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Savage surfaces. (English) Zbl 1529.14022

Let \(G\) be the topological fundamental group of a given nonsingular complex projective surface. The main result of the present paper is the surprising assertion that the Chern slopes \(c_1^2/c_2\) of minimal nonsingular complex projective surfaces \(S\) having topological fundamental group isomorphic to \(G\) are dense in the interval \([1,3]\).
Given a real number \(r\in [2,3]\), in [X. Roulleau and G. Urzúa, Ann. Math. (2) 182, No. 1, 287–306 (2015; Zbl 1346.14097)] it was shown that there exists a sequence of simply connected surfaces \(\{X_p\}\) such that \(c_1^2(X_p)/c_2(X_p)\) approaches \(r\) as \(p\to \infty\), and in the present paper this result is extended for \(r\in [1,3]\), using special configurations of curves in \(\mathbb P^2\). Note that a similar density result for \(r\in [1/5, 2]\) was proven by U. Persson [Compos. Math. 43, 3–58 (1981; Zbl 0479.14018)] with a different type of surfaces.
The sequences of surfaces \(\{X_p\}\) are fundamental for the proof of the result in the present paper, as follows.
Having a minimal surface \(Y\) with fundamental group \(G\) and \(r\in [1,3]\), one takes a sequence \(\{X_p\}\) as above of simply connected surfaces. Then a new sequence of surfaces \(\{S_p\}\) with \(\pi_1(S_p)\simeq G\) and such that \(c_1^2(S_p)/c_2(S_p)\) approaches \(r\) as \(p\to \infty\) is constructed by finding for each \(p\) a suitable surface inside \(X_p\times Y\). The construction of the surfaces \(S_p\) is quite subtle. They are obtained as the intersection of two general sections of a lef line bundle \(M\) on \(X_p\times Y\), which in turn is obtained from a lef line bundle \(\Gamma_p\) in \(X_p\) and a very ample divisor \(B\) in \(Y\) (for the notion of lef line bundle see [M. A. A. de Cataldo and L. Migliorini, Ann. Sci. Éc. Norm. Supér. (4) 35, No. 5, 759–772 (2002; Zbl 1021.14004)]).
The authors explain also that their method of construction does not work for Persson’s sequences of simply connected surfaces, on one hand because it is not clear how to find the suitable lef line bundles and on the other hand because for low values of the slope there are constraints on the kind of fundamental groups that can occur.
The paper finishes with two conjectures on density of the Chern slopes in \([1,3]\). One is similar to the result in the present paper but asking for canonical class ample. The other states that given a Brody hyperbolic non singular surface \(Y\), then the Chern slopes of hyperbolic non singular surfaces with topological fundamental group isomorphic to the fundamental group of \(Y\) are dense in \([1,3]\).

