×

Relative compactified Jacobians of linear systems on Enriques surfaces. (Relative compactified Jacobiansof linear systemson Enriques surfaces.) (English) Zbl 1411.14045

In the paper under review, the author studies certain moduli spaces of sheaves on Enriques surfaces which lead to new examples of Calabi-Yau manifolds. In order to present the main result, let us recall some crucial definitions. Let \((X,H)\) be a smooth projective polarized surface. For a pure \(1\)-dimensional sheaf \(F\) on \(X\) we define Fitting support. Let us recall that such a pure dimension \(1\) sheaf has homological dimension \(1\), i.e., there exists a length one locally free resolution of \(F\), \[ 0 \to L_{1} \stackrel{a}{\rightarrow} L_{0} \rightarrow F \rightarrow 0. \] The Fitting support of \(F\) is the subscheme of \(X\) defined by the equation \(\text{det} \,a = 0\). For the main result, we consider only general Enriques surfaces \(T\).
Main Result. Let \(|C|\) be a genus \(g \geq 2\) linear system on \(T\), \(d= g - 1\) an integer, \(H\) a generic polarization, and let \(N \rightarrow |C|\) be the component of the moduli space of \(H\)-semistable sheaves on \(T\) with Fitting support in \(|C|\) and Euler characteristic equal to \(\chi = d - g + 1\) that contains sheaves supported on irreducible curves. Suppose the divisibility of \(C\) in the Néron-Severi space \(\mathrm{NS}(T)\) is coprime with \(2(d - g + 1)\). Then we have the following:
i) \(N\) is a smooth \((2g - 1)\)-dimensional Calabi-Yau variety.
ii) For \(g = 2\), one gets Calabi-Yau \(3\)-folds with the following Hodge diamond: \[ \begin{gathered} 1\\ 0 \quad 0\\ 0 \quad 10 \quad 0\\ 1 \quad 10 \quad 10 \quad 1. \end{gathered} \]
iii) There is a surjection \(\mathbb{Z}/(2) \rightarrow \pi_{1}(N)\) which, under some natural assumption (please consult Assumption 2.17 therein), turns out to be an isomorphism. Under the same assumption, one can additionally show that for \(g \geq 3\) we have \(h^{2}(N) = 11\).

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14D22 Fine and coarse moduli spaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14K30 Picard schemes, higher Jacobians

References:

[1] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Pillip A., Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 268, xxx+963 pp. (2011), Springer, Heidelberg · Zbl 1235.14002 · doi:10.1007/978-3-540-69392-5
[2] Alexeev, Valery, Compactified Jacobians and Torelli map, Publ. Res. Inst. Math. Sci., 40, 4, 1241-1265 (2004) · Zbl 1079.14019
[3] Arapura, Donu, The Leray spectral sequence is motivic, Invent. Math., 160, 3, 567-589 (2005) · Zbl 1083.14011 · doi:10.1007/s00222-004-0416-x
[4] Artamkin, I. V., On the deformation of sheaves, Izv. Akad. Nauk SSSR Ser. Mat.. Math. USSR-Izv., 52 32, 3, 663-668 (1989) · Zbl 0709.14012 · doi:10.1070/IM1989v032n03ABEH000805
[5] Arbarello, E.; Sacc\`a, G., Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties, Adv. Math., 329, 649-703 (2018) · Zbl 1419.14058 · doi:10.1016/j.aim.2018.02.003
[6] Arbarello, E.; Sacc\`a, G.; Ferretti, A., Relative Prym varieties associated to the double cover of an Enriques surface, J. Differential Geom., 100, 2, 191-250 (2015) · Zbl 1362.14035
[7] Beauville, Arnaud, Vari\'et\'es K\"ahleriennes dont la premi\`ere classe de Chern est nulle, J. Differential Geom., 18, 4, 755-782 (1984) (1983) · Zbl 0537.53056
