×

zbMATH — the first resource for mathematics

Heights of projective varieties and positive Green forms. (English) Zbl 0973.14013
From the introduction: The purpose of this paper is to study analogs of some basic concepts and results of projective geometry in the context of Arakelov geometry [S. J. Arakelov, Proc. int. Congr. Math., Vancouver 1974, Vol 1, 405-408 (1975; Zbl 0351.14003); see also H. Gillet and Ch. Soulé, Publ. Math., Inst. Hautes Étud. Sci. 72, 93-174 (1990; Zbl 0741.14012)]. As was first noticed by G. Faltings in his work on diophantine approximation for abelian varieties [Ann. Math. (2) 133, 549-576 (1991; Zbl 0734.14007)], higher dimensional arithmetic intersection theory can be used to define the height of any (closed integral) projective subscheme \(X\subset \mathbb{P}^N\), where \(\mathbb{P}^N\) is the \(N\)-dimensional projective space over \(\mathbb{Z}\) (or more generally over the integers in a number field). The Faltings height \(h_F(X)\), which is a nonnegative real number, is defined in a similar fashion to the degree of a projective variety over a field. That is, \(h_F(X)\) is the intersection, in the sense of Gillet and Soulé [loc. cit.], of the fundamental class of \(X\) with the first Chern class of the canonical hermitian line bundle on \(\mathbb{P}^N\), raised to the power \(d=\dim(X)\).
In this paper we propose a slightly different definition of the height of \(X\). Namely we denote by \(h(X)\) the intersection of the fundamental class of \(X\) with the \(d\)-th Chern class of the canonical quotient hermitian bundle on \(\mathbb{P}^N\). We prove that \(h(X)\) is nonnegative and smaller than \(h_F(X)\) (except when \(d\leq 1\) or when the generic fiber of \(X\) is empty, in which case \(h(X)=h_F(X))\). Furthermore \(h(X)=0\) if and only if \(X\) is a linear subspace \(\mathbb{P}^{d-1}\subset \mathbb{P}^N\) defined by the vanishing of \(N+1-d\) standard coordinates (theorem 5.2.3).
We obtain several results on the heights of projective varieties, which are inspired by the analogy between heights and degrees. For instance we compute the height of the join of two varieties (proposition 4.2.2) and the behavior of the height under linear projection (3.3.2). We give several proofs of the following arithmetic Bézout theorem. Assume \(X \subset \mathbb{P}^N\) and \(Y\subset\mathbb{P}^N\) are integral projective varieties which meet properly on the generic fiber of \(\mathbb{P}^N\). Their intersection cycle \(X.Y\) can then be defined using W. Fulton’s method [“Intersection theory”, 2nd ed. (Berlin 1998; Zbl 0885.14002)]. It is well defined up to the addition of a cycle linearly equivalent to zero in the closed fibers of \(\mathbb{P}^N\) over \(\mathbb{Z}\), and its height \(h(X.Y)\) (defined by extending by linearity the definition for integral subschemes) does not depend on the choice of representative for \(X.Y\). Denote by \(\deg_\mathbb{Q} (X)\) and \(\deg_\mathbb{Q} (Y)\) the degrees in \(\mathbb{P}^N_\mathbb{Q}\) of the generic fibers of \(X\) and \(Y\) respectively. Then we have \[ h(X.Y)\leq h(X)\deg_\mathbb{Q} (Y)+\deg_\mathbb{Q} (X)h(Y)+ c\deg_\mathbb{Q}(X)\deg_\mathbb{Q} (Y), \] where the constant \(c\) depends only on \(N\), \(\dim(X)\), and \(\dim(Y)\). We give three different proofs of this inequality (theorems 4.2.3, 5.4.4, and 6.1.1), the smallest value of \(c\) being the one in theorem 5.4.4 (we believe that \(c\) can be taken equal to zero, but we cannot prove it except when \(X\) or \(Y\) is a linear subspace).
In transcendental number theory, especially in the work of Nesterenko, and Philippon, another definition of height has been known for some time, which does not use Arakelov theory, and cases of the Bezout theorem have been proved in that context. Namely the height of \(X\subset\mathbb{P}^N\) is defined to be the height of its Chow form, which is a point in a large projective space. The comparison between this definition and \(h_F(X)\) was made by Ch. Soulé [Astérisque 198-200, 355-371 (1991; Zbl 0756.14014)] and P. Philippon [Publ. Math., Inst. Hautes Étud. Sci. 64, 5-52 (1986; Zbl 0615.10044)]. We extend their result to more general Chow forms and not necessarily standard metrics (theorem 4.3.2). As a byproduct we get the following result. Let \(R\) be the resultant of \(N+1\) homogeneous polynomials of degrees \(d_0,\dots,d_N\) in \(N+1\) variables. This is a multihomogeneous polynomial with integral coefficients of multidegree \((\delta_0, \dots,\delta_N)\), where \(\delta_i= \sum^N_{j=0 \atop j\neq i}d_j\). Its variables are the coefficients of the “generic” homogeneous polynomials of degrees \(d_0,\dots, d_N\) in \(N+1\) variables. So \(R\) can be viewed as an element of \(\bigotimes^r_{i=0} S^{\delta_i} (S^{d_i} \mathbb{C}^{\check N+1})^\vee\). Equip this vector space with the hermitian norm \(\|\cdot \|_{\text{Herm}}\) induced by the standard hermitian structure on \(\mathbb{C}^{N+1}\). We prove in lemma 4.3.4 that \[ \log\|R\|_{\text{Herm}}= {1\over 2}\left( \prod^N_{i=0} d_i \right) \cdot\left( (N+1)\left( 1+{1\over 2}+ \cdots+ {1\over N}\right)-N\right) +\varepsilon (d_0,\dots, d_N), \] where \[ \bigl|\varepsilon (d_0, \dots,d_N) \bigr|\leq {1\over 2}N\left( \prod^N_{i=0} d_i\right) \cdot\sum^N_{i=0} {1 \over d_i}\log(d_i+1). \] We also evaluate the size of \(R\) for other norms (theorem 4.3.8). Our main analytic tool is the existence of “positive Green forms” for effective cycles \(Z\) on a complex manifold \(X\). By this we mean a positive \(C^\infty\) form \(\eta\) on \(X-|Z|\) which is locally \(L^1\) on \(X\) and such that the corresponding current \(g=[\eta]\) on \(X\) is a Green current for \(Z\), i.e., such that \(dd^c g+\delta_Z\) is \(C^\infty\) on \(X\) (where \(\delta_Z\) the current given by integration on \(Z\)). An example of such a positive Green form is the Levine form, familiar to Nevanlinna theory, when \(X\) is a complex projective space and \(Z\) a linear subspace. The positivity of these Levine forms has several interesting consequences (proposition 1.4.2, proposition 4.1.3). More generally we give conditions for a given effective cycle (respectively all effective cycles) on \(X\) to have a positive Green form (propositions 6.2.1, 6.2.2, and 6.2.3), and a counterexample showing that some complex manifolds admit effective cycles with no positive Green forms (6.3).

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
11G50 Heights
14G25 Global ground fields in algebraic geometry
PDF BibTeX Cite
Full Text: DOI
References:
[1] Allen B. Altman and Steven L. Kleiman, Joins of schemes, linear projections, Compositio Math. 31 (1975), no. 3, 309 – 343. · Zbl 0337.14004
[2] S. Ju. Arakelov, An intersection theory for divisors on an arithmetic surface, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1179 – 1192 (Russian).
[3] S. J. Arakelov, Theory of intersections on the arithmetic surface, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 405 – 408. · Zbl 0351.14003
[4] Bernard Beauzamy, Enrico Bombieri, Per Enflo, and Hugh L. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), no. 2, 219 – 245. · Zbl 0729.30004
[5] Jean-Michel Bismut and Éric Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), no. 2, 355 – 367. · Zbl 0687.32023
[6] André Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. (3) 73 (1956), 157 – 202 (French). · Zbl 0073.37503
[7] Eduard Bod’a and Wolfgang Vogel, On system of parameters, local intersection multiplicity and Bezout’s theorem, Proc. Amer. Math. Soc. 78 (1980), no. 1, 1 – 7. · Zbl 0436.13012
[8] E. Bombieri and J. Vaaler, On Siegel’s lemma, Invent. Math. 73 (1983), no. 1, 11 – 32. , https://doi.org/10.1007/BF01393823 E. Bombieri and J. Vaaler, Addendum to: ”On Siegel’s lemma”, Invent. Math. 75 (1984), no. 2, 377. · Zbl 0533.10030
[9] A. Borel and R. Remmert, Über kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1961/1962), 429 – 439 (German). · Zbl 0111.18001
[10] Jean-Benoît Bost, Green’s currents and height pairing on complex tori, Duke Math. J. 61 (1990), no. 3, 899 – 912. · Zbl 0742.14006
[11] -, Théorie de l’intersection et théorème de Riemann-Roch arithmétiques, Séminaire Bourbaki no 731, 1990/1991, Astérisque 201-203 (1991), 43-88.
