Hard Lefschetz theorem for simple polytopes
Balin Fleming
and Kalle Karu
DOI: 10.1007/s10801-009-0212-1
Abstract
McMullen's proof of the Hard Lefschetz Theorem for simple polytopes is studied, and a new proof of this theorem that uses conewise polynomial functions on a simplicial fan is provided.
Pages: 227–239
Keywords: simple polytopes; simplicial fans; hard Lefschetz theorem; Hodge-Riemann-Minkowski bilinear relations
Full Text: PDF
References
1. Barthel, G., Brasselet, J.-P., Fieseler, K.-H., Kaup, L.: Combinatorial intersection cohomology for fans. Tôhoku Math. J. 54, 1-41 (2002)
2. Billera, L.J., Lee, C.W.: A proof of the sufficiency of McMullen's conditions for f -vectors of simplicial convex polytopes. J. Comb. Theory Ser. A 31(3), 237-255 (1981)
3. Bressler, P., Lunts, V.A.: Intersection cohomology on nonrational polytopes. Compos. Math. 135(3), 245-278 (2003)
4. Brion, M.: The structure of the polytope algebra. Tôhoku Math. J. 49, 1-32 (1997)
5. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematical Studies, vol.
131. Princeton University Press, Princeton (1993)
6. McMullen, P.: The numbers of faces of simplicial polytopes. Isr. J. Math. 9, 559-570 (1971)
7. McMullen, P.: On simple polytopes. Invent. Math. 113, 419-444 (1993)
8. McMullen, P.: Weights on polytopes. Discrete Comput. Geom. 15(4), 363-388 (1996)
9. Stanley, R.: The number of faces of a simplicial convex polytope. Adv. Math. 35, 236-238 (1980)
10. Timorin, V.A.: An analogue of the Hodge-Riemann relations for simple convex polyhedra. Usp. Mat.
2. Billera, L.J., Lee, C.W.: A proof of the sufficiency of McMullen's conditions for f -vectors of simplicial convex polytopes. J. Comb. Theory Ser. A 31(3), 237-255 (1981)
3. Bressler, P., Lunts, V.A.: Intersection cohomology on nonrational polytopes. Compos. Math. 135(3), 245-278 (2003)
4. Brion, M.: The structure of the polytope algebra. Tôhoku Math. J. 49, 1-32 (1997)
5. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematical Studies, vol.
131. Princeton University Press, Princeton (1993)
6. McMullen, P.: The numbers of faces of simplicial polytopes. Isr. J. Math. 9, 559-570 (1971)
7. McMullen, P.: On simple polytopes. Invent. Math. 113, 419-444 (1993)
8. McMullen, P.: Weights on polytopes. Discrete Comput. Geom. 15(4), 363-388 (1996)
9. Stanley, R.: The number of faces of a simplicial convex polytope. Adv. Math. 35, 236-238 (1980)
10. Timorin, V.A.: An analogue of the Hodge-Riemann relations for simple convex polyhedra. Usp. Mat.
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition