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Complex curves of genus three, Kummer surfaces and Quillen metrics. (English) Zbl 1085.14031

Let \(C\) be a smooth, complex projective curve of genus 3 and \( \lambda(O_C)\) the determinant of the cohomologies of \(C\). The authors define for a pair \((u, \tau) \in (\mathbb{C}^4-0 \times H_2)\), where \(H_2\) is the Siegel upper half-space, whose existence depends on a choice of the double covering \(\tilde{C} \rightarrow C\), a non-zero element \(\varphi(u, \tau) \in \lambda (O_C)\) . Denoting by \(\| \cdot \|\) the Quillen metric on \( \lambda_C\) with respect to \(k_C\), the authors give an explicit formula for \(\| \varphi(u, \tau) \|^2\) and state it as main theorem 0.1. Their main theorem is a generalization of the results presented for example by D. B. Ray and I. M. Singer [Ann. Math. (2), 98, 154–177 (1973; Zbl 0267.32014)], G. Faltings [Ann. Math. 119, 387–424 (1984; Zbl 0559.14005)] and more recently by K. Yoshikawa [J. Differ. Geom. 52, 73–115 (1999; Zbl 1033.58029)].
The paper is organized as follows. In section 1, for a pair \((F, h_F)\) of a hermitian vector bundle on \(X\), for \(X\) a compact Kähler manifold with Kähler metric \(k_X\), they define the Ray-Singer analytic torsion with respect to \(k_X\) in definition 1.1. In definition 1.2 the equivariant Quillen metric is defined. In section 2.3.1 they compute the \(\mu_2\) equivariant Quillen metric on \( \lambda_{\mu_2} (O_{A_{\tau}})\) for \(A_{\tau}\) an abelian surface and on \( \lambda_{\mu_2}({L_{\tau}}^{-2})\). In subsection 2.3.2 they continue with the computation of the Quillen metric on \(\lambda_{\mu_2}( L_{\tau}^2 \otimes K_{A_{\tau}})\). In section 3 subsection 3.1 Kummer’s quartic surface \(R_{\tau}\) is defined. In subsection 3.2, theorem 3.2 they prove that \(R_{\tau}\) is self-dual. In section 4, proposition 4.2 the authors show that every smooth projective curve of genus 3 is isomorphic to a divisor of a Kummer surface constructed in as in section 3. In subsection 5.2 of section 5 they state the main result of the paper as main theorem 5.3 and prove it. The paper ends in section 6 where they prove a technical result stated as proposition 6.1 to prove lemma 3.7 which is used to prove the exact duality of Kummer’s quartic surface (theorem 3.2).

MSC:

