Perverse curves and mirror symmetry
Author:
Helge Ruddat
Journal:
J. Algebraic Geom. 26 (2017), 17-42
DOI:
https://doi.org/10.1090/jag/666
Published electronically:
June 7, 2016
MathSciNet review:
3570582
Full-text PDF
Abstract |
References |
Additional Information
Abstract: This work establishes a subtle connection between mirror symmetry for Calabi-Yau threefolds and that of curves of higher genus. The linking structure is what we call a perverse curve. We show how to obtain such from Calabi-Yau threefolds in the Batyrev mirror construction and prove that their Hodge diamonds are related by the mirror duality.
References
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- M. Abouzaid and D. Auroux, Homological Mirror Symmetry for Hypersurfaces, work in progress.
- Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, and Dmitri Orlov, Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. 26 (2013), no. 4, 1051–1083. MR 3073884, DOI https://doi.org/10.1090/S0894-0347-2013-00770-5
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- Victor V. Batyrev and Lev A. Borisov, On Calabi-Yau complete intersections in toric varieties, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 39–65. MR 1463173
- V. I. Danilov and A. G. Khovanskiĭ, Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 5, 925–945 (Russian). MR 873655
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551
- José Bertin, Jean-Pierre Demailly, Luc Illusie, and Chris Peters, Introduction to Hodge theory, SMF/AMS Texts and Monographs, vol. 8, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2002. Translated from the 1996 French original by James Lewis and Peters. MR 1924513
- Alexander I. Efimov, Homological mirror symmetry for curves of higher genus, Adv. Math. 230 (2012), no. 2, 493–530. MR 2914956, DOI https://doi.org/10.1016/j.aim.2012.02.022
- Mark Gross, Topological mirror symmetry, Invent. Math. 144 (2001), no. 1, 75–137. MR 1821145, DOI https://doi.org/10.1007/s002220000119
- Mark Gross, Toric degenerations and Batyrev-Borisov duality, Math. Ann. 333 (2005), no. 3, 645–688. MR 2198802, DOI https://doi.org/10.1007/s00208-005-0686-7
- M. Gross, L. Katzarkov, and H. Ruddat, Towards Mirror Symmetry for Varieties of General Type, arXiv:math/1202.4042
- 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, From real affine geometry to complex geometry, Ann. of Math. (2) 174 (2011), no. 3, 1301–1428. MR 2846484, DOI https://doi.org/10.4007/annals.2011.174.3.1
- Luc Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, Astérisque 279 (2002), 271–322. Cohomologies $p$-adiques et applications arithmétiques, II. MR 1922832
- Dominic Joyce, A classical model for derived critical loci, J. Differential Geom. 101 (2015), no. 2, 289–367. MR 3399099
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
- Kazuya Kato and Chikara Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over ${\bf C}$, Kodai Math. J. 22 (1999), no. 2, 161–186. MR 1700591, DOI https://doi.org/10.2996/kmj/1138044041
- Anton Kapustin, Ludmil Katzarkov, Dmitri Orlov, and Mirroslav Yotov, Homological mirror symmetry for manifolds of general type, Cent. Eur. J. Math. 7 (2009), no. 4, 571–605. MR 2563433, DOI https://doi.org/10.2478/s11533-009-0056-x
- Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918
- Chikara Nakayama and Arthur Ogus, Relative rounding in toric and logarithmic geometry, Geom. Topol. 14 (2010), no. 4, 2189–2241. MR 2740645, DOI https://doi.org/10.2140/gt.2010.14.2189
- 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
- W.-D. Ruan, Lagrangian Tori Fibration of Toric Calabi-Yau manifold; 3, Symplectic topological SYZ mirror construction for general quintics, Mathematical Physics and Mathematics, 1999, 49 pp.
- Wei-Dong Ruan, Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces. III. Symplectic topological SYZ mirror construction for general quintics, J. Differential Geom. 63 (2003), no. 2, 171–229. MR 2015547
- 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 https://doi.org/10.1090/conm/527/10402
- Helge Ruddat, Nicolò Sibilla, David Treumann, and Eric Zaslow, Skeleta of affine hypersurfaces, Geom. Topol. 18 (2014), no. 3, 1343–1395. MR 3228454, DOI https://doi.org/10.2140/gt.2014.18.1343
- Paul Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), no. 4, 727–769. MR 2819674, DOI https://doi.org/10.1090/S1056-3911-10-00550-3
- Nick Sheridan, On the homological mirror symmetry conjecture for pairs of pants, J. Differential Geom. 89 (2011), no. 2, 271–367. MR 2863919
- Joseph Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229–257. MR 429885, DOI https://doi.org/10.1007/BF01403146
- J. H. M. Steenbrink, Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures, Math. Ann. 301 (1995), no. 1, 105–118. MR 1312571, DOI https://doi.org/10.1007/BF01446621
References
- M. Abouzaid, D. Auroux, and L. Katzarkov, Lagrangian Fibrations on Blowups of Toric Varieties and Mirror Symmetry for Hypersurfaces, arXiv:math/1205.0053
- M. Abouzaid and D. Auroux, Homological Mirror Symmetry for Hypersurfaces, work in progress.
- Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, and Dmitri Orlov, Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. 26 (2013), no. 4, 1051–1083. MR 3073884, DOI https://doi.org/10.1090/S0894-0347-2013-00770-5
- 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)
- Victor V. Batyrev and Lev A. Borisov, On Calabi-Yau complete intersections in toric varieties, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 39–65. MR 1463173 (98j:14052)
- V. I. Danilov and A. G. Khovanskiĭ, Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 5, 925–945 (Russian). MR 873655 (88i:32032)
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 0498551 (58 \#16653a)
- José Bertin, Jean-Pierre Demailly, Luc Illusie, and Chris Peters, Introduction to Hodge theory, SMF/AMS Texts and Monographs, vol. 8, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2002. Translated from the 1996 French original by James Lewis and Peters. MR 1924513
- Alexander I. Efimov, Homological mirror symmetry for curves of higher genus, Adv. Math. 230 (2012), no. 2, 493–530. MR 2914956, DOI https://doi.org/10.1016/j.aim.2012.02.022
- Mark Gross, Topological mirror symmetry, Invent. Math. 144 (2001), no. 1, 75–137. MR 1821145 (2002c:14062), DOI https://doi.org/10.1007/s002220000119
- Mark Gross, Toric degenerations and Batyrev-Borisov duality, Math. Ann. 333 (2005), no. 3, 645–688. MR 2198802 (2007b:14086), DOI https://doi.org/10.1007/s00208-005-0686-7
- M. Gross, L. Katzarkov, and H. Ruddat, Towards Mirror Symmetry for Varieties of General Type, arXiv:math/1202.4042
- Mark Gross and Bernd Siebert, Mirror symmetry via logarithmic degeneration data. I, J. Differential Geom. 72 (2006), no. 2, 169–338. MR 2213573 (2007b:14087)
- 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 https://doi.org/10.4007/annals.2011.174.3.1
- Luc Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, Astérisque 279 (2002), 271–322. Cohomologies $p$-adiques et applications arithmétiques, II. MR 1922832 (2003h:14032)
- Dominic Joyce, A classical model for derived critical loci, J. Differential Geom. 101 (2015), no. 2, 289–367. MR 3399099
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703 (99b:14020)
- Kazuya Kato and Chikara Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $\textbf {C}$, Kodai Math. J. 22 (1999), no. 2, 161–186. MR 1700591 (2000i:14023), DOI https://doi.org/10.2996/kmj/1138044041
- Anton Kapustin, Ludmil Katzarkov, Dmitri Orlov, and Mirroslav Yotov, Homological mirror symmetry for manifolds of general type, Cent. Eur. J. Math. 7 (2009), no. 4, 571–605. MR 2563433 (2010j:53184), DOI https://doi.org/10.2478/s11533-009-0056-x
- Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918 (97f:32040)
- Chikara Nakayama and Arthur Ogus, Relative rounding in toric and logarithmic geometry, Geom. Topol. 14 (2010), no. 4, 2189–2241. MR 2740645 (2012e:14104), DOI https://doi.org/10.2140/gt.2010.14.2189
- 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 (2009c:14018)
- W.-D. Ruan, Lagrangian Tori Fibration of Toric Calabi-Yau manifold; 3, Symplectic topological SYZ mirror construction for general quintics, Mathematical Physics and Mathematics, 1999, 49 pp.
- Wei-Dong Ruan, Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces. III. Symplectic topological SYZ mirror construction for general quintics, J. Differential Geom. 63 (2003), no. 2, 171–229. MR 2015547 (2004k:32043)
- 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 (2011m:14017), DOI https://doi.org/10.1090/conm/527/10402
- Helge Ruddat, Nicolò Sibilla, David Treumann, and Eric Zaslow, Skeleta of affine hypersurfaces, Geom. Topol. 18 (2014), no. 3, 1343–1395. MR 3228454, DOI https://doi.org/10.2140/gt.2014.18.1343
- Paul Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), no. 4, 727–769. MR 2819674 (2012f:53186), DOI https://doi.org/10.1090/S1056-3911-10-00550-3
- Nick Sheridan, On the homological mirror symmetry conjecture for pairs of pants, J. Differential Geom. 89 (2011), no. 2, 271–367. MR 2863919 (2012m:53196)
- Joseph Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229–257. MR 0429885 (55 \#2894)
- J. H. M. Steenbrink, Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures, Math. Ann. 301 (1995), no. 1, 105–118. MR 1312571 (96e:14009), DOI https://doi.org/10.1007/BF01446621
Additional Information
Helge Ruddat
Affiliation:
JGU Mainz, Institut für Mathematik, Staudingerweg 9, 55099 Mainz, Germany
MR Author ID:
912430
Email:
ruddat@uni-mainz.de
Received by editor(s):
September 20, 2013
Received by editor(s) in revised form:
April 26, 2014, and May 12, 2014
Published electronically:
June 7, 2016
Article copyright:
© Copyright 2016
University Press, Inc.
