Abstract
In this paper a monodromy invariant for isotropic classes on generalized Kummer type manifolds is constructed. This invariant is used to determine the polarization type of Lagrangian fibrations on such manifolds—a notion which was introduced in an earlier paper of the author. The result shows that the polarization type of a Lagrangian fibration of generalized Kummer type depends on the connected component of the moduli space.
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Notes
Here we mean with the divisibility \(k = {{\mathrm{Div}}}(\lambda )\), the largest positive number k, such that \((\lambda , \cdot )/k\) is an integral form.
Note that \(v(F) = ({{\mathrm{rk}}}(F), c_1(F), c_1^2(F)/2 - c_2(F))\).
In [3] they use the notation K(L) for \(\ker \phi _L\).
References
Beauville, A.: Variétés kähleriennes dont la premiére classe de chern est nulle. J. Differ. Geom. 18, 755–782 (1984)
Barth, W., Hulek, K., Peters, C., van de Ven, A.: Compact Complex Surfaces. Second Enlarged Edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, vol. 4. Springer, Berlin (2003)
Birkenhake, C., Lange, H.: Complex Abelian Varieties, Second Edition of Grundlehren der mathematischen Wissenschaft, vol. 302. Springer, Berlin (2003)
Campana, F.: Isotrivialité de certaines familles kählériennes de variétés non projectives. Mathematische Zeitschrift 252(1), 147–156 (2005)
Ciliberto, C., van der Geer, G.: On the Jacobian of a hyperplane section of a surface. In: Classification of irregular varieties (Trento, 1990), Lecture Notes in Math., vol. 1515, pp. 33–40. Springer, Berlin (1992)
Eichler, M.: Quadratische Formen und orthogonale Gruppen. Springer, Berlin (1952)
Gross, M., Huybrechts, D., Joyce, D.: Calabi-Yau Manifolds and Related Geometries. Springer, Berlin (2003)
Greb, D., Lehn, C.: Base manifolds for lagrangian fibrations on hyperkähler manifolds. Int. Math. Res. Notices 19, 5483–5487 (2014)
Hwang, J.-M.: Base manifolds for fibrations of projective irreducible symplectic manifolds. Invent. Math. 174(3), 625–644 (2008)
Markman, E.: Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a K3 surface. Int. J. Math. 21(21), 169–223 (2010)
Markman, E.: A survey of Torelli and monodromy results for holomorphic-symplectic varieties. In: Ebeling W et al. (eds.) Complex and Differential Geometry, Proceedings in Math., vol. 8, pp. 257–323. Springer, Berlin (2011)
Markman, E.: Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections. Kyoto J. Math. 53(2), 345–403 (2013)
Markman, E.: Lagrangian fibrations of holomorphic-symplectic varieties of \(\text{K}3^{[n]}\)-type. In: Frühbis-Krüger A, et al. (eds.) Algebraic and Complex Geometry, Proceedings in Math., vol. 71. Springer, Berlin (2014)
Matsushita, D.: On fibre space structures of a projective irreducible symplectic manifold. Topology 38(1), 79–83 (1999)
Matsushita, D.: Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds. Math. Res. Lett. 7, 389–391 (2000)
Matsushita, D.: Addendum to: On fibre space structures of a projective irreducible symplectic manifold. Topology 38(1), 79–83 (2001)
Matsushita, D.: Holomorphic symplectic manifolds and lagrangian fibrations. Acta Appl. Math. 75(1–3), 117–123 (2003)
Matsushita, D.: On isotropic divisors on irreducible symplectic manifolds (2013). arXiv:1310.0896
Mongardi, G.: On the monodromy of irreducible symplectic manifolds. Algebraic Geom. 3(3), 385–391 (2016)
Mongardi, G., Pacienza, G.: Polarized parallel transport and uniruled divisors on deformations of generalized Kummer varieties. Int. Math. Res. Notices (2016). https://doi.org/10.1093/imrn/rnw346
Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77, 101–116 (1984)
Mukai, S.: On the moduli space of bundles on K3 surfaces I. Tata Inst. Fundam. Res. Stud. Math. 11, 341–413 (1987)
Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izvestija 14(1), 103–167 (1980)
O’Grady, K.: The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface. J. Algebraic Geom. 6(4), 599–644 (1997)
O’Grady, K.: Compact Hyperkähler Manifolds: Examples. Online lecture notes (2014). http://www.mimuw.edu.pl/~gael/Document/hk-examples.pdf
Wieneck, B.: On polarization types of Lagrangian fibrations. Manuscr. Math. 151(3–4), 305–327 (2016)
Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001)
Yoshioka, K.: Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface (2012). arXiv:1206.4838v2
Acknowledgements
I thank my advisor Klaus Hulek, Eyal Markman and Giovanni Mongardi for helpful discussions. I thank Olivier Debarre for pointing me out the idea of the proof of Lemma 6.14.
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Wieneck, B. Monodromy invariants and polarization types of generalized Kummer fibrations. Math. Z. 290, 347–378 (2018). https://doi.org/10.1007/s00209-017-2020-y
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DOI: https://doi.org/10.1007/s00209-017-2020-y