## Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface

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Hans-Joachim Hein, Song Sun, Jeff Viaclovsky and Ruobing Zhang
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## Abstract:

We exhibit families of Ricci-flat Kähler metrics on the K3 surface which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the $K3$ surface to the interval, with regular fibers diffeomorphic to either $3$-tori or Heisenberg nilmanifolds.## References

- Michael T. Anderson and Jeff Cheeger,
*$C^\alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below*, J. Differential Geom.**35**(1992), no. 2, 265–281. MR**1158336** - Michael T. Anderson,
*Convergence and rigidity of manifolds under Ricci curvature bounds*, Invent. Math.**102**(1990), no. 2, 429–445. MR**1074481**, DOI 10.1007/BF01233434 - Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima,
*On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth*, Invent. Math.**97**(1989), no. 2, 313–349. MR**1001844**, DOI 10.1007/BF01389045 - Jeff Cheeger and Tobias H. Colding,
*Lower bounds on Ricci curvature and the almost rigidity of warped products*, Ann. of Math. (2)**144**(1996), no. 1, 189–237. MR**1405949**, DOI 10.2307/2118589 - Jeff Cheeger and Tobias H. Colding,
*On the structure of spaces with Ricci curvature bounded below. I*, J. Differential Geom.**46**(1997), no. 3, 406–480. MR**1484888** - Gao Chen and Xiuxiong Chen,
*Gravitational instantons with faster than quadratic curvature decay (III)*, Math. Ann. (2020). - J. Cheeger, T. H. Colding, and G. Tian,
*On the singularities of spaces with bounded Ricci curvature*, Geom. Funct. Anal.**12**(2002), no. 5, 873–914. MR**1937830**, DOI 10.1007/PL00012649 - X.-X. Chen and S. K. Donaldson,
*Volume estimates for Kähler-Einstein metrics and rigidity of complex structures*, J. Differential Geom.**93**(2013), no. 2, 191–201. MR**3024305** - Ronan J. Conlon and Hans-Joachim Hein,
*Asymptotically conical Calabi-Yau manifolds, I*, Duke Math. J.**162**(2013), no. 15, 2855–2902. MR**3161306**, DOI 10.1215/00127094-2382452 - Ronan J. Conlon and Hans-Joachim Hein,
*Asymptotically conical Calabi-Yau metrics on quasi-projective varieties*, Geom. Funct. Anal.**25**(2015), no. 2, 517–552. MR**3334234**, DOI 10.1007/s00039-015-0319-6 - J. Cheeger,
*Integral bounds on curvature elliptic estimates and rectifiability of singular sets*, Geom. Funct. Anal.**13**(2003), no. 1, 20–72. MR**1978491**, DOI 10.1007/s000390300001 - Jeff Cheeger and Aaron Naber,
*Lower bounds on Ricci curvature and quantitative behavior of singular sets*, Invent. Math.**191**(2013), no. 2, 321–339. MR**3010378**, DOI 10.1007/s00222-012-0394-3 - Jeff Cheeger and Aaron Naber,
*Regularity of Einstein manifolds and the codimension 4 conjecture*, Ann. of Math. (2)**182**(2015), no. 3, 1093–1165. MR**3418535**, DOI 10.4007/annals.2015.182.3.5 - Jeff Cheeger and Gang Tian,
*Anti-self-duality of curvature and degeneration of metrics with special holonomy*, Comm. Math. Phys.**255**(2005), no. 2, 391–417. MR**2129951**, DOI 10.1007/s00220-004-1279-0 - Jeff Cheeger and Gang Tian,
*Curvature and injectivity radius estimates for Einstein 4-manifolds*, J. Amer. Math. Soc.**19**(2006), no. 2, 487–525. MR**2188134**, DOI 10.1090/S0894-0347-05-00511-4 - S. Y. Cheng and S. T. Yau,
*Differential equations on Riemannian manifolds and their geometric applications*, Comm. Pure Appl. Math.**28**(1975), no. 3, 333–354. MR**385749**, DOI 10.1002/cpa.3160280303 - S. K. Donaldson,
*Symplectic submanifolds and almost-complex geometry*, J. Differential Geom.**44**(1996), no. 4, 666–705. MR**1438190** - S. K. Donaldson,
*Two-forms on four-manifolds and elliptic equations*, Inspired by S. S. Chern, Nankai Tracts Math., vol. 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 153–172. MR**2313334**, DOI 10.1142/9789812772688_{0}007 - Simon K. Donaldson,
*Calabi-Yau metrics on Kummer surfaces as a model gluing problem*, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 109–118. MR**3077251**, DOI 10.14219/jada.archive.2012.0111 - Lorenzo Foscolo, Mark Haskins, and Johannes Nordström,
*Complete non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds*, Duke Math. J., to appear. - Andreas Floer,
*Self-dual conformal structures on $l\textbf {C}\textrm {P}^2$*, J. Differential Geom.**33**(1991), no. 2, 551–573. MR**1094469** - Joel Fine, Jason D. Lotay, and Michael Singer,
*The space of hyperkähler metrics on a 4-manifold with boundary*, Forum Math. Sigma**5**(2017), Paper No. e6, 50. MR**3631268**, DOI 10.1017/fms.2017.3 - Lorenzo Foscolo,