14J15 Moduli, classification: analytic theory; relations with modular forms
32G20 Period matrices, variation of Hodge structure; degenerations
32N10 Automorphic forms in several complex variables
32N15 Automorphic functions in symmetric domains
14J28 \(K3\) surfaces and Enriques surfaces
58J52 Determinants and determinant bundles, analytic torsion
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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[1] Barth, W., Nieto, I.: Abelian surfaces of type (1,3) and quartic surfaces with 16 skew lines. J. Algebraic Geom. 3, 173–222 (1994) · Zbl 0809.14027
[2] Bismut, J.-M.: Equivariant immersions and Quillen metrics. J. Differential Geom. 41, 53–157 (1995) · Zbl 0826.32024
[3] Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles I,II,III. Commun. Math. Phys. 115, 49–78, 79–126, 301–351 (1988) · Zbl 0651.32017 · doi:10.1007/BF01238853
[4] Bismut, J.-M., Gillet, H., Soulé, C.: Complex immersions and Arakelov geometry. P. Cartier et al. (eds.), The Grothendieck Festschrift, Birkhäuser, Boston 1990, pp. 249–331 · Zbl 0744.14015
[5] Bost, J.-B., Mestre, J.-F., Moret-Bailly, L.: Sur le calcul explicite des classes de Chern des surfaces arithmétiques de genre 2. Astérisque 183, 69–105 (1990)
[6] Faltings, G.: Calculus on arithmetic surfaces. Ann. Math. 119, 387–424 (1984) · Zbl 0559.14005 · doi:10.2307/2007043
[7] Freitag, E.: Siegelsche Modulfunktionen. Springer, Berline, 1983 · Zbl 0498.10016
[8] van der Geer, G.: On the geometry of a Siegel modular threefold. Math. Ann. 260, 317–350 (1982) · Zbl 0488.14007 · doi:10.1007/BF01461467
[9] Gillet, H., Soulé, C.: Analytic torsion and the arithmetic Todd genus. With an appendix by D. Zagier, Topology 30, 21–54 (1991) · Zbl 0787.14005
[10] Gonzalez-Dorrego, M.: (16,6) configurations and geometry of Kummer surfaces in \(\mathbb{P}\)3. Memoirs Am. Math. Soc. 107, 1994 · Zbl 0809.14032
[11] Hudson, R.: Kummer’s Quartic Surface, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990 · Zbl 0716.14025
[12] Igusa, J.: Theta Functions, Springer, Berlin, 1972 · Zbl 0251.14016
[13] Kempf, G.R.: Complex Abelian Varieties and Theta Functions, Springer, Berlin, 1991 · Zbl 0752.14040
[14] Keum, J.H.: Automorphisms of Jacobian Kummer surfaces. Compositio Math. 107, 269–288 (1997) · Zbl 0891.14013 · doi:10.1023/A:1000148907120
[15] Knudsen, F.F., Mumford, D.: The projectivity of the moduli space of stable curves, I. Math. Scand. 39, 19–55 (1976) · Zbl 0343.14008
[16] Köhler, K., Roessler, D.: A fixed point formula of Lefschetz type in Arakelov geometry I. Invent. Math. 145, 333–396 (2001) · Zbl 0999.14002 · doi:10.1007/s002220100151
[17] Köhler, K., Roessler, D.: A fixed point formula of Lefschetz type in Arakelov geometry IV. J. Reine Angew. Math. 556, 127–148 (2003) · Zbl 1032.14004 · doi:10.1515/crll.2003.017
[18] Lange, H., Birkenhale, Ch.: Complex Abelian Varieties. Springer, Berlin, 1992
[19] Masiewicki, L.: Universal properties of Prym varieties with an application to algebraic curves of genus 5. Trans. Am. Math. Soc 222, 221–240 (1976) · Zbl 0333.14012 · doi:10.1090/S0002-9947-1976-0422289-6
[20] Mumford, D.: On the equation defining Abelian varieties. I. Invent. Math. 1, 287–354 (1966) · Zbl 0219.14024 · doi:10.1007/BF01389737
[21] Mumford, D.: Prym varieties I. Contributions to Analysis. Academic Press, New York, 1974, pp. 325–350
[22] Mumford, D.: Tata Lectures on Theta. I Progress in Math. 28, Birkhäuser, Boston, 1983 · Zbl 0509.14049
[23] Nieto, I. : The normalizer of the level (2,2) Heisenberg group. Manuscripta Math. 76, 257–267 (1992) · Zbl 0778.20019 · doi:10.1007/BF02567760
[24] Quillen, D.: Determinants of Cauchy-Riemann operators on Riemann surfaces. Funct. Anal. Appl. 19, 31–34 (1985) · Zbl 0603.32016 · doi:10.1007/BF01086022
[25] Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math. 98, 154–177 (1973) · Zbl 0267.32014 · doi:10.2307/1970909
[26] Ueno, K.: Discriminants of curves of genus 2 and arithmetic surfaces. Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Kinokuniya, Tokyo, 1988, pp. 749–770
[27] Verra, A.: The fiber of the Prym map in genus 3. Math. Ann. 276, 433–448 (1987) · Zbl 0588.14024 · doi:10.1007/BF01450840
[28] Yoshikawa, K.-I.: Discriminant of theta divisors and Quillen metrics. J. Differential Geom. 52, 73–115 (1999) · Zbl 1033.58029
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