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Picard-Fuchs Systems Arising from Toric and Flag Varieties Picard-Fuchs Systems Arising From Toric and Flag Varieties The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:40050104 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Picard-Fuchs systems arising from toric and flag varieties A dissertation presented by Chenglong Yu to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts March 2018 © 2018 Chenglong Yu All rights reserved. Dissertation Advisor: Professor Shing-Tung Yau Author: Chenglong Yu Picard-Fuchs systems arising from toric and flag varieties Abstract This thesis studies the Picard-Fuchs systems for families arising as vector bundles zero loci in toric or partial flag varieties, including Riemann-Hilbert type theorems and arith- metic properties of the differential systems. The theory of tautological systems is proposed by Lian, Song and Yau in [51, 52]. These Picard-Fuchs type differential systems arise from the variation of Hodge structures of complete intersections in variety X with large symmetries, generalizing Gel'fand-Kapranov- Zelevinski systems for toric varieties [25]. The form of tautological systems makes it natural to introduces the powerful tool of D-modules into the study of hypersurface family in X. Following this direction, Riemann-Hilbert type theorems are obtained by Bloch, Huang, Lian, Srinivas, Yau and Zhu in [6, 34, 35], including solution rank formula, completeness and geometric interpretation of solution sheaves. In the first part of this thesis, we generalize these results in two aspects, one is the construction of tautological systems for vector bundles, the other is Riemann-Hilbert type theorems in this case. In the second part of the thesis, we examine an explicit description of Jacobian rings for homogenous vector bundles. This can be viewed as a description of the graded quotients of tautological systems with respect to the natural filtered D-module structure. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these iii conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian ring to study the null varieties of period integrals and their derivatives, generalizing a result in [16] for projective spaces. As an additional application, we prove the Hodge conjecture for very generic hypersurfaces in certain generalized flag varieties. The last part is devoted to the arithmetric properties of fundamental periods near large complex structure limit. Motivated by the work of Candelas, de la Ossa and Rodriguez- Villegas [14], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [67] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [38, 39]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [39] to Dwork's congruence relations [21] about periods. iv Contents 1 Introduction 1 1.1 Tautological systems for homogeneous vector bundles . .3 1.2 Jacobian ring for homogenous bundles . .4 1.3 Hasse-Witt matrices, unit roots and period integrals . .5 1.4 Notations . .8 2 Tautological systems for homogenous vector bundles 8 2.1 Period integrals for zero loci of vector bundles . .8 2.1.1 Calabi-Yau bundles and adjunction formulas . 10 2.1.2 Tautological systems . 13 2.1.3 Examples . 18 2.2 Global residue on P(E_)............................ 19 2.3 Solution rank . 20 2.3.1 Lie algebra homology description . 20 2.3.2 Perverse sheaves description . 21 2.3.3 Irreducible homogeneous vector bundles . 23 2.3.4 Complete intersections . 24 2.4 Chain integrals in log homogenous varieties . 25 3 Jaobian ring for homogenous vector bundles 26 3.1 Line bundles . 26 3.2 Vanishing conditions . 33 3.3 An application to differential zeros of period integrals . 38 3.4 Zero loci of vector bundle sections . 41 v 3.5 Hypersurfaces in log homogenous varieties . 46 3.6 Hodge conjecture for very generic hypersurfaces . 48 4 Arithmetic properties of period integrals 49 4.1 Introduction . 49 4.1.1 Notations . 50 4.1.2 Statement of the theorem . 52 4.2 Local expansions and Hasse-Witt matrices . 54 4.3 Generalized flag vareities . 62 4.4 Complete intersections . 68 4.5 Frobenius matrices of toric hypersurfaces . 70 4.6 Unit root of toric Calabi-Yau hypersurfaces and periods . 79 4.7 Frobenius matrices for Calabi-Yau hypersurfaces . 81 4.7.1 Unit root of Calabi-Yau hypersurfaces in G=P ............ 82 vi Acknowledgements First and foremost I would like to thank my advisor Professor Shing-Tung Yau, for his careful guidance, inspiring discussions, constant support and encouragement. Without his generosity in sharing ideas and his invaluable support I would not be able to finish my thesis work. I would like to thank Professor Bong Lian for sharing his ideas and discussions of all kinds of interesting topics. I thank Professor An Huang for many discussions on our collaborations. I thank Professor Wilfried Schmid for his advises and encouragement. I thank Professor Joe Harris for serving on my thesis committee. I also thank Professor Matt Kerr, Professor Colleen Robles, Professor Mao Sheng and Professor Charles Doran for their interests in my thesis work and helpful conversations. I would also like to thank Professor Gerard van der Geer for his guidance during my last year in Tsinghua and the invitation to Amsterdam, which was a memorable experience for me. I also want to thank Dr. Yu Zhou for encouraging me to switch from chemistry to mathematics. Yau's students seminar opened my eyes to many interesting topics and has been a great source for me to learn mathematics. I owe a lot to the participants, my colleagues and friends Peter Smillie, Jie Zhou, Teng Fei, Yu-Wei Fan, Jonathan Zhu, Baosen Wu, Tristan Collins, Yi Xie, Takahashi Ryosuke, Dingxin Zhang, Zijian Yao, Meng Guo, Boyu Zhang, Ziliang Che, Yusheng Luo, Eduard Duryev, Koji Shimizu, Zhiwei Zheng for helpful and enjoyable discussions on mathematics. Most computation for the last part of this thesis was done in Xiaolin Shi and Yixiang Mao's living room. I appreciate their warm and heartfelt hospitality. My thanks also go to the stuff members in the department and Bok center. I really vii appreciate the help from Jameel Al-Aidroos, Janet Chen, Robin Gottlieb, Yu-Wen Hsu, Oliver Knill, Pamela Pollock and Kate Penner on teaching calculus. I benefited a lot from the support of the teaching team at Harvard. Without the apprentice and coaching system, I would not overcome the language problems and have the confidence to stand in front of my students. I am also grateful for the help from Maureen Armstrong, Pam Brentana, Diana Chen, Susan Gilbert, Larissa Kennedy, Anna Kreslavskaya, Sarah LaBauve, Gaby Leon-Guerrero and Irene Minder. Last but not the least I would like to thank my family for their love. Without their understanding and unconditional support, I would not have the courage to pursue my dream and academic career. I also want to thank Wenyu for all the great memories together. This thesis is dedicated to them. viii 1 Introduction Period integrals connect Hodge theory, number theory, mirror symmetry, and many other important areas of math. The study of periods has a long history dating back to Euler, Legendre, Gauss, Abel, Jacobi and Picard in the form of special functions, which are periods of curves. Period integrals first appears in the form of hypergeometric functions. The name hypergeometric functions first appeared in the book Arithmetica Infinitorum (1655) by John Wallis. Legendre first made the connection to geometry of the hypergeometric functions through the theory of elliptic integrals. On the other hand, Euler studied the Euler-Gauss hypergeometric equations. In special case, it is the Picard-Fuchs equation for the Legendre family of elliptic curves. Euler obtained the power series solutions to the Euler-Gauss equations and also wrote then in terms of period integrals. He introduced the important idea that instead of considering at one elliptic integral alone, one should look at the family of the integrals and consider the corresponding differential equations. This is the first example of variation of Hodge structures and Picard-Fuchs systems. The theory of deformations of higher dimensional complex varieties was pioneered by Kodaira and Spencer [42, 41, 43]. The modern study of variation of Hodge structures starts from the work of Griffiths [29], and continues by Deligne [17, 18, 19] and Schmid [60]. In particular, the periods of hypersurfaces in Pn is studied in [29] from the point of view of variation of Hodge structures. Thanks to the relations between pole order filtrations and Hodge filtrations, there is a reduction of pole method to compute the Picard-Fuchs systems for hypersurfaces in Pn. Two natural generalizations of Pn are toric varieties and homogenous Fano varieties. For toric varieties, Gel'fand, Kapranov and Zelevinski [25] found a hypergeometric sys- tem that annihilates periods of toric hypersurfaces. More recently, Lian, Song, and Yau 1 [51, 52] established a holonomic differential system called the tautological system for vari- eties with large symmetry, which generalizes Gel'fand-Kapranov-Zelevinski systems.
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