*ALF gravitational instantons and collapsing Ricci-flat metrics on the $K3$ surface*, J. Differential Geom.**112**(2019), no. 1, 79–120. MR**3948228**, DOI 10.4310/jdg/1557281007 - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR**1288523**, DOI 10.1002/9781118032527 - Brian R. Greene, Alfred Shapere, Cumrun Vafa, and Shing-Tung Yau,
*Stringy cosmic strings and noncompact Calabi-Yau manifolds*, Nuclear Phys. B**337**(1990), no. 1, 1–36. MR**1059826**, DOI 10.1016/0550-3213(90)90248-C - 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** - Hans-Joachim Hein,
*Gravitational instantons from rational elliptic surfaces*, J. Amer. Math. Soc.**25**(2012), no. 2, 355–393. MR**2869021**, DOI 10.1090/S0894-0347-2011-00723-6 - Mark Haskins, Hans-Joachim Hein, and Johannes Nordström,
*Asymptotically cylindrical Calabi-Yau manifolds*, J. Differential Geom.**101**(2015), no. 2, 213–265. MR**3399097** - Hans-Joachim Hein and Song Sun,
*Calabi-Yau manifolds with isolated conical singularities*, Publ. Math. Inst. Hautes Études Sci.**126**(2017), 73–130. MR**3735865**, DOI 10.1007/s10240-017-0092-1 - Hans-Joachim Hein, Song Sun, Jeff Viaclovsky, and Ruobing Zhang,
*In preparation*. - Chris M. Hull,
*Massive string theories from M-theory and F-theory*, J. High Energy Phys.**11**(1998), Paper 27, 10. MR**1667365**, DOI 10.1088/1126-6708/1998/11/027 - Bert Koehler and Marco Kühnel,
*On asymptotics of complete Ricci-flat Kähler metrics on open manifolds*, Manuscripta Math.**132**(2010), no. 3-4, 431–462. MR**2652441**, DOI 10.1007/s00229-010-0354-4 - Ryoichi Kobayashi,
*Ricci-flat Kähler metrics on affine algebraic manifolds and degenerations of Kähler-Einstein $K3$ surfaces*, Kähler metric and moduli spaces, Adv. Stud. Pure Math., vol. 18, Academic Press, Boston, MA, 1990, pp. 137–228. MR**1145249**, DOI 10.2969/aspm/01820137 - Alexei Kovalev,
*Twisted connected sums and special Riemannian holonomy*, J. Reine Angew. Math.**565**(2003), 125–160. MR**2024648**, DOI 10.1515/crll.2003.097 - A. Kovalev and M. Singer,
*Gluing theorems for complete anti-self-dual spaces*, Geom. Funct. Anal.**11**(2001), no. 6, 1229–1281. MR**1878320**, DOI 10.1007/s00039-001-8230-8 - N. N. Lebedev,
*Special functions and their applications*, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR**0350075** - Chi Li,
*On sharp rates and analytic compactifications of asymptotically conical Kähler metrics*, Duke Math. J.**169**(2020), no. 8, 1397–1483. MR**4101736**, DOI 10.1215/00127094-2019-0073 - John Lott,
*The collapsing geometry of almost Ricci-flat 4-manifolds*, Comment. Math. Helv.**95**(2020), no. 1, 79–98. MR**4082894**, DOI 10.4171/cmh/481 - Claude LeBrun and Michael Singer,
*A Kummer-type construction of self-dual $4$-manifolds*, Math. Ann.**300**(1994), no. 1, 165–180. MR**1289837**, DOI 10.1007/BF01450482 - Peter Li and Luen-Fai Tam,
*Symmetric Green’s functions on complete manifolds*, Amer. J. Math.**109**(1987), no. 6, 1129–1154. MR**919006**, DOI 10.2307/2374588 - Vincent Minerbe,
*Rigidity for multi-Taub-NUT metrics*, J. Reine Angew. Math.**656**(2011), 47–58. MR**2818855**, DOI 10.1515/CRELLE.2011.042 - Aaron Naber and Ruobing Zhang,
*Topology and $\varepsilon$-regularity theorems on collapsed manifolds with Ricci curvature bounds*, Geom. Topol.**20**(2016), no. 5, 2575–2664. MR**3556347**, DOI 10.2140/gt.2016.20.2575 - Hirosi Ooguri and Cumrun Vafa,
*Summing up Dirichlet instantons*, Phys. Rev. Lett.**77**(1996), no. 16, 3296–3298. MR**1411842**, DOI 10.1103/PhysRevLett.77.3296 - Yann Rollin and Michael Singer,
*Non-minimal scalar-flat Kähler surfaces and parabolic stability*, Invent. Math.**162**(2005), no. 2, 235–270. MR**2199006**, DOI 10.1007/s00222-004-0436-6 - Simon Salamon,
*Riemannian geometry and holonomy groups*, Pitman Research Notes in Mathematics Series, vol. 201, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR**1004008** - Peter Scott,
*The geometries of $3$-manifolds*, Bull. London Math. Soc.**15**(1983), no. 5, 401–487. MR**705527**, DOI 10.1112/blms/15.5.401 - Michael Spivak,
*A comprehensive introduction to differential geometry. Vol. I*, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR**532830** - Jingzhou Sun and Song Sun,
*Projective embedding of log Riemann surfaces and K-stability*, J. Geom. Anal. (2020). - G. Tian,
*On Calabi’s conjecture for complex surfaces with positive first Chern class*, Invent. Math.**101**(1990), no. 1, 101–172. MR**1055713**, DOI 10.1007/BF01231499 - G. Tian and Shing-Tung Yau,
*Complete Kähler manifolds with zero Ricci curvature. I*, J. Amer. Math. Soc.**3**(1990), no. 3, 579–609. MR**1040196**, DOI 10.1090/S0894-0347-1990-1040196-6 - Shing Tung Yau,
*On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I*, Comm. Pure Appl. Math.**31**(1978), no. 3, 339–411. MR**480350**, DOI 10.1002/cpa.3160310304

