Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 016, 12 pages      arXiv:1911.00118      https://doi.org/10.3842/SIGMA.2020.016
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Intersections of Hypersurfaces and Ring of Conditions of a Spherical Homogeneous Space

Kiumars Kaveh ^a and Askold G. Khovanskii ^bc
a)  Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA
^b)  Department of Mathematics, University of Toronto, Toronto, Canada
^c)  Moscow Independent University, Moscow, Russia

Received November 04, 2019, in final form March 14, 2020; Published online March 20, 2020

Abstract
We prove a version of the BKK theorem for the ring of conditions of a spherical homogeneous space $G/H$. We also introduce the notion of ring of complete intersections, firstly for a spherical homogeneous space and secondly for an arbitrary variety. Similarly to the ring of conditions of the torus, the ring of complete intersections of $G/H$ admits a description in terms of volumes of polytopes.

Key words: BKK theorem; spherical variety; Newton-Okounkov polytope; ring of conditions.

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References

1. Alexeev V., Brion M., Toric degenerations of spherical varieties, Selecta Math. (N.S.) 10 (2004), 453-478, arXiv:math.AG/0403379.
2. Bernstein D.N., The number of roots of a system of equations, Funct. Anal. Appl. 9 (1975), 183-185.
3. Bifet E., De Concini C., Procesi C., Cohomology of regular embeddings, Adv. Math. 82 (1990), 1-34.
4. Brion M., Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math. J. 58 (1989), 397-424.
5. De Concini C., Procesi C., Complete symmetric varieties. II. Intersection theory, in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., Vol. 6, North-Holland, Amsterdam, 1985, 481-513.
6. Eisenbud D., Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, Vol. 150, Springer-Verlag, New York, 1995.
7. Esterov A., Kazarnovskii B., Khovanskii A.G., Newton polyhedra and tropical geometry, Russian Math. Surveys, to appear.
8. Kaveh K., Note on cohomology rings of spherical varieties and volume polynomial, J. Lie Theory 21 (2011), 263-283, arXiv:math.AG/0312503.
9. Kaveh K., Khovanskii A.G., Mixed volume and an extension of intersection theory of divisors, Mosc. Math. J. 10 (2010), 343-375, arXiv:0812.0433.
10. Kaveh K., Khovanskii A.G., Convex bodies associated to actions of reductive groups, Mosc. Math. J. 12 (2012), 369-396, arXiv:1001.4830.
11. Kaveh K., Khovanskii A.G., Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. 176 (2012), 925-978.
12. Kaveh K., Khovanskii A.G., Note on the Grothendieck group of subspaces of rational functions and Shokurov's Cartier $b$-divisors, Canad. Math. Bull. 57 (2014), 562-572, arXiv:1302.2402.
13. Kaveh K., Villella E., On a notion of anticanonical class for families of convex polytopes, arXiv:1802.06674.
14. Kazarnovskii B., Newton polyhedra and the Bezout formula for matrix-valued functions of finite-dimensional representations, Funct. Anal. Appl. 21 (1987), 319-321.
15. Kazarnovskii B., Khovanskii A.G., Newton polyhedra, tropical geometry and the ring of condition for $({\mathbb C}^*)^n$, arXiv:1705.04248.
16. Khovanskii A.G., Newton polyhedra, and the genus of complete intersections, Funct. Anal. Appl. 12 (1978), 38-46.
17. Knop F., Kraft H., Luna D., Vust T., Local properties of algebraic group actions, in Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., Vol. 13, Birkhäuser, Basel, 1989, 63-75.
18. Kouchnirenko A.G., Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31.
19. Lazarsfeld R., Mustaţă M., Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 783-835, arXiv:0805.4559.
20. Littelmann P., Procesi C., Equivariant cohomology of wonderful compactifications, in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math., Vol. 92, Birkhäuser Boston, Boston, MA, 1990, 219-262.
21. Okunkov A., A remark on the Hilbert polynomial of a spherical manifold, Funct. Anal. Appl. 31 (1997), 138-140.
22. Perrin N., On the geometry of spherical varieties, Transform. Groups 19 (2014), 171-223, arXiv:1211.1277.
23. Popov V.L., Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles, Math. USSR Izv. 8 (1974), 301-327.
24. Strickland E., The ring of conditions of a semisimple group, J. Algebra 320 (2008), 3069-3078.