Compactification of the moduli of polarized abelian varieties and mirror symmetry

Zhu, Yuecheng

doi:10.1090/tran/7008

Compactification of the moduli of polarized abelian varieties and mirror symmetry
HTML articles powered by AMS MathViewer

by Yuecheng Zhu PDF

Trans. Amer. Math. Soc. 370 (2018), 1693-1758 Request permission

Abstract:

We show that Martin Olsson’s compactification of moduli space of polarized abelian varieties can be interpreted in terms of KSBA stable pairs. We find that any degenerating family of polarized abelic sheme over a local normal base is equipped with a canonical set of divisors $S(K_2)$. Choosing any divisor $\Theta$ from the set $S(K_2)$, we get a KSBA stable pair. Then the limit in the moduli space of KSBA pairs $\overline {\mathscr {AP}}_{g,d}$ agrees with the canonical degeneration given by Martin Olsson’s compactification. Moreover, we give an alternative construction of the compactification by using mirror symmetry. We construct a toroidal compactification $\overline {\mathscr {A}}_{g,\delta }^m$ that is isomorphic to Olsson’s compactification over characteristic zero. The collection of fans needed for a toroidal compactification is obtained from the Mori fans of the minimal models of the mirror families.

References

V. Alexeev, Ch. Birkenhake, and K. Hulek, Degenerations of Prym varieties, J. Reine Angew. Math. 553 (2002), 73–116. MR 1944808, DOI 10.1515/crll.2002.103
V. Alexeev, Log canonical singularities and complete moduli of stable pairs, arXiv:9608013, 1996.
Valery Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155 (2002), no. 3, 611–708. MR 1923963, DOI 10.2307/3062130
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
Avner Ash, David Mumford, Michael Rapoport, and Yung-Sheng Tai, Smooth compactifications of locally symmetric varieties, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. With the collaboration of Peter Scholze. MR 2590897, DOI 10.1017/CBO9780511674693
Valery Alexeev and Iku Nakamura, On Mumford’s construction of degenerating abelian varieties, Tohoku Math. J. (2) 51 (1999), no. 3, 399–420. MR 1707764, DOI 10.2748/tmj/1178224770
Dan Abramovich and Angelo Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), no. 1, 27–75. MR 1862797, DOI 10.1090/S0894-0347-01-00380-0
Christina Birkenhake and Herbert Lange, Complex abelian varieties, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673, DOI 10.1007/978-3-662-06307-1
Armand Borel, Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969 (French). MR 0244260
James A. Carlson, Eduardo H. Cattani, and Aroldo G. Kaplan, Mixed Hodge structures and compactifications of Siegel’s space (preliminary report), Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 77–105. MR 605337
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
A. J. de Jong et al., The Stacks Project.
Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353, DOI 10.1007/978-3-662-02632-8
Kenji Fukaya, Mirror symmetry of abelian varieties and multi-theta functions, J. Algebraic Geom. 11 (2002), no. 3, 393–512. MR 1894935, DOI 10.1090/S1056-3911-02-00329-6
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
M. Gross, P. Hacking, and S. Keel, Theta functions for K3 surfaces, preliminary version.
Mark Gross, Paul Hacking, and Sean Keel, Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65–168. MR 3415066, DOI 10.1007/s10240-015-0073-1
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
V. Golyshev, V. Lunts, and D. Orlov, Mirror symmetry for abelian varieties, J. Algebraic Geom. 10 (2001), no. 3, 433–496. MR 1832329
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI 10.1307/mmj/1030132722
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
Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. MR 1296725, DOI 10.2307/2374941
Yujiro Kawamata, On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997), no. 5, 665–687. MR 1468356, DOI 10.1142/S0129167X97000354
Yujiro Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423. MR 2426353, DOI 10.2977/prims/1210167332
János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
Seán Keel and James McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, viii+153. MR 1610249, DOI 10.1090/memo/0669
Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, 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], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
Laurent Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985), 266 (French, with English summary). MR 797982
David R. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218 (1993), 243–271. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265317
D. Mumford, On the equations defining abelian varieties. I, Invent. Math. 1 (1966), 287–354. MR 204427, DOI 10.1007/BF01389737
David Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239–272. MR 352106
Grigory Mikhalkin and Ilia Zharkov, Tropical curves, their Jacobians and theta functions, Curves and abelian varieties, Contemp. Math., vol. 465, Amer. Math. Soc., Providence, RI, 2008, pp. 203–230. MR 2457739, DOI 10.1090/conm/465/09104
I. Nakamura, Compactification by GIT-stability of the moduli space of Abelian varieties, arXiv:1406.0174.
Iku Nakamura, Stability of degenerate abelian varieties, Invent. Math. 136 (1999), no. 3, 659–715. MR 1695209, DOI 10.1007/s002220050322
Iku Nakamura, Another canonical compactification of the moduli space of abelian varieties, Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), Adv. Stud. Pure Math., vol. 58, Math. Soc. Japan, Tokyo, 2010, pp. 69–135. MR 2676158, DOI 10.2969/aspm/05810069
Yukihiko Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Mathematics, vol. 812, Springer, Berlin, 1980. MR 584625
A. Ogus, Lectures on Logarithmic Algebraic Geometry, preprint, 2006.
Martin C. Olsson, Compactifying moduli spaces for abelian varieties, Lecture Notes in Mathematics, vol. 1958, Springer-Verlag, Berlin, 2008. MR 2446415, DOI 10.1007/978-3-540-70519-2
Martin Olsson, Algebraic spaces and stacks, American Mathematical Society Colloquium Publications, vol. 62, American Mathematical Society, Providence, RI, 2016. MR 3495343, DOI 10.1090/coll/062
Alexander Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Mathematics, vol. 153, Cambridge University Press, Cambridge, 2003. MR 1987784, DOI 10.1017/CBO9780511546532
Igor R. Shafarevich, Basic algebraic geometry. 1, Translated from the 2007 third Russian edition, Springer, Heidelberg, 2013. Varieties in projective space. MR 3100243
Sug Woo Shin, Abelian varieties and Weil representations, Algebra Number Theory 6 (2012), no. 8, 1719–1772. MR 3033526, DOI 10.2140/ant.2012.6.1719
A. Tyurin, Geometric quantization and mirror symmetry, arXiv:9902027.
Paolo Valabrega, A few theorems on completion of excellent rings, Nagoya Math. J. 61 (1976), 127–133. MR 407007