Abstract
Given an integrable connection on a smooth quasi-projective algebraic surface U over a subfield k of the complex numbers, we define rapid decay homology groups with respect to the associated analytic connection which pair with the algebraic de Rham cohomology in terms of period integrals. These homology groups generalize the analogous groups in the same situation over curves defined by Bloch and Esnault. In dimension two, however, new features appear in this context which we explain in detail. Assuming a conjecture of Sabbah on the formal classification of meromorphic connections on surfaces (known to be true if the rank is lower than or equal to 5), we prove perfectness of the period pairing in dimension two.
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References
André Y. (2004) Comparison theorems between algebraic and analytic De Rham cohomology (with emphasis on the p-adic case). J. Théorie des Nombres Bordeaux 16, 335–355
Beilinson A., Bloch S., Esnault H. (2002) \(\varepsilon\)-factors for Gauss–Manin determinants. Mosc. Math. J. 2(3): 477–532
Bloch S., Esnault H. (2004) Homology for irregular connections. J. Théorie des Nombres Bordeaux 16, 65–78
Bloch, S., Esnault, H.: Gauss-Manin determinant connections and periods for irregular connections, GAFA 2000 (Tel Aviv 1999), Geom. Funct. Anal. Special Volume, Part I, 1–31 (2000)
Bloch S., Esnault H. (2001) Gauss-Manin determinants for rank 1 irregular connections on curves. With an appendix in French by P. Deligne. Math. Ann. 321(1): 15–87
Deligne P. (1970) Equations Différentielles à Points Singuliers Réguliers, LNS 163. Springer, Berlin Heidelberg New York
Grothendieck A. (1966) On the de Rham cohomology of algebraic varieties. Publ. Math. IHES 29, 93–103
Hörmander L. (1990) The Analysis of Linear Partial Differential Operators, I. Springer, Berlin-Heidelberg New York
Kashiwara M., Schapira P. (1990) Sheaves on manifolds, Grundlehren der mathem. Wissenschaft 292. Springer, Berlin Heidelberg New York
Majima H. (1984) Asymptotic analysis for integrable connections with irregular singular points, LNS 1075. Springer, Berlin Heidelberg New York
Malgrange B. (1991) Equations Différentielles à Coefficients Polynomiaux, Prog. in Math. 96. Birkhäuser, Basel
Malgrange B. (1966) Ideals of Differentiable Functions. Oxford University Press, Oxford
Mebkhout Z. (1989) Le théorème de comparaison entre cohomologies de de Rham d’une variété algébrique complexe et le théorème d’existence de Riemann. Publ.Math. IHES 69, 47–89
Mebkhout Z. (1989) Le formalisme de Six Opérations de Grothendieck pour les \(\mathcal D_{X}\)-modules cohérents, Travaux en Cours 35.Hermann, Paris
Narváez Macarro L. (2004) The local duality theorem in \(\mathcal{D}\)-module theory. In: Eléments de la théorie des systèmes ifférentiells géométrique, Sémin.Congr. 8, Soc. Math. France, Paris
Ramis J.-P., Sibuya Y. (1989) Hukuhara domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type. Asymp. Anal. 2(1): 39–94
Sabbah, C.: Equations Différentielles à Points Singuliers Irréguliers et Phénomène de Stokes en Dimension 2. Astérisque 263, Soc. Math. de France (2000)
Sabbah C. (1993) Equations Différentielles à Points Singuliers Irréguliers en Dimension 2. Ann. Inst. Fourier (Grenoble) 43, 1619–1688
Sabbah C. (1996) On the comparison theorem for elementary irregular \(\mathcal {D}\)-modules. Nagoya Math. J. 141, 107–124
Saito T., Terasoma T. (1997) Determinant of period integrals. J. AMS 10(4): 865–937
Swan R.G. (1964) The theory of sheaves, Chicago Lectures in Mathematics. The University of Chicago Press, Chicago/London
Terasoma T. (1996) Confluent hypergeometric functions and wild ramification. J. Algebra 185, 1–18
Verdier J.-L. (1976) Classe d’homologie associée a un cycle, in ‘Séminaire de géométrie analytique’. Astérisque 36/37: 101–151