Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Logarithmic Frobenius manifolds, hypergeometric systems and quantum $ \mathcal D$-modules

Authors: Thomas Reichelt and Christian Sevenheck
Journal: J. Algebraic Geom. 24 (2015), 201-281
Published electronically: February 12, 2014
MathSciNet review: 3311584
Full-text PDF

Abstract | References | Additional Information

Abstract: We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered $ \mathcal {D}$-modules. We discuss in particular the logarithmic degeneration behavior at the large radius limit point and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauß-Manin system of the mirror Landau-Ginzburg model with a hypergeometric $ \mathcal {D}$-module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined by this filtration and show the existence of a primitive form, which yields the construction of Frobenius structures with logarithmic poles associated to the mirror Laurent polynomial. As a final application, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski open subset of the complexified Kähler moduli space of the variety.

References [Enhancements On Off] (What's this?)

  • [Ado94] Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), no. 2, 269-290. MR 1262208 (96c:33020),
  • [AS12] Alan Adolphson and Steven Sperber, $ A$-hypergeometric systems that come from geometry, Proc. Amer. Math. Soc. 140 (2012), no. 6, 2033-2042. MR 2888191,
  • [Bar00] Serguei Barannikov, Semi-infinite Hodge structures and mirror symmetry for projective spaces, Preprint math.AG/0010157, 2000.
  • [Bat91] Victor V. Batyrev, On the classification of smooth projective toric varieties, Tohoku Math. J. (2) 43 (1991), no. 4, 569-585. MR 1133869 (92j:14065),
  • [Bat93] Victor V. Batyrev, Quantum cohomology rings of toric manifolds, Astérisque 218 (1993), 9-34. Journées de Géométrie Algébrique d'Orsay (Orsay, 1992). MR 1265307 (95b:32034)
  • [Bat94] Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493-535. MR 1269718 (95c:14046)
  • [BH93] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993.
  • [BH06] Lev A. Borisov and R. Paul Horja, Mellin-Barnes integrals as Fourier-Mukai transforms, Adv. Math. 207 (2006), no. 2, 876-927. MR 2271990 (2007m:14056),
  • [Bri70] Egbert Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math. 2 (1970), 103-161 (German, with English summary). MR 0267607 (42 #2509)
  • [CK99] David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117 (2000d:14048)
  • [CL55] Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338 (16,1022b)
  • [CvR09] David A. Cox and Christine von Renesse, Primitive collections and toric varieties, Tohoku Math. J. (2) 61 (2009), no. 3, 309-332. MR 2568257 (2010k:14095),
  • [Dim04] Alexandru Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072 (2005j:55002)
  • [DL91] J. Denef and F. Loeser, Weights of exponential sums, intersection cohomology, and Newton polyhedra, Invent. Math. 106 (1991), no. 2, 275-294. MR 1128216 (93a:14019),
  • [DM09] Antoine Douai, A canonical Frobenius structure, Math. Z. 261 (2009), no. 3, 625-648. MR 2471092 (2010h:32038),
  • [Dou05] Antoine Douai, Construction de variétés de Frobenius via les polynômes de Laurent: une autre approche, Singularités, Inst. Élie Cartan, vol. 18, Univ. Nancy, Nancy, 2005, pp. 105-123 (French). MR 2205838 (2007c:53128)
  • [Dou09] Antoine Douai, A canonical Frobenius structure, Math. Z. 261 (2009), no. 3, 625-648. MR 2471092 (2010h:32038),
  • [DS03] A. Douai and C. Sabbah, Gauss-Manin systems, Brieskorn lattices and Frobenius structures. I, Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), 2003, pp. 1055-1116 (English, with English and French summaries). MR 2033510 (2005h:32073)
  • [DS04] A. Douai and C. Sabbah, Gauss-Manin systems, Brieskorn lattices and Frobenius structures. II, Frobenius manifolds, Aspects Math., E36, Vieweg, Wiesbaden, 2004, pp. 1-18.
  • [Ful93] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037 (94g:14028)
  • [Giv98] Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., vol. 160, Birkhäuser Boston, Boston, MA, 1998, pp. 141-175. MR 1653024 (2000a:14063)
