Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Genera of algebraic varieties and counting of lattice points

Authors: Sylvain E. Cappell and Julius L. Shaneson
Journal: Bull. Amer. Math. Soc. 30 (1994), 62-69
MSC (2000): Primary 14F45; Secondary 11P21, 14M25, 32S60
MathSciNet review: 1217352
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper announces results on the behavior of some important algebraic and topological invariants -- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. -- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of Todd classes of toric varieties, and an explicit formula for the number of integral points in a polytope in Euclidean space with integral vertices.

References [Enhancements On Off] (What's this?)

  • [BBD] A. A. Beilinson, J. Berstein, and P. Deligne, Faisceaux pervers, Analyse et topologique sur les espaces singulièrs, Asterisque 100 (1982), 1-171. MR 751966 (86g:32015)
  • [BFM] P. Baum, W. Fulton, and R. MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101-145. MR 0412190 (54:317)
  • [B] M. Brion, Points entière dans les polyedres convexes, Ann. Sci. École Norm. Sup. (4$ ^{e}$) 21 (1988), 653-663. MR 982338 (90d:52020)
  • [CS] S. E. Cappell and J. L. Shaneson, Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991), 521-551. MR 1102578 (92d:57024)
  • [CSW] S. E. Cappell, J. L. Shaneson, and S. Weinberger, Classes characteristiques pour les actions des groupes sur les espaces singulières, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 293-295. MR 1126399 (92f:57035)
  • [D] V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97-154. MR 495499 (80g:14001)
  • [DK] P. Deligne and N. Katz, Groupes de mondromies en geometrie algebraique, Lecture Notes in Math., vol. 340, Springer-Verlag, New York, 1973, Expose XVI, 2, pp. 268-271 (by P. Deligne). MR 0354657 (50:7135)
  • [E] E. Ehrhart, Sur un problème de géometrie diophantine linéaire, J. Reine Angew Math. 227 (1967), 1-29. MR 0213320 (35:4184)
  • [Fu] J. H. Fu, Curvature measures and Chern classes of singular analytic varieties (to appear).
  • [F1] W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, New York, 1984. MR 732620 (85k:14004)
  • [F2] -, Introduction to toric varieties, Princeton Univ. Press, Princeton, NJ (to appear). MR 1234037 (94g:14028)
  • [GM1] M. Goresky and R. MacPherson, Stratified Morse theory, Erbgeb. Math. Grenzgeb. (3), Springer-Verlag, New York (1980). MR 713089 (84k:58017)
  • [GM2] -, Intersection homology. I, Topology 19 (1980), 135-162. MR 572580 (82b:57010)
  • [H] F. Hirzebruch, Topological methods in algebraic geometry (3rd ed.), Grundlehren Math. Wiss., vol. 131, Springer-Verlag, New York, 1978. MR 1335917 (96c:57002)
  • [HZ] F. Hirzebruch and D. Zagier, The Atiyah-Singer index theorem and elementary number theory, Publish or Perish Press, Boston, MA, 1974.
  • [In] S. S. Infirri, Lefschetz fixed point theorem and number of lattice points in convex polytopes, (to appear).
  • [I] B. Iversen, Critical points of an algebraic function, Invent. Math. 12 (1971), 210-224. MR 0342512 (49:7258)
  • [KK] J. M. Kantor and A. Khovanskii, On the number of integral points in polyhedra with integral vertices, C. R. Acad. Sci. Paris Sér. I Math. (to appear).
  • [KS] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren Math. Wiss., vol. 292, Springer-Verlag, New York, 1990. MR 1074006 (92a:58132)
  • [Kh] A. Khovanskii, Newton polyhedra and toric varieties, Func. Anal. Appl. 4 (1977), 56-67.
  • [K] S. Kleiman, The enumerative theory of singularities, Real and Complex Singularities (P. Holm, ed.), Sijthoff and Noordhoff, 1976, pp. 298-384. MR 0568897 (58:27960)
  • [M] L. J. Mordell, Lattice points in a tetrahedron and generalized Dedekind sums, J. Indian. Math. 15 (1951), 41-46. MR 0043815 (13:322b)
  • [Mo] R. Morelli, Pick's Theorem and the Todd class of toric variety, Adv. in Math. 100 (1993), 183-231. MR 1234309 (94j:14048)
  • [O] T. Oda, Convex bodies and algebraic geometry, Springer-Verlag, New York, 1987. MR 922894 (88m:14038)
  • [Pi] G. Pick, Geometrisches zur Zahlenlehre, Sitzungsber. Lotos Prag. (2) 19 (1870), 311-319.
  • [P] J. E. Pommerscheim, Toric varieties, lattice points, and Dedekind sums, Math. Ann. (to appear). MR 1198839 (94c:14043)
  • [R1] H. Rademacher, On Dedekind sums and lattice points in a tetrahedron, Stud. Math. Mech., Academic Press, New York, 1954, pp. 49-53, Reprinted in Collected Papers of H. Rademacher, Vol. II (E. Grosswald, ed.), M.I.T. Press, 1974, pp. 391-398. MR 0064824 (16:341b)
  • [R2] H. Rademacher and E. Grosswald, Dedekind sums, Carus Math. Monographs Vol. 16, Math. Assoc. America, Washington, DC, 1972. MR 0357299 (50:9767)
  • [S] M. Saito, Modules de Hodge polarizables, Publ. Res. Inst. Math. Sci. 24 (1988), 849-995. MR 1000123 (90k:32038)
  • [V] J. L. Verdier, Specialisation de faisceaux et monodromie moderee, Asterisque 101-102 (1982), 332-364. MR 737938 (86f:32010)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 14F45, 11P21, 14M25, 32S60

Retrieve articles in all journals with MSC (2000): 14F45, 11P21, 14M25, 32S60

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society