[12] Jean-Benoît Bost, Henri Gillet, and Christophe Soulé, Un analogue arithmétique du théorème de Bézout, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 11, 845 – 848 (French, with English summary). · Zbl 0756.14012
[13] Raoul Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965), 71 – 112. · Zbl 0148.31906
[14] Wei-Liang Chow and B. L. van der Waerden, Zur algebraischen Geometrie. IX, Math. Ann. 113 (1937), no. 1, 692 – 704 (German). · Zbl 0016.04004
[15] Maurizio Cornalba and Phillip Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties, Invent. Math. 28 (1975), 1 – 106. · Zbl 0293.32026
[16] Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387 – 424. · Zbl 0559.14005
[17] Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549 – 576. · Zbl 0734.14007
[18] Gerd Faltings, Lectures on the arithmetic Riemann-Roch theorem, Annals of Mathematics Studies, vol. 127, Princeton University Press, Princeton, NJ, 1992. Notes taken by Shouwu Zhang. · Zbl 0744.14016
[19] Gerd Faltings and Gisbert Wüstholz , Rational points, Aspects of Mathematics, E6, Friedr. Vieweg & Sohn, Braunschweig; distributed by Heyden & Son, Inc., Philadelphia, PA, 1984. Papers from the seminar held at the Max-Planck-Institut für Mathematik, Bonn, 1983/1984. · Zbl 0753.14019
[20] John Fogarty, Truncated Hilbert functors, J. Reine Angew. Math. 234 (1969), 65 – 88. · Zbl 0197.17101
[21] William Fulton, Rational equivalence on singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 147 – 167. · Zbl 0332.14002
[22] -, Intersection theory, Ergeb. Math. Grenzgeb. (3), Band 2, Springer-Verlag, Berlin, Heidelberg, and New York, 1984. · Zbl 0541.14005
[23] William Fulton, Introduction to intersection theory in algebraic geometry, CBMS Regional Conference Series in Mathematics, vol. 54, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1984. · Zbl 0541.14005
[24] Federico Gaeta, Sul calcolo effettivo della forma associata \?(\?_{\?+\?-\?}^{\?\?}) all’intersezione di due cicle effettivi puri \?_{\?}^{\?}, \?\?^{\?} di \?_{\?}, in funzione delle \?(\?_{\?}^{\?}), \?(\?\?^{\?}) relative ai cicli secanti. I, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 24 (1958), 269 – 276 (Italian). · Zbl 0083.16104
[25] I. M. Gelfand, Collected papers , Springer-Verlag, Berlin, 1987-1989.
[26] Henri Gillet, An introduction to higher-dimensional Arakelov theory, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 209 – 228.
[27] H. Gillet and C. Soulé, Intersection theory using Adams operations, Invent. Math. 90 (1987), no. 2, 243 – 277. · Zbl 0632.14009
[28] Henri Gillet and Christophe Soulé, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93 – 174 (1991). · Zbl 0741.14012
[29] Henri Gillet and Christophe Soulé, Characteristic classes for algebraic vector bundles with Hermitian metric. I, Ann. of Math. (2) 131 (1990), no. 1, 163 – 203. , https://doi.org/10.2307/1971512 Henri Gillet and Christophe Soulé, Characteristic classes for algebraic vector bundles with Hermitian metric. II, Ann. of Math. (2) 131 (1990), no. 2, 205 – 238. · Zbl 0715.14006
[30] Henri Gillet and Christophe Soulé, Amplitude arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 17, 887 – 890 (French, with English summary). · Zbl 0676.14007
[31] Henri Gillet and Christophe Soulé, Un théorème de Riemann-Roch-Grothendieck arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 17, 929 – 932 (French, with English summary). · Zbl 0732.14002
[32] Henri Gillet and Christophe Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), no. 3, 473 – 543. · Zbl 0777.14008
[33] H. Gillet and C. Soulé, On the number of lattice points in convex symmetric bodies and their duals, Israel J. Math. 74 (1991), no. 2-3, 347 – 357. · Zbl 0752.52008
[34] Daniel R. Grayson, Reduction theory using semistability, Comment. Math. Helv. 59 (1984), no. 4, 600 – 634. · Zbl 0564.20027
[35] W. Gubler, Höhentheorie, Dissertation, ETH, Zurich, 1992.
[36] B. Wang and B. Harris, Archimedean height pairing of intersecting cycles, Internat. Math. Res. Notices 4 (1993), 107 – 111. · Zbl 0787.32029
[37] Reese Harvey and A. W. Knapp, Positive (\?,\?) forms, Wirtinger’s inequality, and currents, Value distribution theory (Proc. Tulane Univ. Program, Tulane Univ., New Orleans, La., 1972 – 1973) Dekker, New York, 1974, pp. 43 – 62. · Zbl 0287.53046
[38] Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. · Zbl 0543.58001
[39] M. Hindry, Letter to J.-B. Bost, January 1990.
[40] James R. King, The currents defined by analytic varieties, Acta Math. 127 (1971), no. 3-4, 185 – 220. · Zbl 0224.32008
[41] Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on ”det” and ”Div”, Math. Scand. 39 (1976), no. 1, 19 – 55. · Zbl 0343.14008
[42] J. Kramer, Néron-Tate height for cycles on Abelian varieties (d’après Faltings), preprint.
[43] L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraischen Grössen, Crelle J. Reine Angew. Math. 92 (1881-1882), 1-122; Werke, Vol. II, Leipzig, Teubner, 1897, pp. 237-287.
[44] Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. · Zbl 0667.14001
[45] Michel Laurent, Hauteur de matrices d’interpolation, Approximations diophantiennes et nombres transcendants (Luminy, 1990) de Gruyter, Berlin, 1992, pp. 215 – 238 (French, with English summary). · Zbl 0773.11047
[46] P. Lelong,Plurisubharmonic functions and positive differential forms, Gordon and Breach, New York, 1969. · Zbl 0195.11604
[47] Pierre Lelong, Mesure de Mahler des polynômes et majoration par convexité, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 2, 139 – 142 (French, with English and French summaries). · Zbl 0763.32003
[48] Harold I. Levine, A theorem on holomorphic mappings into complex projective space, Ann. of Math. (2) 71 (1960), 529 – 535. · Zbl 0142.04801
[49] L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985). · Zbl 0595.14032
[50] -, Métriques permises, Séminaire sur les Pinceaux Arithmétiques; la Conjecture de Mordell, Astérisque 127 (1985), 29-87. · Zbl 1182.11028
[51] Laurent Moret-Bailly, Hauteurs et classes de Chern sur les surfaces arithmétiques, Astérisque 183 (1990), 37 – 58 (French). Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Jean-Benoît Bost, Jean-François Mestre, and Laurent Moret-Bailly, Sur le calcul explicite des ”classes de Chern” des surfaces arithmétiques de genre 2, Astérisque 183 (1990), 69 – 105 (French). Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988).
[52] F. Mertens, Über die bestimmenden Eigenschaften der Resultante von \( n\)-Formen mit \( n\) Veränderlichen, Sitzungsber. Wiener Akad. 93 (1886), 527-566. · JFM 18.0105.02
[53] David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. · Zbl 0504.14008
[54] Ju. V. Nesterenko, Estimate of the orders of the zeroes of functions of a certain class, and their application in the theory of transcendental numbers, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 253 – 284, 477 (Russian). · Zbl 0354.10026
[55] Yu. V. Nesterenko, Estimates for the characteristic function of a prime ideal, Mat. Sb. (N.S.) 123(165) (1984), no. 1, 11 – 34 (Russian).