## Additional Information

**Hans-Joachim Hein**- Affiliation: Department of Mathematics, Fordham University, Bronx, New York 10458; and Mathematisches Institut, WWU Münster, 48149 Münster, Germany
- MR Author ID: 938594
- ORCID: 0000-0002-3719-9549
- Email: hhein@uni-muenster.de
**Song Sun**- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- MR Author ID: 879901
- Email: sosun@berkeley.edu
**Jeff Viaclovsky**- Affiliation: Department of Mathematics, University of California, Irvine, California 92697
- MR Author ID: 648525
- Email: jviaclov@uci.edu
**Ruobing Zhang**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 1181360
- Email: ruobingz@princeton.edu
- Received by editor(s): July 24, 2018
- Received by editor(s) in revised form: June 5, 2020, January 15, 2021, and January 30, 2021
- Published electronically: June 25, 2021
- Additional Notes: The first author was partially supported by NSF Grant DMS-1745517. The second author was supported by NSF Grant DMS-1708420, an Alfred P. Sloan Fellowship, and a grant from the Simons Foundation ($\sharp$488633, S.S.). The third author was partially supported by NSF Grant DMS-1811096. The fourth author was partially supported by NSF Grant DMS-1906265.
- © Copyright 2021 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**35**(2022), 123-209 - MSC (2020): Primary 53C25, 53C26, 53C55
- DOI: https://doi.org/10.1090/jams/978
- MathSciNet review: 4322391