Boundedness questions for Calabi–Yau threefolds

Author:
P. M. H. Wilson

Journal:
J. Algebraic Geom. **30** (2021), 631-684

DOI:
https://doi.org/10.1090/jag/781

Published electronically:
June 10, 2021

Full-text PDF

Abstract |
References |
Additional Information

Abstract:

In this paper, we study boundedness questions for (simply connected) smooth Calabi–Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second Chern class. The motivating question for this paper is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowledge of these two forms.

Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. If for instance there are no such surfaces, then the answer to the motivating question is yes (Theorem 0.1). In particular, for given cubic and linear forms on the second cohomology, there must exist such surfaces for large enough third Betti number (Corollary 0.2).

The paper starts by proving general results on these rigid non-movable surfaces and boundedness of the family of threefolds. The basic principle is that if the cohomology classes of these surfaces are also known, then boundedness should hold (Theorem 4.5). The second half of the paper restricts to the case of Picard number 2, where it is shown that knowledge of the cubic and linear forms does indeed bound the family of Calabi–Yau threefolds (Theorem 0.3). This appears to be the first non-trivial case where a general boundedness result for Calabi–Yau threefolds has been proved (without the assumption of a special structure).

References
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*Boundedness and $K^2$ for log surfaces*, Internat. J. Math. **5** (1994), no. 6, 779–810. MR **1298994**, DOI 10.1142/S0129167X94000395
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*Deforming Calabi-Yau threefolds*, Math. Ann. **308** (1997), no. 2, 187–220. MR **1464900**, DOI 10.1007/s002080050072
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*Primitive Calabi-Yau threefolds*, J. Differential Geom. **45** (1997), no. 2, 288–318. MR **1449974**
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*Calabi-Yau threefolds with $\rho >13$*, Math. Ann. **294** (1992), no. 1, 49–57. MR **1180449**, DOI 10.1007/BF01934312
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*Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups*, J. Algebraic Geom. **1** (1992), no. 3, 449–530. MR **1158626**
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*Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces*, Ann. of Math. (2) **127** (1988), no. 1, 93–163. MR **924674**, DOI 10.2307/1971417
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*General hyperplane sections of nonsingular flops in dimension $3$*, Math. Res. Lett. **1** (1994), no. 1, 49–52. MR **1258489**, DOI 10.4310/MRL.1994.v1.n1.a6
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*Flops*, Nagoya Math. J. **113** (1989), 15–36. MR **986434**, DOI 10.1017/S0027763000001240
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*Flips, flops, minimal models, etc*, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 113–199. MR **1144527**
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*Classification of three-dimensional flips*, J. Amer. Math. Soc. **5** (1992), no. 3, 533–703. MR **1149195**, DOI 10.1090/S0894-0347-1992-1149195-9
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*Birational geometry of algebraic varieties*, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR **1658959**, DOI 10.1017/CBO9780511662560
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*Positivity in algebraic geometry. II*, 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. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR **2095472**, DOI 10.1007/978-3-642-18808-4
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*The Chern classes and Kodaira dimension of a minimal variety*, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 449–476. MR **946247**, DOI 10.2969/aspm/01010449
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*Zariski-decomposition and abundance*, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR **2104208**
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*On algebraic fiber space structures on a Calabi-Yau $3$-fold*, Internat. J. Math. **4** (1993), no. 3, 439–465. With an appendix by Noboru Nakayama. MR **1228584**, DOI 10.1142/S0129167X93000248
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*On the finiteness of fiber-space structures on a Calabi-Yau 3-fold*, J. Math. Sci. (New York) **106** (2001), no. 5, 3320–3335. Algebraic geometry, 11. MR **1878052**, DOI 10.1023/A:1017959510524
- Keiji Oguiso and Thomas Peternell,
*On polarized canonical Calabi-Yau threefolds*, Math. Ann. **301** (1995), no. 2, 237–248. MR **1314586**, DOI 10.1007/BF01446628
- Keiji Oguiso and Thomas Peternell,
*Calabi-Yau threefolds with positive second Chern class*, Comm. Anal. Geom. **6** (1998), no. 1, 153–172. MR **1619841**, DOI 10.4310/CAG.1998.v6.n1.a5
- Henry C. Pinkham,
*Factorization of birational maps in dimension $3$*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 343–371. MR **713260**
- Miles Reid,
*Minimal models of canonical $3$-folds*, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131–180. MR **715649**, DOI 10.2969/aspm/00110131
- Miles Reid,
*Nonnormal del Pezzo surfaces*, Publ. Res. Inst. Math. Sci. **30** (1994), no. 5, 695–727. MR **1311389**, DOI 10.2977/prims/1195165581
- Dennis Sullivan,
*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math. **47** (1977), 269–331 (1978). MR **646078**, DOI 10.1007/BF02684341
- C. T. C. Wall,
*Classification problems in differential topology. V. On certain $6$-manifolds*, Invent. Math. **1** (1966), 355–374; corrigendum, ibid. 2 (1966), 306. MR **215313**, DOI 10.1007/BF01425407
- Hugh E. Warren,
*Lower bounds for approximation by nonlinear manifolds*, Trans. Amer. Math. Soc. **133** (1968), 167–178. MR **226281**, DOI 10.1090/S0002-9947-1968-0226281-1
- P. M. H. Wilson,
*Calabi-Yau manifolds with large Picard number*, Invent. Math. **98** (1989), no. 1, 139–155. MR **1010159**, DOI 10.1007/BF01388848
- P. M. H. Wilson,
*The Kähler cone on Calabi-Yau threefolds*, Invent. Math. **107** (1992), no. 3, 561–583. MR **1150602**, DOI 10.1007/BF01231902
- P. M. H. Wilson,
*Erratum: “The Kähler cone on Calabi-Yau threefolds” [Invent. Math. 107 (1992), no. 3, 561–583; MR1150602 (93a:14037)]*, Invent. Math. **114** (1993), no. 1, 231–233. MR **1235026**, DOI 10.1007/BF01232669
- P. M. H. Wilson,
*Symplectic deformations of Calabi-Yau threefolds*, J. Differential Geom. **45** (1997), no. 3, 611–637. MR **1472891**
- P. M. H. Wilson,
*Flops, Type III contractions and Gromov-Witten invariants on Calabi-Yau threefolds*, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 465–484. MR **1714834**, DOI 10.1017/CBO9780511721540.018
- P. M. H. Wilson,
*Calabi–Yau threefolds with Picard number three*, arXiv:2011.12876.
- Shing Tung Yau,
*Calabi’s conjecture and some new results in algebraic geometry*, Proc. Nat. Acad. Sci. U.S.A. **74** (1977), no. 5, 1798–1799. MR **451180**, DOI 10.1073/pnas.74.5.1798

References
- Valery Alexeev,
*Boundedness and $K^2$ for log surfaces*, Internat. J. Math. **5** (1994), no. 6, 779–810. MR **1298994**, DOI 10.1142/S0129167X94000395
- C. Birkar,
*Lectures on birational geometry*, arXiv:1210.2670.
- Robert Friedman,
*On threefolds with trivial canonical bundle*, Complex geometry and Lie theory (Sundance, UT, 1989) Proc. Sympos. Pure Math., vol. 53, Amer. Math. Soc., Providence, RI, 1991, pp. 103–134. MR **1141199**, DOI 10.1090/pspum/053/1141199
- Robert Friedman,
*Simultaneous resolution of threefold double points*, Math. Ann. **274** (1986), no. 4, 671–689. MR **848512**, DOI 10.1007/BF01458602
- Jixiang Fu, Jun Li, and Shing-Tung Yau,
*Balanced metrics on non-Kähler Calabi-Yau threefolds*, J. Differential Geom. **90** (2012), no. 1, 81–129. MR **2891478**
- Mark Gross,
*A finiteness theorem for elliptic Calabi-Yau threefolds*, Duke Math. J. **74** (1994), no. 2, 271–299. MR **1272978**, DOI 10.1215/S0012-7094-94-07414-0
- Mark Gross,
*Deforming Calabi-Yau threefolds*, Math. Ann. **308** (1997), no. 2, 187–220. MR **1464900**, DOI 10.1007/s002080050072
- Mark Gross,
*Primitive Calabi-Yau threefolds*, J. Differential Geom. **45** (1997), no. 2, 288–318. MR **1449974**
- D. R. Heath-Brown and P. M. H. Wilson,
*Calabi-Yau threefolds with $\rho >13$*, Math. Ann. **294** (1992), no. 1, 49–57. MR **1180449**, DOI 10.1007/BF01934312
- Marc Hindry and Joseph H. Silverman,
*Diophantine geometry*, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR **1745599**, DOI 10.1007/978-1-4612-1210-2
- Atsushi Kanazawa and P. M. H. Wilson,
*Trilinear forms and Chern classes of Calabi-Yau threefolds*, Osaka J. Math. **51** (2014), no. 1, 203–213. MR **3192539**
- Sheldon Katz and David R. Morrison,
*Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups*, J. Algebraic Geom. **1** (1992), no. 3, 449–530. MR **1158626**
- Yujiro Kawamata,
*Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces*, Ann. of Math. (2) **127** (1988), no. 1, 93–163. MR **924674**, DOI 10.2307/1971417
- Yujiro Kawamata,
*General hyperplane sections of nonsingular flops in dimension $3$*, Math. Res. Lett. **1** (1994), no. 1, 49–52. MR **1258489**, DOI 10.4310/MRL.1994.v1.n1.a6
- János Kollár,
*Flops*, Nagoya Math. J. **113** (1989), 15–36. MR **986434**, DOI 10.1017/S0027763000001240
- János Kollár,
*Flips, flops, minimal models, etc*, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 113–199. MR **1144527**
- János Kollár and Shigefumi Mori,
*Classification of three-dimensional flips*, J. Amer. Math. Soc. **5** (1992), no. 3, 533–703. MR **1149195**, DOI 10.2307/2152704
- János Kollár and Shigefumi Mori,
*Birational geometry of algebraic varieties*, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR **1658959**, DOI 10.1017/CBO9780511662560
- Robert Lazarsfeld,
*Positivity in algebraic geometry. II*: *Positivity for vector bundles, and multiplier ideals*, 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. 49, Springer-Verlag, Berlin, 2004. MR **2095472**, DOI 10.1007/978-3-642-18808-4
- Yoichi Miyaoka,
*The Chern classes and Kodaira dimension of a minimal variety*, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 449–476. MR **946247**, DOI 10.2969/aspm/01010449
- Noboru Nakayama,
*Zariski-decomposition and abundance*, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR **2104208**