MSC:

14J29 Surfaces of general type
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
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[1] Amorós, J., Burger, M., Corlette, K., Kotschick, D., Toledo, D.: Fundamental Groups of Com-pact Kähler Manifolds. Math. Surveys Monogr. 44, Amer. Math. Soc., Providence, RI (1996) Zbl 0849.32006 MR 1379330 · Zbl 0849.32006
[2] Aprodu, M., Nagel, J.: Koszul Cohomology and Algebraic Geometry. Univ. Lecture Ser. 52, Amer. Math. Soc., Providence, RI (2010) Zbl 1189.14001 MR 2573635 · Zbl 1189.14001
[3] Arapura, D.: Fundamental groups of smooth projective varieties. In: Current Topics in Com-plex Algebraic Geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ. 28, Cambridge Univ. Press, Cambridge, 1-16 (1995) Zbl 0873.14021 MR 1397055 · Zbl 0873.14021
[4] Baldridge, S., Kirk, P.: Symplectic 4-manifolds with arbitrary fundamental group near the Bogomolov-Miyaoka-Yau line. J. Symplectic Geom. 4, 63-70 (2006) Zbl 1105.53067 MR 2240212 · Zbl 1105.53067
[5] Baldridge, S., Kirk, P.: On symplectic 4-manifolds with prescribed fundamental group. Com-ment. Math. Helv. 82, 845-875 (2007) Zbl 1155.57024 MR 2341842 · Zbl 1155.57024
[6] Barth, W. P., Hulek, K., Peters, C. A. M., Van de Ven, A.: Compact Complex Surfaces. 2nd ed., Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin (2004) Zbl 1036.14016 · Zbl 1036.14016
[7] Beauville, A.: Complex Algebraic Surfaces. 2nd ed., London Math. Soc. Student Texts 34, Cambridge Univ. Press, Cambridge (1996) Zbl 0849.14014 MR 1406314 · Zbl 0849.14014
[8] Catanese, F.: Fibred surfaces, varieties isogenous to a product and related moduli spaces. Amer. J. Math. 122, 1-44 (2000) Zbl 0983.14013 MR 1737256 · Zbl 0983.14013
[9] Chen, Z. J.: On the geography of surfaces. Simply connected minimal surfaces with positive index. Math. Ann. 277, 141-164 (1987) Zbl 0595.14027 MR 884652 · Zbl 0595.14027
[10] Cornalba, M.: A remark on the topology of cyclic coverings of algebraic varieties. Boll. Un. Mat. Ital. A (5) 18, 323-328 (1981) Zbl 0462.14007 MR 618353 · Zbl 0462.14007
[11] Cutkosky, S. D.: Introduction to Algebraic Geometry. Grad. Stud. Math. 188, Amer. Math. Soc., Providence, RI (2018) Zbl 1396.14001 MR 3791837 · Zbl 1396.14001
[12] de Cataldo, M. A. A., Migliorini, L.: The hard Lefschetz theorem and the topology of semi-small maps. Ann. Sci. École Norm. Sup. (4) 35, 759-772 (2002) Zbl 1021.14004 MR 1951443 · Zbl 1021.14004
[13] Deligne, P.: Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après W. Fulton). In: Séminaire Bourbaki, exp. 543 (1979), Lecture Notes in Math. 842, Springer, 1-10 (1981) Zbl 0478.14008 MR 0429435 · Zbl 0478.14008
[14] Fulton, W., Lang, S.: Riemann-Roch Algebra. Grundlehren Math. Wiss. 277, Springer, New York (1985) Zbl 0579.14011 MR 801033 · Zbl 0579.14011
[15] Gieseker, D.: Global moduli for surfaces of general type. Invent. Math. 43, 233-282 (1977) Zbl 0389.14006 MR 498596 · Zbl 0389.14006
[16] Gompf, R. E.: A new construction of symplectic manifolds. Ann. of Math. (2) 142, 527-595 (1995) Zbl 0849.53027 MR 1356781 · Zbl 0849.53027
[17] Goresky, M., MacPherson, R.: Stratified Morse Theory. Ergeb. Math. Grenzgeb. (3) 14, Springer, Berlin (1988) Zbl 0639.14012 MR 932724 · Zbl 0639.14012
[18] Horikawa, E.: On deformations of quintic surfaces. Invent. Math. 31, 43-85 (1975) Zbl 0317.14018 MR 1573789 · Zbl 0317.14018
[19] Horikawa, E.: Algebraic surfaces of general type with small c 2 1 . I: Ann. of Math. (2) 104, 357-387 (1976); · Zbl 0339.14024
[20] II: Invent. Math. 37, 121-155 (1976); III: Invent. Math. 47, 209-248 (1978); IV: Invent. Math. 50, 103-128 (1978/79);
[21] V: J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, 745-755 · Zbl 0505.14028
[22] Kapovich, M.: Dirichlet fundamental domains and topology of projective varieties. Invent. Math. 194, 631-672 (2013) Zbl 1348.20045 MR 3127064 · Zbl 1348.20045
[23] Kapovich, M., Kollár, J.: Fundamental groups of links of isolated singularities. J. Amer. Math. Soc. 27, 929-952 (2014) Zbl 1307.14005 MR 3230815 · Zbl 1307.14005
[24] Kirk, P., Livingston, C.: The geography problem for 4-manifolds with specified fundamental group. Trans. Amer. Math. Soc. 361, 4091-4124 (2009) Zbl 1177.57020 MR 2500880 · Zbl 1177.57020
[25] Lazarsfeld, R.: Positivity in Algebraic Geometry. I. Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin (2004) Zbl 1093.14501 MR 2095471 · Zbl 1093.14501
[26] Mendes Lopes, M., Pardini, R.: On the algebraic fundamental group of surfaces with K 2 Ä 3 . J. Differential Geom. 77, 189-199 (2007) Zbl 1143.14032 MR 2355783 · Zbl 1143.14032
[27] Migliorini, L.: A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial. J. Algebraic Geom. 4, 353-361 (1995) Zbl 0834.14021 MR 1311355 · Zbl 0834.14021
[28] Miyaoka, Y.: The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann. 268, 159-171 (1984) Zbl 0521.14013 MR 744605 · Zbl 0521.14013
[29] Pardini, R.: The Severi inequality K 2 4 for surfaces of maximal Albanese dimension. Invent. Math. 159, 669-672 (2005) Zbl 1082.14041 MR 2125737 · Zbl 1082.14041
[30] Park, J.: The geography of symplectic 4-manifolds with an arbitrary fundamental group. Proc. Amer. Math. Soc. 135, 2301-2307 (2007) Zbl 1116.57023 MR 2299508 · Zbl 1116.57023
[31] Persson, U.: Chern invariants of surfaces of general type. Compos. Math. 43, 3-58 (1981) Zbl 0479.14018 MR 631426 · Zbl 0479.14018
[32] Persson, U., Peters, C., Xiao, G.: Geography of spin surfaces. Topology 35, 845-862 (1996) Zbl 0874.14031 MR 1404912 · Zbl 0874.14031
[33] Roulleau, X., Urzúa, G.: Chern slopes of simply connected complex surfaces of general type are dense in OE2; 3. Ann. of Math. (2) 182, 287-306 (2015) Zbl 1346.14097 MR 3374961 · Zbl 1346.14097
[34] Serre, J.-P.: Sur la topologie des variétés algébriques en caractéristique p. In: Symposium internacional de topología algebraica, Univ. Nacional Autónoma de México and UNESCO, Mexico City, 24-53 (1958) Zbl 0098.13103 MR 0098097 · Zbl 0088.38402
[35] Siu, Y. T.: The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. of Math. (2) 112, 73-111 (1980) MR 584075 · Zbl 0517.53058
[36] Stover, M., Toledo, D.: Residually finite lattices in PU(2,1) and fundamental groups of smooth projective surfaces. Michigan Math. J. 72, 559-597 (2022) MR 4460264 · Zbl 1506.14042
[37] Urzúa, G.: Arrangements of curves and algebraic surfaces. J. Algebraic Geom. 19, 335-365 (2010) Zbl 1192.14033 MR 2580678 · Zbl 1192.14033
[38] Urzúa, G.: Chern slopes of surfaces of general type in positive characteristic. Duke Math. J. 166, 975-1004 (2017) Zbl 1390.14105 MR 3626568 · Zbl 1390.14105
[39] Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann. 276, 449-466 (1987) Zbl 0596.14028 MR 875340 · Zbl 0596.14028
[40] Yau, S. T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. U.S.A. 74, 1798-1799 (1977) Zbl 0355.32028 MR 451180 · Zbl 0355.32028
[41] Yeung, S.-K.: Classification of fake projective planes. In: Handbook of Geometric Analysis, No. 2, Adv. Lect. Math. 13, Int. Press, Somerville, MA, 391-431 (2010) Zbl 1218.14025 MR 2761486 · Zbl 1218.14025
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