an extension of $${\mathcal O}_X$$-modules $${\mathcal E}: 0 \to G \to E \to F \to 0$$ together with a complex-conjugation-invariant $${\mathcal C}^\infty$$-splitting $$s: F_{\mathbb C} \to E_{\mathbb C}$$ of the associated extension of complex vector bundles over the manifold of complex points $$X(\mathbb C)$$. The set of isomorphism classes of arithmetic extensions, denoted $$\widehat{\text{Ext}}^1_X(F, G)$$, becomes an Abelian group when equipped with the Baer sum operation.
Arithmetic extensions arise naturally in Arakelov geometry via admissible extensions: if $$\overline{F}$$ and $$\overline{G}$$ are Hermitian vector bundles on $$X$$, then an admissible extension of $$\overline{F}$$ by $$\overline{G}$$ is an exact sequence of Hermitian vector bundles $$\overline{{\mathcal E}}: 0 \to \overline{G} \to \overline{E} \to \overline{F} \to 0$$ for which the metrics on $$\overline{F}$$ and $$\overline{G}$$ agree with the quotient and subspace metrics induced by $$\overline{E}$$, respectively. The authors observe that, in this context, isomorphism classes of arithmetic extensions of $$F$$ by $$G$$ are in bijective correspondence with isomorphism classes of admissible extensions of $$\overline{F}$$ by $$\overline{G}$$. As an important special case, if $${\mathcal O}_K$$ is the ring of integers in a number field $$K$$ and $$X = \text{Spec} \, {\mathcal O}_K$$ is the associated arithmetic curve, then any locally free coherent $${\mathcal O}_X$$-module can be endowed with a Hermitian metric (by choosing a basis for the associated complex vector space), and hence every arithmetic extension of $${\mathcal O}_X$$-modules arises as an admissible extension.
The authors explain how the group of arithmetic extensions $$\widehat{\text{Ext}}^1_X(F, G)$$ itself arises as an extension of the “classical extension group” $$\text{Ext}^1_{{\mathcal O}_X}(F, G)$$ by a group of analytic type. They give various functorial properties of this group, and discuss the relationship between their notion of arithmetic extension and the “arithmetic torsors” of A. Chambert-Loir and Y. Tschinkel [Prog. Math. 199, 37–70 (2001; Zbl 1001.14005)]. The later part of the paper is devoted to studying slopes of arithmetic extensions over arithmetic curves. A number of classical results in the geometry of numbers and the reduction theory of quadratic forms are examined in this framework. For example, Theorem 4.3.1 exhibits the phenomenon of “bounded distortion” for the splitting of lattices: any $${\mathcal O}_K$$-lattice in $$\mathbb C^n$$ differs from a rectangular lattice by a bounded amount, depending only on $$K$$ and $$n$$.
The authors plan to extend their results in two subsequent papers to discuss the arithmetic Atiyah extension, the arithmetic Hodge extension, and the arithmetic Schwarz extension. A sketch of these ideas is given in the introduction to the present paper.

### MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11H31 Lattice packing and covering (number-theoretic aspects) 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)

### Citations:

Zbl 0588.14026; Zbl 1001.14005; Zbl 1182.11029
Full Text:

### References:

 [1] Baer, R., Erweiterung von gruppen und ihren isomorphismen, Math. Z., 38, 375-416, (1934) · JFM 60.0079.04 [2] Banaszczyk, W., New bounds in some transference theorems in the geometry of numbers, Math. ann., 296, 4, 625-635, (1993) · Zbl 0786.11035 [3] Banaszczyk, W., Inequalities for convex bodies and polar reciprocal lattices in $$\mathbb{R}^n$$, Discrete comput. geom., 13, 2, 217-231, (1995) · Zbl 0824.52011 [4] Berthelot, P.; Breen, L.; Messing, W., Théorie de Dieudonné cristalline. II, Lecture notes in math., vol. 930, (1982), Springer-Verlag Berlin · Zbl 0516.14015 [5] Bost, J.-B.; Gillet, H.; Soulé, C., Heights of projective varieties and positive Green forms, J. amer. math. soc., 7, 4, 903-1027, (1994) · Zbl 0973.14013 [6] J.-B. Bost, K. Künnemann, Hermitian vector bundles and extension groups on arithmetic schemes. II, The arithmetic Atiyah class, in: X. Ma (Ed.), From probability to geometry, Volume dedicated to J.M. Bismut for his 60th birthday, Astérisque, in press [7] J.-B. Bost, K. Künnemann, Hermitian vector bundles and extension groups on arithmetic schemes. III, in preparation [8] Borel, A., Introduction aux groupes arithmétiques, (1969), Hermann & Cie Paris, 125 p · Zbl 0186.33202 [9] Borek, T., Successive minima and slopes of Hermitian vector bundles over number fields, J. number theory, 113, 2, 380-388, (2005) · Zbl 1100.14513 [10] Bost, J.-B., Périodes et isogénies des variétés abéliennes sur LES corps de nombres (d’après D. masser et G. Wüstholz), Astérisque, 795, 4, 237, 115-161, (1996), Séminaire Bourbaki, vol. 1994/1995 · Zbl 0936.11042 [11] Cartan, H.; Eilenberg, S., Homological algebra, Princeton math. ser., vol. 19, (1956), Princeton University Press XV Princeton, NJ, 390 p · Zbl 0075.24305 [12] () [13] Chambert-Loir, A.; Tschinkel, Y., Torseurs arithmétiques et espaces fibrés, (), 37-70 · Zbl 1001.14005 [14] Conrad, B., Grothendieck duality and base change, Lecture notes in math., vol. 1750, (2000), Springer-Verlag Berlin · Zbl 0992.14001 [15] Coxeter, H.S.M., Extreme forms, Canadian J. math., 3, 391-441, (1951) · Zbl 0044.04201 [16] Conway, J.H.; Sloane, N.J.A., The cell structures of certain lattices, (), 71-107 · Zbl 0738.52014 [17] Conway, J.H.; Sloane, N.J.A., Low-dimensional lattices. VI. voronoĭ reduction of three-dimensional lattices, Proc. R. soc. lond. ser. A, 436, 1896, 55-68, (1992) · Zbl 0747.11027 [18] Conway, J.H.; Sloane, N.J.A., Sphere packings, lattices and groups, Grundlehren math. wiss., vol. 290, (1999), Springer New York, NY, lxxiv, 703 p · Zbl 0915.52003 [19] Deligne, P., Équations différentielles à points singuliers réguliers, Lecture notes in math., vol. 163, (1970), Springer-Verlag Berlin · Zbl 0244.14004 [20] Deligne, P., Exposé XVII: cohomologie à support propre, (), 250-461 [21] Eilenberg, S.; MacLane, S., Group extensions and homology, Ann. math., 43, 757-831, (1942) · Zbl 0061.40602 [22] Ford, L.R., Fractions, Amer. math. monthly, 45, 586-601, (1938) · Zbl 0019.39505 [23] Griffiths, P.; Harris, J., Principles of algebraic geometry, Pure appl. math., (1978), Wiley-Interscience New York · Zbl 0408.14001 [24] Grayson, D.R., Reduction theory using semistability, Comment. math. helv., 59, 4, 600-634, (1984) · Zbl 0564.20027 [25] Graftieaux, P., Formal groups and the isogeny theorem, Duke math. J., 106, 1, 81-121, (2001) · Zbl 1064.14045 [26] Griffiths, P.A., The extension problem in complex analysis. II: embeddings with positive normal bundle, Amer. J. math., 88, 366-446, (1966) · Zbl 0147.07502 [27] Griffiths, P.A., Hermitian differential geometry, Chern classes, and positive vector bundles, (), 185-251 · Zbl 0201.24001 [28] Grothendieck, A., On the de Rham cohomology of algebraic varieties, Inst. hautes études sci. publ. math., 29, 95-103, (1966) · Zbl 0145.17602 [29] Grünbaum, B., Convex polytopes, Pure appl. math., vol. 16, (1967), Interscience Publishers John Wiley & Sons, Inc. New York, with the cooperation of Victor Klee, M.A. Perles and G.C. Shephard · Zbl 0163.16603 [30] Gillet, H.; Soulé, C., Arithmetic intersection theory, Inst. hautes études sci. publ. math., 72, 93-174, (1990) · Zbl 0741.14012 [31] Gillet, H.; Soulé, C., On the number of lattice points in convex symmetric bodies and their duals, Israel J. math., 74, 2-3, 347-357, (1991) · Zbl 0752.52008 [32] Gillet, H.; Soulé, C., An arithmetic Riemann-Roch theorem, Invent. math., 110, 3, 473-543, (1992) · Zbl 0777.14008 [33] Hartshorne, R., Residues and duality, (), with an appendix by P. Deligne · Zbl 0196.24301 [34] Hartshorne, R., Algebraic geometry, Grad. texts in math., vol. 52, (1977), Springer-Verlag New York · Zbl 0367.14001 [35] Illusie, L., Exposé II: existence de résolutions globales, (), 160-221 [36] Illusie, L., Frobenius et dégénérescence de Hodge, () [37] Kitaoka, Y., Arithmetic of quadratic forms, Cambridge tracts in math., vol. 106, (1993), Cambridge University Press Cambridge, x, 268 p · Zbl 0785.11021 [38] Lang, S., Introduction to Arakelov theory, (1988), Springer-Verlag New York · Zbl 0667.14001 [39] Lagarias, J.C.; Lenstra, H.W.; Schnorr, C.-P., Korkin-zolotarev bases and successive minima of a lattice and its reciprocal lattice, Combinatorica, 10, 4, 333-348, (1990) · Zbl 0723.11029 [40] Martinet, J., Perfect lattices in Euclidean spaces, Grundlehren math. wiss., vol. 327, (2003), Springer-Verlag Berlin · Zbl 1017.11031 [41] Milnor, J.; Husemoller, D., Symmetric bilinear forms, Ergeb. math. grenzgeb., vol. 73, (1980), Springer-Verlag Berlin-Heidelberg-New York [42] Mac Lane, S., Homology, Classics math., (1995), Springer-Verlag Berlin · Zbl 0818.18001 [43] Mochizuki, S., A theory of ordinary p-adic curves, Publ. res. inst. math. sci., 32, 6, 957-1152, (1996) · Zbl 0879.14009 [44] S. Mochizuki, The Hodge-Arakelov theory of elliptic curves: Global discretization of local Hodge theories, RIMS Preprint Nos. 1255, 1256, October, 1999 [45] Neukirch, J., Algebraic number theory, Grundlehren math. wiss., vol. 322, (1999), Springer-Verlag Berlin, Translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder [46] Rademacher, H., Lectures on elementary number theory, (), IX, 146 p · Zbl 0119.27803 [47] Selling, E., Über die binären und ternären quadratischen formen, J. reine angew. math., 77, 143-229, (1874) · JFM 06.0128.01 [48] Revêtements étales et groupe fondamental (SGA 1), Doc. math. (Paris), Lecture notes in math., vol. 224, (1971), Springer Berlin, Séminaire de géométrie algébrique du Bois Marie 1960-1961, directed by A. Grothendieck, with two papers by M. Raynaud, updated and annotated reprint of the 1971 original: [49] Soulé, C., Hermitian vector bundles on arithmetic varieties, (), 383-419 · Zbl 0926.14011 [50] Stuhler, U., Eine bemerkung zur reduktionstheorie quadratischer formen, Arch. math. (basel), 27, 6, 604-610, (1976) · Zbl 0338.10024 [51] Szpiro, L., Degrés, intersections, hauteurs, (), 11-28 · Zbl 1182.11029 [52] Tougeron, J.-C., Idéaux de fonctions différentiables, Ergeb. math. grenzgeb., vol. 71, (1972), Springer-Verlag Berlin · Zbl 0251.58001 [53] Thomason, R.W.; Trobaugh, Th., Higher algebraic K-theory of schemes and of derived categories, (), 247-435 [54] van der Waerden, B.L., Die reduktionstheorie der positiven quadratischen formen, Acta math., 96, 265-309, (1956) · Zbl 0072.03601 [55] Voronoï, G., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. I. premier mémoire : sur quelques propriétés des formes positives parfaites, J. reine angew. math., 133, 97-178, (1908) · JFM 38.0261.01 [56] Voronoï, G., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. III. deuxième mémoire : recherches sur LES paralléloèdres primitifs - seconde partie - domaines de formes quadratiques correspondant aux différents types de paralléloèdres primitifs, J. reine angew. math., 136, 67-181, (1909) · JFM 40.0267.17 [57] Weibel, C.A., An introduction to homological algebra, Camb. stud. adv. math., vol. 38, (1994), Cambridge University Press Cambridge · Zbl 0797.18001 [58] Witt, E., Spiegelungsgruppen und aufzählung halbeinfacher liescher ringe, Abh. math. semin. hansische univ., 14, 289-322, (1941) · JFM 67.0077.03 [59] Ziegler, G., Lectures on polytopes, Grad. texts in math., vol. 152, (1995), Springer-Verlag New York · Zbl 0823.52002
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