- Keiji Oguiso,
*On algebraic fiber space structures on a Calabi-Yau $3$-fold*, Internat. J. Math. **4** (1993), no. 3, 439–465. With an appendix by Noboru Nakayama. MR **1228584**, DOI 10.1142/S0129167X93000248
- Keiji Oguiso,
*On the finiteness of fiber-space structures on a Calabi-Yau 3-fold*, J. Math. Sci. (New York) **106** (2001), no. 5, 3320–3335. MR **1878052**, DOI 10.1023/A:1017959510524
- Keiji Oguiso and Thomas Peternell,
*On polarized canonical Calabi-Yau threefolds*, Math. Ann. **301** (1995), no. 2, 237–248. MR **1314586**, DOI 10.1007/BF01446628
- Keiji Oguiso and Thomas Peternell,
*Calabi-Yau threefolds with positive second Chern class*, Comm. Anal. Geom. **6** (1998), no. 1, 153–172. MR **1619841**, DOI 10.4310/CAG.1998.v6.n1.a5
- Henry C. Pinkham,
*Factorization of birational maps in dimension $3$*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 343–371. MR **713260**
- Miles Reid,
*Minimal models of canonical $3$-folds*, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131–180. MR **715649**, DOI 10.2969/aspm/00110131
- Miles Reid,
*Nonnormal del Pezzo surfaces*, Publ. Res. Inst. Math. Sci. **30** (1994), no. 5, 695–727. MR **1311389**, DOI 10.2977/prims/1195165581
- Dennis Sullivan,
*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math. **47** (1977), 269–331 (1978). MR **646078**
- C. T. C. Wall,
*Classification problems in differential topology. V. On certain $6$-manifolds*, Invent. Math. **1** (1966), 355–374; corrigendum, ibid. 2 (1966), 306. MR **215313**, DOI 10.1007/BF01425407
- Hugh E. Warren,
*Lower bounds for approximation by nonlinear manifolds*, Trans. Amer. Math. Soc. **133** (1968), 167–178. MR **226281**, DOI 10.2307/1994937
- P. M. H. Wilson,
*Calabi-Yau manifolds with large Picard number*, Invent. Math. **98** (1989), no. 1, 139–155. MR **1010159**, DOI 10.1007/BF01388848
- P. M. H. Wilson,
*The Kähler cone on Calabi-Yau threefolds*, Invent. Math. **107** (1992), no. 3, 561–583. MR **1150602**, DOI 10.1007/BF01231902
- P. M. H. Wilson,
*Erratum: “The Kähler cone on Calabi-Yau threefolds” [Invent. Math. ***107** (1992), no. 3, 561–583; MR1150602 (93a:14037)], Invent. Math. **114** (1993), no. 1, 231–233. MR **1235026**, DOI 10.1007/BF01232669
- P. M. H. Wilson,
*Symplectic deformations of Calabi-Yau threefolds*, J. Differential Geom. **45** (1997), no. 3, 611–637. MR **1472891**
- P. M. H. Wilson,
*Flops, Type III contractions and Gromov-Witten invariants on Calabi-Yau threefolds*, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 465–484. MR **1714834**, DOI 10.1017/CBO9780511721540.018
- P. M. H. Wilson,
*Calabi–Yau threefolds with Picard number three*, arXiv:2011.12876.
- Shing Tung Yau,
*Calabi’s conjecture and some new results in algebraic geometry*, Proc. Nat. Acad. Sci. U.S.A. **74** (1977), no. 5, 1798–1799. MR **451180**, DOI 10.1073/pnas.74.5.1798

Additional Information

**P. M. H. Wilson**

Affiliation:
Department of Pure Mathematics, University of Cambridge, 16 Wilberforce Road, Cambridge CB3 0WB, United Kingdom

Email:
pmhw@dpmms.cam.ac.uk

Received by editor(s):
May 29, 2018

Received by editor(s) in revised form:
November 24, 2020, and January 28, 2021

Published electronically:
June 10, 2021

Article copyright:
© Copyright 2021
University Press, Inc.