Hasse-Witt matrices, unit roots and period integrals. (English) Zbl 1527.14052
This paper is concerned with Calabi-Yau hypersurfaces in both toric varieties and flag varieties. In particular, the paper studies relations among Hasse-Witt matrices, unit root parts of the zeta-functions, and period integrals.
Such relations were studied by Dwork in his work on the Dwork families of hypersurfaces.
This paper generalizes the abvoe relations to hypersurfaces of Calabi-Yau, or of general type in a complete toric variety \(X\), or flag variety of dimension \(n\).
The first main result is about Hasse-Witt matrices. Consider the cases of the universal Calabi-Yau family of hypersurfaces in a toric variety \(X\). There is a canonical degeneration \(Y_{s_0}\) (called the large complex structure limit) such that there is a canonical invariant cycle \(\gamma_0\in H_{n-1}(Y)\) near \(Y_{s_0}\). There is also a canonical trivialization \(\omega\) of the holomorphic \(n\)-form of the Calabi-Yau family. Tje period integral \(\int_{\gamma_0}\omega\) has a power series expansion at \(s_0\) with integer coefficients.
Theorem 1. (a) A certain trancation of the period integral \(\int_{\gamma_0}\omega\) is equal to the Hasse-Witt matrix of the reduction of the family modulo a prime \(p\).
(b) If the Hasse-Witt matrix is nondegenerate, there is a unique \(p\)-adic unit root of the zeta function. There is an explicit formula for the \(p\)-adic unit root in terms of the period integral.
An algorithm to compute the Hasse-Witt matrix is presented relating the Hasse-Witt operator to the Cartier operator (as classically done). Then the algorithm is applied to the computation of the Hasse-Witt matriix of a general toric hypersurface, which, in turn, yields the relation to the period integral.
The second main result is to give relations between the unit root part of the zeta function and period integral. This is done first proving Vlasenko’s conjecture [M. Vlasenko, Indag. Math., New Ser. 29, No. 5, 1411–1424 (2018; Zbl 1423.11199)] about the Frobenius map and the connecting map for toric hypersurfaces, applying Katz’s local expansion method. Then it is shown that the Frobenius matrices (which is nothing but the Hasse-Witt matrices) of unit root \(F\)-crystals have close relations with the period integrals.
The above methods are also applied to flag varieties to prove the corresponding results.
The first main result is about Hasse-Witt matrices. Consider the cases of the universal Calabi-Yau family of hypersurfaces in a toric variety \(X\). There is a canonical degeneration \(Y_{s_0}\) (called the large complex structure limit) such that there is a canonical invariant cycle \(\gamma_0\in H_{n-1}(Y)\) near \(Y_{s_0}\). There is also a canonical trivialization \(\omega\) of the holomorphic \(n\)-form of the Calabi-Yau family. Tje period integral \(\int_{\gamma_0}\omega\) has a power series expansion at \(s_0\) with integer coefficients.
Theorem 1. (a) A certain trancation of the period integral \(\int_{\gamma_0}\omega\) is equal to the Hasse-Witt matrix of the reduction of the family modulo a prime \(p\).
(b) If the Hasse-Witt matrix is nondegenerate, there is a unique \(p\)-adic unit root of the zeta function. There is an explicit formula for the \(p\)-adic unit root in terms of the period integral.
An algorithm to compute the Hasse-Witt matrix is presented relating the Hasse-Witt operator to the Cartier operator (as classically done). Then the algorithm is applied to the computation of the Hasse-Witt matriix of a general toric hypersurface, which, in turn, yields the relation to the period integral.
The second main result is to give relations between the unit root part of the zeta function and period integral. This is done first proving Vlasenko’s conjecture [M. Vlasenko, Indag. Math., New Ser. 29, No. 5, 1411–1424 (2018; Zbl 1423.11199)] about the Frobenius map and the connecting map for toric hypersurfaces, applying Katz’s local expansion method. Then it is shown that the Frobenius matrices (which is nothing but the Hasse-Witt matrices) of unit root \(F\)-crystals have close relations with the period integrals.
The above methods are also applied to flag varieties to prove the corresponding results.
Reviewer: Noriko Yui (Kingston)
MSC:
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 11G25 | Varieties over finite and local fields |
| 14D07 | Variation of Hodge structures (algebro-geometric aspects) |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 32G20 | Period matrices, variation of Hodge structure; degenerations |
Keywords:
Calabi-Yau hypersurface; toric variety; flag variety; Hasse-Witt matrix; unit root; zeta function; period integralCitations:
Zbl 1423.11199References:
| [1] | Dwork, B.: A deformation theory for the zeta function of a hypersurface. In: Proceedings of the International Congress of Mathematics, Stockholm, pp. 247-259 (1962) · Zbl 0196.53302 |
| [2] | Dwork, B., \(p\)-adic cycles, Inst. Hautes Études Sci. Publ. Math., 37, 27-115 (1969) · Zbl 0284.14008 · doi:10.1007/BF02684886 |
| [3] | Dwork, B., On Hecke polynomials, Invent. Math., 12, 249-256 (1971) · Zbl 0219.14014 · doi:10.1007/BF01418784 |
| [4] | Katz, N.: Travaux de Dwork, 167-200317 (1973) · Zbl 0259.14007 |
| [5] | Katz, N., Algebraic solutions of differential equations \((p\)-curvature and the Hodge filtration), Invent. Math., 18, 1, 1-118 (1972) · Zbl 0278.14004 · doi:10.1007/BF01389714 |
| [6] | Yu, J-D, Variation of the unit root along the Dwork family of Calabi-Yau varieties, Math. Ann., 343, 1, 53-78 (2009) · Zbl 1158.14035 · doi:10.1007/s00208-008-0265-9 |
| [7] | Miyatani, K., Monomial deformations of certain hypersurfaces and two hypergeometric functions, In. J. Number Theory, 11, 8, 2405-2430 (2015) · Zbl 1332.14028 · doi:10.1142/S1793042115501122 |
| [8] | Adolphson, A.; Sperber, S., Distinguished-root formulas for generalized Calabi-Yau hypersurfaces, Algebra Number Theory, 11, 6, 1317-1356 (2017) · Zbl 1394.11054 · doi:10.2140/ant.2017.11.1317 |
| [9] | Wan, D.: Mirror symmetry for zeta functions. arXiv:math/0411464 (arXiv preprint) (2004) · Zbl 1116.11044 |
| [10] | Doran, CF; Kelly, TL; Salerno, A.; Sperber, S.; Voight, J.; Whitcher, U., Zeta functions of alternate mirror Calabi-Yau families, Israel J. Math., 228, 2, 665-705 (2018) · Zbl 1403.14055 · doi:10.1007/s11856-018-1783-0 |
| [11] | Kloosterman, R., Zeta functions of monomial deformations of Delsarte hypersurfaces. Symmetry, integrability and geometry, Methods Appl., 13, 25 (2017) |
| [12] | Vlasenko, M., Higher Hasse-Witt matrices, Indag. Math., 29, 5, 1411-1424 (2018) · Zbl 1423.11199 · doi:10.1016/j.indag.2018.07.004 |
| [13] | Katz, N., Internal reconstruction of unit-root \(F \)-crystals via expansion-coefficients With an appendix by Luc Illusie, Ann. Sci. Ecole Norm. Sup., 245-285 (1985), New York: Elsevier, New York · Zbl 0592.14021 |
| [14] | Adolphson, A.; Sperber, S., A-hypergeometric series and the Hasse-Witt matrix of a hypersurface, Finite Fields Appl., 41, 55-63 (2016) · Zbl 1416.11101 · doi:10.1016/j.ffa.2016.05.001 |
| [15] | Katz, N., Expansion-coefficients as approximate solution of differential-equations, Astérisque, 119, 183-189 (1984) · Zbl 0561.14010 |
| [16] | Knutson, A.; Lam, T.; Speyer, DE, Projections of Richardson varieties, J. Reine Angewa. Mat. (Crelles J.), 2014, 687, 133-157 (2014) · Zbl 1345.14047 · doi:10.1515/crelle-2012-0045 |
| [17] | Huang, A.; Lian, BH; Zhu, X., Period integrals and the Riemann-Hilbert correspondence, J. Differ. Geometry, 104, 2, 325-369 (2016) · Zbl 1387.14042 · doi:10.4310/jdg/1476367060 |
| [18] | Fulton, W., A fixed point formula for varieties over finite fields, Math. Scand., 42, 2, 189-196 (1978) · Zbl 0417.14006 · doi:10.7146/math.scand.a-11747 |
| [19] | Candelas, P., de la Ossa, X., Rodriguez-Villegas, F.: Calabi-Yau manifolds over finite fields, I. arXiv:hep-th/0012233 (2000) · Zbl 1100.14032 |
| [20] | Beukers, F., Vlasenko, M.: Dwork crystals I. arXiv:1903.11155 (arXiv preprint) (2019) · Zbl 1492.14039 |
| [21] | Beukers, F., Vlasenko, M.: Dwork crystals II. arXiv:1907.10390 (arXiv preprint) (2019) · Zbl 1501.11051 |
| [22] | Huang, A., Stoica, B., Yau, S.-T.: General relativity from \(p \)-adic strings. arXiv:1901.02013 (arXiv preprint) (2019) · Zbl 1521.83006 |
| [23] | Brion, M.; Kumar, S., Frobenius Splitting Methods in Geometry and Representation Theory (2007), Berlin: Springer, Berlin · Zbl 1072.14066 |
| [24] | Adolphson, A.; Sperber, S., Hasse invariants and mod \(p\) solutions of A-hypergeometric systems, J. Number Theory, 142, 183-210 (2014) · Zbl 1295.11134 · doi:10.1016/j.jnt.2014.02.010 |
| [25] | Lusztig, G., Total positivity in partial flag manifolds, Represent. Theory Am. Math. Soc., 2, 3, 70-78 (1998) · Zbl 0895.14014 · doi:10.1090/S1088-4165-98-00046-6 |
| [26] | Rietsch, K., Closure relations for totally nonnegative cells in \(g/p \), Math. Res. Lett., 13, 5, 775-786 (2006) · Zbl 1107.14040 · doi:10.4310/MRL.2006.v13.n5.a8 |
| [27] | Goodearl, K.; Yakimov, M., Poisson structures on affine spaces and flag varieties. II, Trans. Am. Math. Soc., 361, 11, 5753-5780 (2009) · Zbl 1179.53087 · doi:10.1090/S0002-9947-09-04654-6 |
| [28] | Brion, M., Lectures on the geometry of flag varieties, Topics in Cohomological Studies of Algebraic Varieties, 33-85 (2005), Berlin: Springer, Berlin · Zbl 1487.14105 · doi:10.1007/3-7643-7342-3_2 |
| [29] | Brion, M.; Lakshmibai, V., A geometric approach to standard monomial theory, Represent. Theory Am. Math. Soc., 7, 25, 651-680 (2003) · Zbl 1053.14056 · doi:10.1090/S1088-4165-03-00211-5 |
| [30] | Rietsch, K., A mirror symmetric construction of \(qHT_*(G/P)(q)\), Adv. Math., 217, 6, 2401-2442 (2008) · Zbl 1222.14107 · doi:10.1016/j.aim.2007.08.010 |
| [31] | Rietsch, K., A mirror symmetric solution to the quantum toda lattice, Commun. Math. Phys., 309, 1, 23-49 (2012) · Zbl 1256.14042 · doi:10.1007/s00220-011-1308-8 |
| [32] | Marsh, R., Rietsch, K.: The B-model connection and mirror symmetry for Grassmannians. arXiv:1307.1085 (arXiv preprint) (2013) · Zbl 1453.14104 |
| [33] | Lam, T., Templier, N.: The mirror conjecture for minuscule flag varieties. arXiv:1705.00758 (arXiv preprint) (2017) |
| [34] | Stienstra, J., Formal group laws arising from algebraic varieties, Am. J. Math., 109, 5, 907-925 (1987) · Zbl 0633.14022 · doi:10.2307/2374494 |
| [35] | Stienstra, J., Formal groups and congruences for L-functions, Am. J. Math., 109, 6, 1111-1127 (1987) · Zbl 0645.14011 · doi:10.2307/2374587 |
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.
