Geodesic
Generalized Virtual Polytopes and Quasitoric Manifolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 139-165.

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We develop a theory of volume polynomials of generalized virtual polytopes based on the study of topology of affine subspace arrangements in a real Euclidean space. We apply this theory to obtain a topological version of the Bernstein–Kushnirenko theorem as well as Stanley–Reisner and Pukhlikov–Khovanskii type descriptions for the cohomology rings of generalized quasitoric manifolds.
Keywords: quasitoric manifold, star-shaped sphere, virtual polytope, multi-polytope, Stanley–Reisner ring.
Mots-clés : multi-fan, moment–angle complex 
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Ivan Yu. Limonchenko; Leonid V. Monin; Askold G. Khovanskii. Generalized Virtual Polytopes and Quasitoric Manifolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 139-165. http://geodesic.mathdoc.fr/item/TM_2022_318_a8/

[1] Ayzenberg A., Masuda M., “Volume polynomials and duality algebras of multi-fans”, Arnold Math. J., 2:3 (2016), 329–381 | DOI | MR | Zbl

[2] Baralić Ð., Grbić J., Limonchenko I., Vučić A., “Toric objects associated with the dodecahedron”, Filomat, 34:7 (2020), 2329–2356 | DOI | MR | Zbl

[3] V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, and S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russ. Math. Surv., 72:2 (2017), 199–256 | DOI | MR | Zbl

[4] V. M. Buchstaber and T. E. Panov, “On manifolds defined by 4-colourings of simple 3-polytopes”, Russ. Math. Surv., 71:6 (2016), 1137–1139 | DOI | MR | Zbl

[5] Buchstaber V.M., Panov T.E., Ray N., “Spaces of polytopes and cobordism of quasitoric manifolds”, Moscow Math. J., 7:2 (2007), 219–242 | DOI | MR | Zbl

[6] Choi S., Masuda M., Suh D.Y., “Quasitoric manifolds over a product of simplices”, Osaka J. Math., 47:1 (2010), 109–129 | MR | Zbl

[7] Davis M.W., Januszkiewicz T., “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Math. J., 62:2 (1991), 417–451 | DOI | MR | Zbl

[8] Hasui S., Kishimoto D., “$p$-Local stable cohomological rigidity of quasitoric manifolds”, Osaka J. Math., 54:2 (2017), 343–350 | MR | Zbl

[9] Hasui S., Kishimoto D., Sato T., “$p$-Local stable splitting of quasitoric manifolds”, Osaka J. Math., 53:3 (2016), 843–854 | MR | Zbl

[10] Hattori A., Masuda M., “Theory of multi-fans”, Osaka J. Math., 40:1 (2003), 1–68 | MR | Zbl

[11] Hofscheier J., Khovanskii A., Monin L., Cohomology rings of toric bundles and the ring of conditions, E-print, 2020, arXiv: 2006.12043 [math.AG]

[12] Ishida H., Fukukawa Y., Masuda M., “Topological toric manifolds”, Moscow Math. J., 13:1 (2013), 57–98 | DOI | MR | Zbl

[13] Kaveh K., “Note on cohomology rings of spherical varieties and volume polynomial”, J. Lie Theory, 21:2 (2011), 263–283 | MR | Zbl

[14] A. G. Khovanskii, “Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in Lobachevskii space”, Funct. Anal. Appl., 20:1 (1986), 41–50 | DOI | MR | Zbl

[15] Khovanskii A., Limonchenko I., Monin L., Cohomology rings of quasitoric bundles, E-print, 2021, arXiv: 2112.14970 [math.AT]

[16] Khovanskii A., Monin L., Gorenstein algebras and toric bundles, E-print, 2021, arXiv: 2106.15562 [math.AC]

[17] I. Yu. Limonchenko, Z. Lü, and T. E. Panov, “Calabi–Yau hypersurfaces and SU-bordism”, Proc. Steklov Inst. Math., 302 (2018), 270–278 | DOI | MR | Zbl

[18] Lü Z., Panov T., “On toric generators in the unitary and special unitary bordism rings”, Algebr. Geom. Topol., 16:5 (2016), 2865–2893 | DOI | MR | Zbl

[19] Lü Z., Wang W., “Examples of quasitoric manifolds as special unitary manifolds”, Math. Res. Lett., 23:5 (2016), 1453–1468 | DOI | MR | Zbl

[20] Masuda M., “Unitary toric manifolds, multi-fans and equivariant index”, Tohoku Math. J. Ser. 2, 51:2 (1999), 237–265 | MR | Zbl

[21] Masuda M., Panov T., “On the cohomology of torus manifolds”, Osaka J. Math., 43:3 (2006), 711–746 | MR | Zbl

[22] McMullen P., “On simple polytopes”, Invent. math., 113:2 (1993), 419–444 | DOI | MR | Zbl

[23] Panov T., Ustinovsky Yu., “Complex-analytic structures on moment–angle manifolds”, Moscow Math. J., 12:1 (2012), 149–172 | DOI | MR | Zbl

[24] A. V. Pukhlikov and A. G. Khovanskii, “Finitely additive measures of virtual polytopes”, St. Petersbg. Math. J., 4:2 (1993), 337–356 | MR

[25] A. V. Pukhlikov and A. G. Khovanskii, “A Riemann–Roch theorem for integrals and sums of quasipolynomials over virtual polytopes”, St. Petersbg. Math. J., 4:4 (1993), 789–812 | MR | Zbl

[26] V. A. Timorin, “An analogue of the Hodge–Riemann relations for simple convex polytopes”, Russ. Math. Surv., 54:2 (1999), 381–426 | DOI | MR | Zbl