[56] D. G. Northcott, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949), 502 – 509. · Zbl 0035.30701
[57] Patrice Philippon, Critères pour l’indépendance algébrique, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 5 – 52 (French). · Zbl 0615.10044
[58] Patrice Philippon, Sur des hauteurs alternatives. I, Math. Ann. 289 (1991), no. 2, 255 – 283 (French). · Zbl 0726.14017
[59] Patrice Philippon, Sur des hauteurs alternatives. I, Math. Ann. 289 (1991), no. 2, 255 – 283 (French). · Zbl 0726.14017
[60] Pierre Samuel, Méthodes d’algèbre abstraite en géométrie algébrique, Seconde édition, corrigée. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 4, Springer-Verlag, Berlin-New York, 1967 (French). · Zbl 0146.16901
[61] Wolfgang M. Schmidt, On heights of algebraic subspaces and diophantine approximations, Ann. of Math. (2) 85 (1967), 430 – 472. · Zbl 0152.03602
[62] B. V. Shabat, Distribution of values of holomorphic mappings, Translations of Mathematical Monographs, vol. 61, American Mathematical Society, Providence, RI, 1985. Translated from the Russian by J. R. King; Translation edited by Lev J. Leifman. · Zbl 0564.32016
[63] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abhandlungen Preussischen Akad. Wiss. (1929), 41-69. · JFM 56.0180.05
[64] C. Soulé, Théorie de Nevanlinna et théorie d’Arakelov, Astérisque 183 (1990), 127 – 135 (French). Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988).
[65] -, Géométrie d’Arakelov et théorie des nombres transcendants, J. Arithmétiques de Luminy (17-21 juillet 1989), Astérisque 198-200 (1991), 355-371. · Zbl 0756.14014
[66] -, Opérations en \( K\)-théorie algébrique, Canad. J. Math. 27 (1985), 488-550. · Zbl 0575.14015
[67] C. Soulé, Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. · Zbl 0812.14015
[68] Wilhelm Stoll, The continuity of the fiber integral, Math. Z. 95 (1967), 87 – 138. · Zbl 0148.31904
[69] Wilhelm Stoll, About the value distribution of holomorphic maps into the projective space, Acta Math. 123 (1969), 83 – 114. · Zbl 0177.11302
[70] Wilhelm Stoll, Value distribution of holomorphic maps into compact complex manifolds., Lecture Notes in Mathematics, Vol. 135, Springer-Verlag, Berlin-New York, 1970. · Zbl 0195.36702
[71] Wilhelm Stoll, Aspects of value distribution theory in several complex variables, Bull. Amer. Math. Soc. 83 (1977), no. 2, 166 – 183. · Zbl 0344.32015
[72] Ulrich Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischer Formen, Arch. Math. (Basel) 27 (1976), no. 6, 604 – 610 (German). · Zbl 0338.10024
[73] Thomas Struppeck and Jeffrey D. Vaaler, Inequalities for heights of algebraic subspaces and the Thue-Siegel principle, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 493 – 528. · Zbl 0722.11033
[74] Lucien Szpiro, Degrés, intersections, hauteurs, Astérisque 127 (1985), 11 – 28 (French). Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). · Zbl 1182.11029
[75] W. Vogel, Lectures on results on Bezout’s theorem, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 74, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1984. Notes by D. P. Patil. · Zbl 0553.14022
[76] Paul Vojta, Applications of arithmetic algebraic geometry to Diophantine approximations, Arithmetic algebraic geometry (Trento, 1991) Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 164 – 208. · Zbl 0846.14009
[77] Bartel Leendert van der Waerden, Moderne Algebra, J. Springer, Berlin, 1940 (German).
[78] André Weil, Number-theory and algebraic geometry, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R. I., 1952, pp. 90 – 100.
[79] André Weil, Arithmetic on algebraic varieties, Ann. of Math. (2) 53 (1951), 412 – 444. · Zbl 0043.27002
[80] G. Wüstholz, Über das Abelsche Analogon des Lindemannschen Satzes. I, Invent. Math. 72 (1983), no. 3, 363 – 388 (German). · Zbl 0528.10024
[81] Shouwu Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569 – 587. · Zbl 0788.14017
[82] -, Positive line bundles on arithmetic varieties, preprint, February 1992.
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.