Skeleta of affine hypersurfaces

Ruddat, Helge¹ ; Sibilla, Nicolò² ; Treumann, David³ ; Zaslow, Eric⁴

¹ Mathematisches Institut, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, D-55099 Mainz, Germany
² Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany
³ Department of Mathematics, Boston College, Carney Hall, Room 301, Chestnut Hill, Boston, MA 02467-3806, USA
⁴ Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA

Geometry & topology, Tome 18 (2014) no. 3, pp. 1343-1395.

Résumé

A smooth affine hypersurface $Z$ of complex dimension $n$ is homotopy equivalent to an $n$–dimensional cell complex. Given a defining polynomial $f$ for $Z$ as well as a regular triangulation ${\mathcal{T}}_{△}$ of its Newton polytope $△$, we provide a purely combinatorial construction of a compact topological space $S$ as a union of components of real dimension $n$, and prove that $S$ embeds into $Z$ as a deformation retract. In particular, $Z$ is homotopy equivalent to $S$.

Détail

DOI : 10.2140/gt.2014.18.1343

Classification : 14J70, 14R99
Keywords: skeleton, retraction, hypersurface, homotopy equivalence, affine, toric degeneration, Kato–Nakayama space, log geometry, Newton polytope, triangulation

Bibliographie

[1] M Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006) 1097 | DOI

[2] M Abouzaid, Symplectic perspectives on tropical geometry (2009)

[3] M Abouzaid, D Auroux, L Katzarkov, Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces,

[4] A Andreotti, T Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. 69 (1959) 713 | DOI

[5] V V Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993) 349 | DOI

[6] V G Berkovich, Smooth p–adic analytic spaces are locally contractible, II, from: "Geometric aspects of Dwork theory, Vol. I, II" (editors A Adolphson, F Baldassarri, P Berthelot, N Katz, F Loeser), de Gruyter (2004) 293

[7] H Bruggesser, P Mani, Shellable decompositions of cells and spheres, Math. Scand. 29 (1971) 197

[8] K Cieliebak, Y Eliashberg, Symplectic geometry of Stein manifolds, in preparation

[9] V I Danilov, A G Khovanskiǐ, Newton polyhedra and an algorithm for calculating Hodge–Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 925

[10] P Deligne, Hodge cycles on abelian varieties, from: "Hodge cycles, motives, and Shimura varieties", Lecture Notes in Math. 900 (1982) 9

[11] B Fang, C C M Liu, D Treumann, E Zaslow, The coherent–constructible correspondence for toric Deligne–Mumford stacks,

[12] B Fang, C C M Liu, D Treumann, E Zaslow, A categorification of Morelli’s theorem, Invent. Math. 186 (2011) 79 | DOI

[13] R H Fox, Covering spaces with singularities, from: "A symposium in honor of S Lefschetz", Princeton Univ. Press (1957) 243

[14] W Fulton, Introduction to toric varieties, 131, Princeton Univ. Press (1993)

[15] I M Gelfand, M M Kapranov, A V Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser (1994) | DOI

[16] M Gross, L Katzarkov, H Ruddat, Towards mirror symmetry for varieties of general type,

[17] M Gross, B Siebert, From real affine geometry to complex geometry, Ann. of Math. 174 (2011) 1301 | DOI

[18] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[19] F Kato, Log smooth deformation theory, Tohoku Math. J. 48 (1996) 317 | DOI

[20] K Kato, Logarithmic structures of Fontaine–Illusie, from: "Algebraic analysis, geometry, and number theory" (editor J I Igusa), Johns Hopkins Univ. Press (1989) 191

[21] K Kato, C Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C, Kodai Math. J. 22 (1999) 161 | DOI

[22] M Kontsevich, Y Soibelman, Affine structures and non-Archimedean analytic spaces, from: "The unity of mathematics" (editors P Etingof, V Retakh, I M Singer), Progr. Math. 244, Birkhäuser (2006) 321 | DOI

[23] D Kozlov, Combinatorial algebraic topology, 21, Springer, Berlin (2008) | DOI

[24] J P Mccammond, A general small cancellation theory, Internat. J. Algebra Comput. 10 (2000) 1 | DOI

[25] G Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004) 1035 | DOI

[26] C Nakayama, A Ogus, Relative rounding in toric and logarithmic geometry, Geom. Topol. 14 (2010) 2189 | DOI

[27] H Ruddat, Log Hodge groups on a toric Calabi–Yau degeneration, from: "Mirror symmetry and tropical geometry" (editors R Castaño-Bernard, Y Soibelman, I Zharkov), Contemp. Math. 527, Amer. Math. Soc. (2010) 113 | DOI

[28] N Sibilla, D Treumann, E Zaslow, Ribbon graphs and mirror symmetry, I,

[29] D Treumann, E Zaslow, Polytopes and skeleta,

[30] J Woolf, The fundamental category of a stratified space, J. Homotopy Relat. Struct. 4 (2009) 359