Periods of tropical Calabi–Yau hypersurfaces
Author:
Yuto Yamamoto
Journal:
J. Algebraic Geom. 31 (2022), 303-343
DOI:
https://doi.org/10.1090/jag/778
Published electronically:
December 20, 2021
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Abstract |
References |
Additional Information
Abstract: We consider the residual B-model variation of Hodge structure of Iritani defined by a family of toric Calabi–Yau hypersurfaces over a punctured disk $D \setminus \left \{ 0\right \}$. It is naturally extended to a logarithmic variation of polarized Hodge structure of Kato–Usui on the whole disk $D$. By restricting it to the origin, we obtain a polarized logarithmic Hodge structure (PLH) on the standard log point. In this paper, we describe the PLH in terms of the integral affine structure of the dual intersection complex of the toric degeneration in the Gross–Siebert program.
References
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- Arthur Ogus, On the logarithmic Riemann-Hilbert correspondence, Doc. Math. Extra Vol. (2003), 655–724. Kazuya Kato’s fiftieth birthday. MR 2046612
- Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, 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. 52, Springer-Verlag, Berlin, 2008. MR 2393625
- Helge Ruddat, Log Hodge groups on a toric Calabi-Yau degeneration, Mirror symmetry and tropical geometry, Contemp. Math., vol. 527, Amer. Math. Soc., Providence, RI, 2010, pp. 113–164. MR 2681794, DOI 10.1090/conm/527/10402
- Helge Ruddat, A homology theory for tropical cycles on integral affine manifolds and a perfect pairing, Preprint, 2020.
- Helge Ruddat and Bernd Siebert, Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations, arXiv:1907.03794, 2019.
- Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 382272, DOI 10.1007/BF01389674
- Yuki Tsutsui, Radiance obstructions of tropical Kummer surfaces and their quotients, Preprint, 2020.
- Kazushi Ueda, Mirror symmetry and $K3$ surfaces, arXiv:1407.1566, 2014.
References
- Mohammed Abouzaid, Sheel Ganatra, Hiroshi Iritani, and Nick Sheridan, The Gamma and Strominger-Yau-Zaslow conjectures: a tropical approach to periods, arXiv:1809.02177, 2018.
- Victor V. Batyrev, Quantum cohomology rings of toric manifolds, Astérisque 218 (1993), 9–34. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265307
- 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
- Victor V. Batyrev and Lev A. Borisov, Mirror duality and string-theoretic Hodge numbers, Invent. Math. 126 (1996), no. 1, 183–203. MR 1408560, DOI 10.1007/s002220050093
- Victor V. Batyrev and Dimitrios I. Dais, Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology 35 (1996), no. 4, 901–929. MR 1404917, DOI 10.1016/0040-9383(95)00051-8
- 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, DOI 10.1090/surv/068
- V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
- Simon Felten, Matej Filip, and Helge Ruddat, Smoothing toroidal crossing spaces, arXiv:1908.11235, 2019.
- William Goldman and Morris W. Hirsch, The radiance obstruction and parallel forms on affine manifolds, Trans. Amer. Math. Soc. 286 (1984), no. 2, 629–649. MR 760977, DOI 10.2307/1999812
- Mark Gross, Toric degenerations and Batyrev-Borisov duality, Math. Ann. 333 (2005), no. 3, 645–688. MR 2198802, DOI 10.1007/s00208-005-0686-7
- Mark Gross and Bernd Siebert, Mirror symmetry via logarithmic degeneration data. I, J. Differential Geom. 72 (2006), no. 2, 169–338. MR 2213573
- Mark Gross and Bernd Siebert, Mirror symmetry via logarithmic degeneration data, II, J. Algebraic Geom. 19 (2010), no. 4, 679–780. MR 2669728, DOI 10.1090/S1056-3911-2010-00555-3
- Mark Gross and Bernd Siebert, From real affine geometry to complex geometry, Ann. of Math. (2) 174 (2011), no. 3, 1301–1428. MR 2846484, DOI 10.4007/annals.2011.174.3.1
- Mark Gross and P. M. H. Wilson, Large complex structure limits of $K3$ surfaces, J. Differential Geom. 55 (2000), no. 3, 475–546. MR 1863732
- Mark Gross, Paul Hacking, and Bernd Siebert, Theta functions on varieties with effective anti-canonical class, arXiv:1601.07081, 2016.