[8] Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 4, xii+436 pp. (2004), Springer-Verlag, Berlin · Zbl 1036.14016 · doi:10.1007/978-3-642-57739-0
[9] Caporaso, Lucia; Coelho, Juliana; Esteves, Eduardo, Abel maps of Gorenstein curves, Rend. Circ. Mat. Palermo (2), 57, 1, 33-59 (2008) · Zbl 1139.14025 · doi:10.1007/s12215-008-0002-y
[10] Cossec, Fran\c{c}ois R.; Dolgachev, Igor V., Enriques surfaces. I, Progress in Mathematics 76, x+397 pp. (1989), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0665.14017 · doi:10.1007/978-1-4612-3696-2
[11] Chen, Dawei; Kass, Jesse Leo, Moduli of generalized line bundles on a ribbon, J. Pure Appl. Algebra, 220, 2, 822-844 (2016) · Zbl 1372.14018 · doi:10.1016/j.jpaa.2015.07.019
[12] Chaudouard, Pierre-Henri; Laumon, G\'erard, Un th\'eor\`eme du support pour la fibration de Hitchin, Ann. Inst. Fourier (Grenoble), 66, 2, 711-727 (2016) · Zbl 1375.14069
[13] Cossec, Fran\c{c}ois R., Projective models of Enriques surfaces, Math. Ann., 265, 3, 283-334 (1983) · Zbl 0501.14021 · doi:10.1007/BF01456021
[14] Deligne, Pierre, \'Equations diff\'erentielles \`“a points singuliers r\'”eguliers, Lecture Notes in Mathematics, Vol. 163, iii+133 pp. (1970), Springer-Verlag, Berlin-New York · Zbl 0244.14004
[15] Deligne, Pierre, Th\'eorie de Hodge. II, Inst. Hautes \'Etudes Sci. Publ. Math., 40, 5-57 (1971) · Zbl 0219.14007
[16] Dimca, Alexandru, Singularities and topology of hypersurfaces, Universitext, xvi+263 pp. (1992), Springer-Verlag, New York · Zbl 0753.57001 · doi:10.1007/978-1-4612-4404-2
[17] Dolgachev, Igor V., A brief introduction to Enriques surfaces. Development of moduli theory-Kyoto 2013, Adv. Stud. Pure Math. 69, 1-32 (2016), Math. Soc. Japan, [Tokyo] · Zbl 1369.14048
[18] G\"ottsche, L.; Huybrechts, D., Hodge numbers of moduli spaces of stable bundles on \(K3\) surfaces, Internat. J. Math., 7, 3, 359-372 (1996) · Zbl 0879.14013 · doi:10.1142/S0129167X96000219
[19] Gieseker, D., On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math., 101, 1, 77-85 (1979) · Zbl 0431.14005 · doi:10.2307/2373939
[20] Goresky, Mark; MacPherson, Robert, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 14, xiv+272 pp. (1988), Springer-Verlag, Berlin · Zbl 0639.14012 · doi:10.1007/978-3-642-71714-7
[21] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. II. \'Etude globale \'el\'ementaire de quelques classes de morphismes, Inst. Hautes \'Etudes Sci. Publ. Math., 8, 222 pp. (1961)
[22] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. IV. \'Etude locale des sch\'emas et des morphismes de sch\'emas IV, Inst. Hautes \'Etudes Sci. Publ. Math., 32, 361 pp. (1967) · Zbl 0153.22301
[23] Hauzer, Marcin, On moduli spaces of semistable sheaves on Enriques surfaces, Ann. Polon. Math., 99, 3, 305-321 (2010) · Zbl 1238.14008 · doi:10.4064/ap99-3-7
[24] Huybrechts, Daniel; Lehn, Manfred, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, xiv+269 pp. (1997), Friedr. Vieweg & Sohn, Braunschweig · Zbl 0872.14002 · doi:10.1007/978-3-663-11624-0
[25] Huybrechts, Daniel, Birational symplectic manifolds and their deformations, J. Differential Geom., 45, 3, 488-513 (1997) · Zbl 0917.53010
[26] Katz, Nicholas M., The regularity theorem in algebraic geometry. Actes du Congr\`“es International des Math\'”ematiciens, Nice, 1970, 437-443 (1971), Gauthier-Villars, Paris · Zbl 0235.14006
[27] Kim, Hoil, Moduli spaces of stable vector bundles on Enriques surfaces, Nagoya Math. J., 150, 85-94 (1998) · Zbl 0931.14002 · doi:10.1017/S002776300002506X
[28] Kim, Hoil, Stable vector bundles of rank two on Enriques surfaces, J. Korean Math. Soc., 43, 4, 765-782 (2006) · Zbl 1113.14029 · doi:10.4134/JKMS.2006.43.4.765
[29] Knutsen, Andreas Leopold, On \(k\) th-order embeddings of \(K3\) surfaces and Enriques surfaces, Manuscripta Math., 104, 2, 211-237 (2001) · Zbl 1017.14015 · doi:10.1007/s002290170040
[30] Koll\'ar, J\'anos, Higher direct images of dualizing sheaves. II, Ann. of Math. (2), 124, 1, 171-202 (1986) · Zbl 0605.14014 · doi:10.2307/1971390
[31] Koll\'ar, J\'anos, Higher direct images of dualizing sheaves. I, Ann. of Math. (2), 123, 1, 11-42 (1986) · Zbl 0598.14015 · doi:10.2307/1971351
[32] Koll\'ar, J\'anos, Singularities of the minimal model program, Cambridge Tracts in Mathematics 200, x+370 pp. (2013), Cambridge University Press, Cambridge, England · Zbl 1282.14028 · doi:10.1017/CBO9781139547895
[33] Laumon, G\'erard, Un analogue global du c\^one nilpotent, Duke Math. J., 57, 2, 647-671 (1988) · Zbl 0688.14023 · doi:10.1215/S0012-7094-88-05729-8
[34] Lazarsfeld, Robert, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 49, xviii+385 pp. (2004), Springer-Verlag, Berlin · Zbl 1093.14500 · doi:10.1007/978-3-642-18808-4
[35] Leibman, A., Fiber bundles with degenerations and their applications to computing fundamental groups, Geom. Dedicata, 48, 1, 93-126 (1993) · Zbl 0812.57004 · doi:10.1007/BF01265678
[36] Le Potier, J., Faisceaux semi-stables de dimension \(1\) sur le plan projectif, Rev. Roumaine Math. Pures Appl., 38, 7-8, 635-678 (1993) · Zbl 0815.14029
[37] Matsushita, Daisuke, On fibre space structures of a projective irreducible symplectic manifold, Topology, 38, 1, 79-83 (1999) · Zbl 0932.32027 · doi:10.1016/S0040-9383(98)00003-2
[38] Matsushita, Daisuke, Higher direct images of dualizing sheaves of Lagrangian fibrations, Amer. J. Math., 127, 2, 243-259 (2005) · Zbl 1069.14011
[39] M. Melo, A. Rapagnetta, and F. Viviani, Fourier\textendash Mukai and autoduality for compactified Jacobians. I, arXiv:1207.7233 (2013). · Zbl 1458.14044
[40] Melo, Margarida; Rapagnetta, Antonio; Viviani, Filippo, Fine compactified Jacobians of reduced curves, Trans. Amer. Math. Soc., 369, 8, 5341-5402 (2017) · Zbl 1364.14007 · doi:10.1090/tran/6823
[41] Markushevich, D.; Tikhomirov, A. S., New symplectic \(V\)-manifolds of dimension four via the relative compactified Prymian, Internat. J. Math., 18, 10, 1187-1224 (2007) · Zbl 1138.14032 · doi:10.1142/S0129167X07004503
[42] Mukai, Shigeru, Symplectic structure of the moduli space of sheaves on an abelian or \(K3\)surface, Invent. Math., 77, 1, 101-116 (1984) · Zbl 0565.14002 · doi:10.1007/BF01389137
[43] Mumford, David, Prym varieties. I. Contributions to analysis (a collection of papers dedicated to Lipman Bers), 325-350 (1974), Academic Press, New York · Zbl 0299.14018
[44] Namikawa, Yukihiko, Periods of Enriques surfaces, Math. Ann., 270, 2, 201-222 (1985) · Zbl 0536.14024 · doi:10.1007/BF01456182
[45] Nuer, Howard, A note on the existence of stable vector bundles on Enriques surfaces, Selecta Math. (N.S.), 22, 3, 1117-1156 (2016) · Zbl 1364.14008 · doi:10.1007/s00029-015-0218-6
[46] Nuer, Howard, Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface, Proc. Lond. Math. Soc. (3), 113, 3, 345-386 (2016) · Zbl 1361.14007 · doi:10.1112/plms/pdw033
[47] O’Grady, Kieran G., The weight-two Hodge structure of moduli spaces of sheaves on a \(K3\) surface, J. Algebraic Geom., 6, 4, 599-644 (1997) · Zbl 0916.14018
[48] Oda, Tadao; Seshadri, C. S., Compactifications of the generalized Jacobian variety, Trans. Amer. Math. Soc., 253, 1-90 (1979) · Zbl 0418.14019 · doi:10.2307/1998186
[49] Oguiso, Keiji; Schr\"oer, Stefan, Enriques manifolds, J. Reine Angew. Math., 661, 215-235 (2011) · Zbl 1272.14026 · doi:10.1515/CRELLE.2011.077
[50] Peters, Chris A. M.; Steenbrink, Joseph H. M., Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 52, xiv+470 pp. (2008), Springer-Verlag, Berlin · Zbl 1138.14002
[51] Rapagnetta, Antonio, On the Beauville form of the known irreducible symplectic varieties, Math. Ann., 340, 1, 77-95 (2008) · Zbl 1156.14008 · doi:10.1007/s00208-007-0139-6
[52] Sacc\`a, Giulia, Fibrations in abelian varieties associated to Enriques surfaces, 159 pp. (2013), ProQuest LLC, Ann Arbor, MI
[53] Spanier, Edwin H., Algebraic topology, xiv+528 pp. (1966), McGraw-Hill Book Co., New York\textendash Toronto, Ont.\textendash London · Zbl 0810.55001
[54] Steenbrink, J. H. M., Mixed Hodge structure on the vanishing cohomology. Real and complex singularities, Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976, 525-563 (1977), Sijthoff and Noordhoff, Alphen aan den Rijn · Zbl 0373.14007
[55] Steenbrink, Joseph; Zucker, Steven, Variation of mixed Hodge structure. I, Invent. Math., 80, 3, 489-542 (1985) · Zbl 0626.14007 · doi:10.1007/BF01388729
[56] Takemoto, Fumio, Stable vector bundles on algebraic surfaces. II, Nagoya Math. J., 52, 173-195 (1973) · Zbl 0296.14012
[57] Verra, Alessandro, On Enriques surface as a fourfold cover of \({\bf P}^2\), Math. Ann., 266, 2, 241-250 (1983) · Zbl 0506.14032 · doi:10.1007/BF01458446
[58] Voisin, Claire, Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics 77, x+351 pp. (2003), Cambridge University Press, Cambridge, England · Zbl 1032.14002 · doi:10.1017/CBO9780511615177
[59] Yoshioka, K\=ota, Some examples of Mukai’s reflections on \(K3\) surfaces, J. Reine Angew. Math., 515, 97-123 (1999) · Zbl 0940.14026 · doi:10.1515/crll.1999.080
[60] Yoshioka, K\=ota, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann., 321, 4, 817-884 (2001) · Zbl 1066.14013 · doi:10.1007/s002080100255
[61] Yoshioka, K\=ota, Stability and the Fourier-Mukai transform. II, Compos. Math., 145, 1, 112-142 (2009) · Zbl 1165.14033 · doi:10.1112/S0010437X08003758
[62] Yoshioka, K\=ota, A note on stable sheaves on Enriques surfaces, Tohoku Math. J. (2), 69, 3, 369-382 (2017) · Zbl 1401.14064 · doi:10.2748/tmj/1505181622
[63] M. Zowislok, Subvarieties of moduli spaces of sheaves via finite coverings, arXiv:1210.4794 (2013).
[64] Zucker, Steven, Degeneration of Hodge bundles (after Steenbrink). Topics in transcendental algebraic geometry, Princeton, N.J., 1981/1982, Ann. of Math. Stud. 106, 121-141 (1984), Princeton Univ. Press, Princeton, NJ · Zbl 0574.14008
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.