  • [GKZ90] I. M. Gel'fand, M. M. Kapranov, and A. V. Zelevinsky, Generalized Euler integrals and $ A$-hypergeometric functions, Adv. Math. 84 (1990), no. 2, 255-271.
  • [GMS09] Ignacio de Gregorio, David Mond, and Christian Sevenheck, Linear free divisors and Frobenius manifolds, Compos. Math. 145 (2009), no. 5, 1305-1350. MR 2551998 (2011b:32049),
  • [Gue08] Martin A. Guest, From quantum cohomology to integrable systems, Oxford Graduate Texts in Mathematics, vol. 15, Oxford University Press, Oxford, 2008. MR 2391365 (2009f:53144)
  • [Her02] Claus Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, vol. 151, Cambridge University Press, Cambridge, 2002. MR 1924259 (2004a:32043)
  • [Her03] Claus Hertling, $ tt^*$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77-161. MR 1956595 (2005f:32049),
  • [HM04] Claus Hertling and Yuri Manin, Unfoldings of meromorphic connections and a construction of Frobenius manifolds, Frobenius manifolds, Aspects Math., E36, Vieweg, Wiesbaden, 2004, pp. 113-144. MR 2115768 (2005k:32013)
  • [Hoc72] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318-337. MR 0304376 (46 #3511)
  • [Hot98] Ryoshi Hotta, Holonomic $ {\mathcal {D}}$-modules in representation theory, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 87-102. MR 933353 (89g:22020)
  • [HS07] Claus Hertling and Christian Sevenheck, Nilpotent orbits of a generalization of Hodge structures, J. Reine Angew. Math. 609 (2007), 23-80. MR 2350780 (2009c:32036),
  • [HS10] Claus Hertling and Christian Sevenheck, Limits of families of Brieskorn lattices and compactified classifying spaces, Adv. Math. 223 (2010), no. 4, 1155-1224. MR 2581368 (2010m:32017),
  • [HTT08] Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $ D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361 (2008k:32022)
  • [Iri06] Hiroshi Iritani, Quantum $ D$-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (2006), no. 3, 577-622. MR 2207760 (2007e:53118),
  • [Iri07] Hiroshi Iritani, Convergence of quantum cohomology by quantum Lefschetz, J. Reine Angew. Math. 610 (2007), 29-69. MR 2359850 (2008k:14104),
  • [Iri09a] Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016-1079. MR 2553377 (2010j:53182),
  • [Iri09b] Hiroshi Iritani, $ tt^*$-geometry in quantum cohomology, Preprint math.AG/0906.1307, 2009.
  • [Kho77] A. G. Hovanskiĭ, Newton polyhedra, and toroidal varieties, Funkcional. Anal. i Priložen. 11 (1977), no. 4, 56-64, 96 (Russian). MR 0476733 (57 #16291)
  • [KKP08] L. Katzarkov, M. Kontsevich, and T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78, Amer. Math. Soc., Providence, RI, 2008, pp. 87-174. MR 2483750 (2009j:14052)
  • [Kou76] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1-31 (French). MR 0419433 (54 #7454)
  • [Man99] Yuri I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, vol. 47, American Mathematical Society, Providence, RI, 1999. MR 1702284 (2001g:53156)
  • [Man08] Etienne Mann, Orbifold quantum cohomology of weighted projective spaces, J. Algebraic Geom. 17 (2008), no. 1, 137-166. MR 2357682 (2008k:14106),
  • [MMW05] Laura Felicia Matusevich, Ezra Miller, and Uli Walther, Homological methods for hypergeometric families, J. Amer. Math. Soc. 18 (2005), no. 4, 919-941 (electronic). MR 2163866 (2007d:13027),
  • [Moc02] Takuro Mochizuki, Asymptotic behaviour of tame nilpotent harmonic bundles with trivial parabolic structure, J. Differential Geom. 62 (2002), no. 3, 351-559. MR 2005295 (2005f:32032)
  • [Moc09] Takuro Mochizuki, Good formal structure for meromorphic flat connections on smooth projective surfaces, Algebraic analysis and around, Adv. Stud. Pure Math., vol. 54, Math. Soc. Japan, Tokyo, 2009, pp. 223-253. MR 2499558 (2010h:14027)
  • [Moc10] Takuro Mochizuki, Holonomic D-module with Betti structure, Preprint math/1001.2336, 2010.
  • [Moc11a] Takuro Mochizuki, Asymptotic behavior of variation of pure polarized TERP structure, Publ. Res. Inst. Math. Sci. 47 (2011), no. 2, 419-534. MR 2849639 (2012i:32022),
  • [Moc11b] Takuro Mochizuki, Wild harmonic bundles and wild pure twistor $ D$-modules, Astérisque 340 (2011), x+607 (English, with English and French summaries). MR 2919903
  • [MS05] Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098 (2006d:13001)
  • [Pan98] Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), Astérisque 252 (1998), Exp. No. 848, 5, 307-340. Séminaire Bourbaki. Vol. 1997/98. MR 1685628 (2000e:14094)
  • [Pha79] Frédéric Pham, Singularités des systèmes différentiels de Gauss-Manin, Progress in Mathematics, vol. 2, Birkhäuser Boston, Mass., 1979 (French). With contributions by Lo Kam Chan, Philippe Maisonobe and Jean-Étienne Rombaldi. MR 553954 (81h:32015)
  • [Rei09] Thomas Reichelt, A construction of Frobenius manifolds with logarithmic poles and applications, Comm. Math. Phys. 287 (2009), no. 3, 1145-1187. MR 2486676 (2009m:53232),
  • [Sab97] Claude Sabbah, Monodromy at infinity and Fourier transform, Publ. Res. Inst. Math. Sci. 33 (1997), no. 4, 643-685. MR 1489993 (99f:14011),
  • [Sab02] Claude Sabbah, Déformations isomonodromiques et variétés de Frobenius, Savoirs Actuels, EDP Sciences, Les Ulis, 2002, Mathématiques.
  • [Sab06] Claude Sabbah, Hypergeometric periods for a tame polynomial, Port. Math. (N.S.) 63 (2006), no. 2, 173-226. MR 2229875 (2007i:32014)
  • [Sab08] Claude Sabbah, Fourier-Laplace transform of a variation of polarized complex Hodge structure, J. Reine Angew. Math. 621 (2008), 123-158. MR 2431252 (2010b:32017),
  • [Sab11] Claude Sabbah, Non-commutative Hodge structures, Ann. Inst. Fourier (Grenoble), 61 (2011), no. 7, 2681-2717. MR 3112504
  • [Sai89] Morihiko Saito, On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble) 39 (1989), no. 1, 27-72 (English, with French summary). MR 1011977 (91i:32035)
  • [Sai94] Morihiko Saito, On the theory of mixed Hodge modules, Selected papers on number theory, algebraic geometry, and differential geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 160, Amer. Math. Soc., Providence, RI, 1994, Translated from Sūgaku, Translation edited by Katsumi Nomizu, pp. 47-61.
  • [Sai01] Mutsumi Saito, Isomorphism classes of $ A$-hypergeometric systems, Compositio Math. 128 (2001), no. 3, 323-338. MR 1858340 (2003f:33019),
  • [Sch85] Pierre Schapira, Microdifferential systems in the complex domain, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 269, Springer-Verlag, Berlin, 1985. MR 774228 (87k:58251)
  • [Sev11] Christian Sevenheck, Bernstein polynomials and spectral numbers for linear free divisors, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 1, 379-400 (English, with English and French summaries). MR 2828135,
  • [Sim88] Carlos T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867-918. MR 944577 (90e:58026),
  • [SST00] Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000. MR 1734566 (2001i:13036)
  • [SW09] Mathias Schulze and Uli Walther, Hypergeometric D-modules and twisted Gauß-Manin systems, J. Algebra 322 (2009), no. 9, 3392-3409. MR 2567427 (2010m:14028),
  • [Wal07] Uli Walther, Duality and monodromy reducibility of $ A$-hypergeometric systems, Math. Ann. 338 (2007), no. 1, 55-74. MR 2295504 (2008e:32043),
  • [Wiś02] Geometry of toric varieties, Séminaires et Congrès [Seminars and Congresses], vol. 6, Société Mathématique de France, Paris, 2002. Lectures from the Summer School held in Grenoble, June 19-July 7, 2000; Edited by Laurent Bonavero and Michel Brion. MR 2072676 (2005a:14001)

Additional Information

Thomas Reichelt
Affiliation: Lehrstuhl für Mathematik VI, Institut für Mathematik, Universität Mannheim, A 5, 6, 68131 Mannheim, Germany

Christian Sevenheck
Affiliation: Lehrstuhl für Mathematik VI, Institut für Mathematik, Universität Mannheim, A 5, 6, 68131 Mannheim, Germany

Received by editor(s): August 12, 2011
Received by editor(s) in revised form: September 20, 2012
Published electronically: February 12, 2014
Additional Notes: The first author was supported by the DFG grant He 2287/2-2.
The second author was supported by a DFG Heisenberg fellowship (Se 1114/2-1).
Article copyright: © Copyright 2014 University Press, Inc.

American Mathematical Society