- Christian Haase and Ilia Zharkov, Integral affine structures on spheres and torus fibrations of Calabi–Yau toric hypersurfaces I, arXiv:math/0205321, 2002.
- Christian Haase and Ilia Zharkov, Integral affine structures on spheres: complete intersections, Int. Math. Res. Not. (2005), no. 51, 3153–3167. MR 2187503
- Kenji Hashimoto and Kazushi Ueda, Reconstruction of general elliptic K3 surfaces from their Gromov-Hausdorff limits, arXiv:1805.01719v1, 2018.
- Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. MR 2553377, DOI 10.1016/j.aim.2009.05.016
- Hiroshi Iritani, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958 (English, with English and French summaries). MR 3112512
- Shinsuke Iwao, Integration over tropical plane curves and ultradiscretization, Int. Math. Res. Not. IMRN 1 (2010), 112–148. MR 2576286, DOI 10.1093/imrn/rnp129
- Eric Katz, Hannah Markwig, and Thomas Markwig, The $j$-invariant of a plane tropical cubic, J. Algebra 320 (2008), no. 10, 3832–3848. MR 2457725, DOI 10.1016/j.jalgebra.2008.08.018
- Eric Katz, Hannah Markwig, and Thomas Markwig, The tropical $j$-invariant, LMS J. Comput. Math. 12 (2009), 275–294. MR 2570928
- Maxim Kontsevich and Yan Soibelman, Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 321–385. MR 2181810, DOI 10.1007/0-8176-4467-9_9
- Kazuya Kato and Chikara Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $\mathbf {C}$, Kodai Math. J. 22 (1999), no. 2, 161–186. MR 1700591, DOI 10.2996/kmj/1138044041
- Kazuya Kato and Sampei Usui, Classifying spaces of degenerating polarized Hodge structures, Annals of Mathematics Studies, vol. 169, Princeton University Press, Princeton, NJ, 2009. MR 2465224, DOI 10.1515/9781400837113
- 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
- Yuji Odaka and Yoshiki Oshima, Collapsing K3 surfaces and Moduli compactification, Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 8, 81–86. MR 3859764, DOI 10.3792/pjaa.94.81
- Yuji Odaka and Yoshiki Oshima, Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen type, arXiv:1810.07685, 2018.
- Arthur Ogus, On the logarithmic Riemann-Hilbert correspondence, Doc. Math. Extra Vol. (2003), 655–724. Kazuya Kato’s fiftieth birthday. MR 2046612
- Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, 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. 52, Springer-Verlag, Berlin, 2008. MR 2393625
- Helge Ruddat, Log Hodge groups on a toric Calabi-Yau degeneration, Mirror symmetry and tropical geometry, Contemp. Math., vol. 527, Amer. Math. Soc., Providence, RI, 2010, pp. 113–164. MR 2681794, DOI 10.1090/conm/527/10402
- Helge Ruddat, A homology theory for tropical cycles on integral affine manifolds and a perfect pairing, Preprint, 2020.
- Helge Ruddat and Bernd Siebert, Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations, arXiv:1907.03794, 2019.
- Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 382272, DOI 10.1007/BF01389674
- Yuki Tsutsui, Radiance obstructions of tropical Kummer surfaces and their quotients, Preprint, 2020.
- Kazushi Ueda, Mirror symmetry and $K3$ surfaces, arXiv:1407.1566, 2014.
Additional Information
Yuto Yamamoto
Affiliation:
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic of Korea
MR Author ID:
1166484
Email:
yuto@ibs.re.kr
Received by editor(s):
February 11, 2020
Received by editor(s) in revised form:
August 3, 2020, September 10, 2020, and September 14, 2020
Published electronically:
December 20, 2021
Additional Notes:
This work was supported by IBS-R003-D1, Grant-in-Aid for JSPS Research Fellow (18J11281), and the Program for Leading Graduate Schools, MEXT, Japan. The author was financially supported by Sam Payne for the visit to Yale University. The visit to University of Cambridge was supported by JSPS Overseas Challenge Program for Young Researchers.
Article copyright:
© Copyright 2021
University Press, Inc.
