## Calabi-Yau three-folds and moduli of abelian surfaces II

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- by Mark Gross and Sorin Popescu PDF
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**363**(2011), 3573-3599 Request permission

## Abstract:

We give explicit descriptions of the moduli spaces of abelian surfaces with polarizations of type $(1,d)$, for $d=12,14,16,18$ and $20$. More precisely, in each case we show that a certain choice of moduli space of such abelian surfaces with a*partial level structure*can be described explicitly and is unirational, and in some cases rational. These moduli spaces with partial level structure are covers of the ordinary moduli spaces, so the Kodaira dimension of the ordinary moduli spaces in these cases is $-\infty$. In addition, we give a few new examples of Calabi-Yau three-folds fibred in abelian surfaces. In the case of $d=20$, such Calabi-Yau three-folds play a key role in the description of the abelian surfaces.

## References

- Alf Bjørn Aure,
*Surfaces on quintic threefolds associated to the Horrocks-Mumford bundle*, Arithmetic of complex manifolds (Erlangen, 1988) Lecture Notes in Math., vol. 1399, Springer, Berlin, 1989, pp. 1–9. MR**1034252**, DOI 10.1007/BFb0095964 - Alf Aure, Wolfram Decker, Sorin Popescu, Klaus Hulek, and Kristian Ranestad,
*The geometry of bielliptic surfaces in $\textbf {P}^4$*, Internat. J. Math.**4**(1993), no. 6, 873–902. MR**1250253**, DOI 10.1142/S0129167X93000406 - Alf Aure, Wolfram Decker, Klaus Hulek, Sorin Popescu, and Kristian Ranestad,
*Syzygies of abelian and bielliptic surfaces in $\textbf {P}^4$*, Internat. J. Math.**8**(1997), no. 7, 849–919. MR**1482969**, DOI 10.1142/S0129167X97000421 - Bak, A., Bouchard, V., Donagi, R., “Exploring a new peak in the heterotic landscape”, preprint, arXiv:0811.1242.
- Bayer, D., Stillman, M., “Macaulay: A system for computation in algebraic geometry and commutative algebra source and object code available for Unix and Macintosh computers”. Contact the authors, or download from
**math.harvard.edu**via anonymous ftp. - Arnaud Beauville,
*Variétés de Prym et jacobiennes intermédiaires*, Ann. Sci. École Norm. Sup. (4)**10**(1977), no. 3, 309–391 (French). MR**472843** - Ch. Birkenhake and H. Lange,
*Fixed-point free automorphisms of abelian varieties*, Geom. Dedicata**51**(1994), no. 3, 201–213. MR**1293798**, DOI 10.1007/BF01263993 - Lev Borisov and Andrei Căldăraru,
*The Pfaffian-Grassmannian derived equivalence*, J. Algebraic Geom.**18**(2009), no. 2, 201–222. MR**2475813**, DOI 10.1090/S1056-3911-08-00496-7 - Philip Candelas and Rhys Davies,
*New Calabi-Yau manifolds with small Hodge numbers*, Fortschr. Phys.**58**(2010), no. 4-5, 383–466. MR**2662012**, DOI 10.1002/prop.200900105 - Grayson, D., Stillman, M., “Macaulay 2: A computer program designed to support computations in algebraic geometry and computer algebra”. Source and object code available from http://www.math.uiuc.edu/Macaulay2/.
- Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR**507725** - Valeri Gritsenko,
*Irrationality of the moduli spaces of polarized abelian surfaces*, Abelian varieties (Egloffstein, 1993) de Gruyter, Berlin, 1995, pp. 63–84. With an appendix by the author and K. Hulek. MR**1336601** - Valeri Gritsenko,
*Irrationality of the moduli spaces of polarized abelian surfaces*, Internat. Math. Res. Notices**6**(1994), 235 ff., approx. 9 pp.}, issn=1073-7928, review= MR**1277050**, doi=10.1155/S1073792894000267, - Mark Gross and Simone Pavanelli,
*A Calabi-Yau threefold with Brauer group $(\Bbb Z/8\Bbb Z)^2$*, Proc. Amer. Math. Soc.**136**(2008), no. 1, 1–9. MR**2350382**, DOI 10.1090/S0002-9939-07-08840-5 - Mark Gross and Sorin Popescu,
*Equations of $(1,d)$-polarized abelian surfaces*, Math. Ann.**310**(1998), no. 2, 333–377. MR**1602020**, DOI 10.1007/s002080050151 - Mark Gross and Sorin Popescu,
*The moduli space of $(1,11)$-polarized abelian surfaces is unirational*, Compositio Math.**126**(2001), no. 1, 1–23. MR**1827859**, DOI 10.1023/A:1017518027822 - Mark Gross and Sorin Popescu,
*Calabi-Yau threefolds and moduli of abelian surfaces. I*, Compositio Math.**127**(2001), no. 2, 169–228. MR**1845899**, DOI 10.1023/A:1012076503121 - Kentaro Hori and David Tong,
*Aspects of non-abelian gauge dynamics in two-dimensional $\scr N=(2,2)$ theories*, J. High Energy Phys.**5**(2007), 079, 41. MR**2318130**, DOI 10.1088/1126-6708/2007/05/079 - Klaus Hulek, Constantin Kahn, and Steven H. Weintraub,
*Moduli spaces of abelian surfaces: compactification, degenerations, and theta functions*, De Gruyter Expositions in Mathematics, vol. 12, Walter de Gruyter & Co., Berlin, 1993. MR**1257185**, DOI 10.1515/9783110891928 - K. Hulek and G. K. Sankaran,
*The Kodaira dimension of certain moduli spaces of abelian surfaces*, Compositio Math.**90**(1994), no. 1, 1–35. MR**1266492** - Kuznetsov, A., “Homological projective duality for Grassmannians of lines”, preprint, arXiv:math/0610957.
- Herbert Lange and Christina Birkenhake,
*Complex abelian varieties*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR**1217487**, DOI 10.1007/978-3-662-02788-2 - F. Melliez and K. Ranestad,
*Degenerations of $(1,7)$-polarized abelian surfaces*, Math. Scand.**97**(2005), no. 2, 161–187. MR**2191701**, DOI 10.7146/math.scand.a-14970 - D. Mumford,
*On the equations defining abelian varieties. I*, Invent. Math.**1**(1966), 287–354. MR**204427**, DOI 10.1007/BF01389737 - David Mumford,
*Abelian varieties*, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR**0282985** - Einar Andreas Rødland,
*The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian $G(2,7)$*, Compositio Math.**122**(2000), no. 2, 135–149. MR**1775415**, DOI 10.1023/A:1001847914402 - Popescu, S.,
*On smooth surfaces of degree $\ge 11$ in the projective fourspace,*Ph.D. Thesis, Saarbücken, 1993. - Semple, G., Roth, L.,
*Algebraic Geometry*, Chelsea 1937.

## Additional Information

**Mark Gross**- Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
- MR Author ID: 308804
- Email: mgross@math.ucsd.edu
**Sorin Popescu**- Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
- Address at time of publication: Renaissance Technologies, 600 Route 25A, East Setauket, New York 11733
- Email: sorin@rentec.com
- Received by editor(s): April 21, 2009
- Received by editor(s) in revised form: August 4, 2009
- Published electronically: January 28, 2011
- Additional Notes: This work was partially supported by NSF grants DMS-0805328, DMS-0502070, DMS-0083361, and MSRI, Berkeley.
- © Copyright 2011 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**363**(2011), 3573-3599 - MSC (2000): Primary 14K10, 14J32
- DOI: https://doi.org/10.1090/S0002-9947-2011-05179-2
- MathSciNet review: 2775819