Skip to main content Accessibility help

The essential skeleton of a product of degenerations

  • Morgan V. Brown (a1) and Enrica Mazzon (a2)


We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration $\mathscr{X}_{R}$ changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to $X_{K}$ . Using the Kato fan, we define a skeleton $\text{Sk}(\mathscr{X}_{R})$ when the model $\mathscr{X}_{R}$ is log-regular. We show that if $\mathscr{X}_{R}$ and $\mathscr{Y}_{R}$ are log-smooth, and at least one is semistable, then $\text{Sk}(\mathscr{X}_{R}\times _{R}\mathscr{Y}_{R})\simeq \text{Sk}(\mathscr{X}_{R})\times \text{Sk}(\mathscr{Y}_{R})$ . The essential skeleton $\text{Sk}(X_{K})$ , defined by Mustaţă and Nicaise, is a birational invariant of $X_{K}$ and is independent of the choice of $R$ -model. We extend their definition to pairs, and show that if both $X_{K}$ and $Y_{K}$ admit semistable models, $\text{Sk}(X_{K}\times _{K}Y_{K})\simeq \text{Sk}(X_{K})\times \text{Sk}(Y_{K})$ . As an application, we compute the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. We consider both the case of the Hilbert scheme of a semistable degeneration of K3 surfaces, and the generalized Kummer construction applied to a semistable degeneration of abelian surfaces. In both cases we find that the dual complex of the $2n$ -dimensional degeneration is homeomorphic to a point, $n$ -simplex, or $\mathbb{C}\mathbb{P}^{n}$ , depending on the type of the degeneration.



Hide All
[ACMUW16] Abramovich, D., Chen, Q., Marcus, S., Ulirsch, M. and Wise, J., Skeletons and fans of logarithmic structures , in Nonarchimedean and tropical geometry, Simons Symposia, eds Baker, M. and Payne, S. (Springer, Cham, 2016).
[Ale96] Alexeev, V., Moduli spaces M g, n(W) for surfaces , in Higher-dimensional complex varieties (Trento, 1994) (de Gruyter, Berlin, 1996), 122.
[Bea83] Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle , J. Differential Geom. 18 (1983), 755782.10.4310/jdg/1214438181
[Ber90] Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990).
[Ber95] Berkovich, V. G., The automorphism group of the Drinfeld half-plane , C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 11271132.
[BCHM10] Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J., Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), 405468.10.1090/S0894-0347-09-00649-3
[BK02] Bouchiba, S. and Kabbaj, S., Tensor products of Cohen-Macaulay rings: solution to a problem of Grothendieck , J. Algebra 252 (2002), 6573.10.1016/S0021-8693(02)00019-4
[Bul15] Bultot, E., Motivic integration and logarithmic geometry, Preprint (2015), arXiv:1505.05688 [math.AG].
[dKX17] de Fernex, T., Kollár, J. and Xu, C., The dual complex of singularities , in Higher dimensional algebraic geometry, in honour of Professor Yujiro Kawamata’s 60th birthday, Advanced Studies in Pure Mathematics, vol. 74 (Mathematical Society of Japan, 2017), 103130.10.2969/aspm/07410103
[Fog68] Fogarty, J., Algebraic families on an algebraic surface , Amer. J. Math. 90 (1968), 511521.10.2307/2373541
[FM83] Friedman, R. and Morrison, D. R., The birational geometry of degenerations: an overview , in The birational geometry of degenerations, Progress in Mathematics, vol. 29 (Birkhäuser, Boston, MA, 1983), 132.
[Ful93] Fulton, W., Introduction to toric varieties, The William H. Roever Lectures in Geometry, vol. 131 (Princeton University Press, Princeton, NJ, 1993).
[GR04] Gabber, O. and Ramero, L., Foundations for almost ring theory – release 6.95, Preprint (2004), arXiv:math/0409584.10.1007/b10047
[GRW16] Gubler, W., Rabinoff, J. and Werner, A., Skeletons and tropicalizations , Adv. Math. 294 (2016), 150215.10.1016/j.aim.2016.02.022
[GHH15] Gulbrandsen, M. G., Halle, L. H. and Hulek, K., A relative Hilbert-Mumford criterion , Manuscripta Math. 148 (2015), 283301.10.1007/s00229-015-0744-8
[GHH16] Gulbrandsen, M. G., Halle, L. H. and Hulek, K., A GIT construction of degenerations of Hilbert schemes of points, Doc. Math., to appear. Preprint (2016), arXiv:1604.00215 [math.AG].
[GHHZ18] Gulbrandsen, M. G., Halle, L. H., Hulek, K. and Zhang, Z., The geometry of degenerations of Hilbert schemes of points, Preprint (2018), arXiv:1802.00622 [math.AG].
[HMX18] Hacon, C. D., McKernan, J. and Xu, C., Boundedness of varieties of log general type , in Algebraic geometry: Salt Lake City 2015, Proceedings of Symposia in Pure Mathematics, vol. 97 (American Mathematical Society, Providence, RI, 2018), 309348.10.1090/pspum/097.1/01677
[HN17] Halle, L. H. and Nicaise, J., Motivic zeta functions of degenerating Calabi-Yau varieties , Math. Ann. 370 (2017), 12771320.10.1007/s00208-017-1578-3
[HM98] Harris, J. and Morrison, I., Moduli of curves, Graduate Texts in Mathematics, vol. 187 (Springer, New York, 1998).
[Har77] Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).10.1007/978-1-4757-3849-0
[Hat02] Hatcher, A., Algebraic topology (Cambridge University Press, Cambridge, 2002).
[Hwa08] Hwang, J.-M., Base manifolds for fibrations of projective irreducible symplectic manifolds , Invent. Math. 174 (2008), 625644.10.1007/s00222-008-0143-9
[Kat89] Kato, K., Logarithmic structures of Fontaine-Illusie , in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) (Johns Hopkins University Press, Baltimore, MD, 1989), 191224.
[Kat94] Kato, K., Toric singularities , Amer. J. Math. 116 (1994), 10731099.10.2307/2374941
[KKMS73] Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B., Toroidal embeddings. I (Springer, Berlin, 1973).10.1007/BFb0070318
[KLSV17] Kollár, J., Laza, R., Saccà, G. and Voisin, C., Remarks on degenerations of hyper-Kähler manifolds, Ann. Inst. Fourier, to appear. Preprint (2017), arXiv:1704.02731 [math.AG].10.5802/aif.3228
[KM08] Kollár, J. and Mori, S., Birational geometry of algebraic varieties (Cambridge University Press, Cambridge, 2008).
[KNX17] Kollár, J., Nicaise, J. and Xu, C., Semi-stable extensions over 1-dimensional bases , Acta Math. Sin. (Engl. Ser.) (2017).
[KS88] Kollár, J. and Shepherd-Barron, N. I., Threefolds and deformations of surface singularities , Invent. Math. 91 (1988), 299338.10.1007/BF01389370
[KX16] Kollár, J. and Xu, C., The dual complex of Calabi–Yau pairs , Invent. Math. 205 (2016), 527557.10.1007/s00222-015-0640-6
[KS01] Kontsevich, M. and Soibelman, Y., Homological mirror symmetry and torus fibrations , in Symplectic geometry and mirror symmetry (Seoul, 2000) (World Scientific, River Edge, NJ, 2001), 203263.10.1142/9789812799821_0007
[KS06] Kontsevich, M. and Soibelman, Y., Affine structures and non-Archimedean analytic spaces , in The unity of mathematics: in honor of the ninetieth birthday of I.M. Gelfand, eds Etingof, P., Retakh, V. and Singer, I. M. (Birkhäuser, Boston, MA, 2006), 321385.
[Kul77] Kulikov, V. S., Degenerations of K3 surfaces and Enriques surfaces , Math. USSR-Izv. 11 (1977), 957989.10.1070/IM1977v011n05ABEH001753
[Li01] Li, J., Stable morphisms to singular schemes and relative stable morphisms , J. Differential Geom. 57 (2001), 509578.10.4310/jdg/1090348132
[Liu02] Liu, Q., Algebraic geometry and arithmetic curves, Vol. 6 (Oxford University Press, Oxford, 2002); translated from the French by Reinie Erné.
[Loo76/77] Looijenga, E., Root systems and elliptic curves , Invent. Math. 38 (1976/77), 1732.10.1007/BF01390167
[Mor67] Morton, H. R., Symmetric products of the circle , Proc. Cambridge Philos. Soc. 63 (1967), 349352.10.1017/S0305004100041256
[MN15] Mustaţă, M. and Nicaise, J., Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton , Algebr. Geom. 2 (2015), 365404.10.14231/AG-2015-016
[NX16] Nicaise, J. and Xu, C., The essential skeleton of a degeneration of algebraic varieties , Amer. J. Math. 138 (2016), 16451667.10.1353/ajm.2016.0049
[Niz06] Nizioł, W., Toric singularities: log-blow-ups and global resolutions , J. Algebraic Geom. 15 (2006), 129.10.1090/S1056-3911-05-00409-1
[Now97] Nowak, K. J., Flat morphisms between regular varieties , Univ. Iagel. Acta Math. 35 (1997), 243246.
[O’Gr99] O’Grady, K. G., Desingularized moduli spaces of sheaves on a K3 , J. Reine Angew. Math. 512 (1999), 49117.
[O’Gr03] O’Grady, K. G., A new six-dimensional irreducible symplectic variety , J. Algebraic Geom. 12 (2003), 435505.10.1090/S1056-3911-03-00323-0
[PP81] Persson, U. and Pinkham, H., Degeneration of surfaces with trivial canonical bundle , Ann. of Math. (2) 113 (1981), 4566.10.2307/1971133
[Sai04] Saito, T., Log smooth extension of a family of curves and semi-stable reduction , J. Algebraic Geom. 13 (2004), 287321.10.1090/S1056-3911-03-00338-2
[SYZ96] Strominger, A., Yau, S.-T. and Zaslow, E., Mirror symmetry is T-duality , Nuclear Phys. B 479 (1996), 243259.10.1016/0550-3213(96)00434-8
[Tem16] Temkin, M., Metrization of differential pluriforms on Berkovich analytic spaces , in Nonarchimedean and tropical geometry, Simons Symposia, eds Baker, M. and Payne, S. (Springer, Cham, 2016), 195285.10.1007/978-3-319-30945-3_8
[Thu07] Thuillier, A., Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels , Manuscripta Math. 123 (2007), 381451.10.1007/s00229-007-0094-2
[Uli17] Ulirsch, M., Functorial tropicalization of logarithmic schemes: the case of constant coefficients , Proc. Lond. Math. Soc. 114 (2017), 10811113.10.1112/plms.12031
[Uli19] Ulirsch, M., Non-Archimedean geometry of Artin fans , Adv. Math. 345 (2019), 346381.10.1016/j.aim.2019.01.008
[Vid04] Vidal, I., Monodromie locale et fonctions zeta des log schémas , in Geometric aspects of Dwork theory. Vols I, II, eds Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N. and Loeser, F. (Walter de Gruyter, Berlin, 2004), 9831038.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed