Inden for matematik er Hodges formodning en stort uløst problem inden for algebraisk geometri, der relaterer algebraisk topologi til ikke-singulære kompleks Algebraisk varietet med dets undervariater. Mere specifikt siger formodningen at visse de Rhamkohomologi-klasser er algebraiske, dvs. at de er summer af Poincarés dualitet af homologiske klasser af undervariaterne.
Det blev formuleret af den skotske matematiker William Vallance Douglas Hodge som et resultat af hans arbejde mellem 1930 og 1940 for at uddybe beskrivelsen af de Rhamkohomologi til at inkludere ekstra strukture, der eksisterer for komplekse algebraiske varieteter. Det modtog kun ringe opmærksomhed før Hodge præsenterede i 1950 på International Congress of Mathematicians, der blev afholdt i Cambridge, Massachusetts.
Hodges formodning er en af Clay Mathematics Institutes millenniumproblemer, hvor der gives en præmie på $1 mio. for at bevise eller modbevise formodningen.
- Atiyah, M. F.; Hirzebruch, F. (1961), "Vector bundles and homogeneous spaces", Proc. Sympos. Pure Math., 3: 7-38
- Cattani, Eduardo; Deligne, Pierre; Kaplan, Aroldo (1995), "On the locus of Hodge classes", Journal of the American Mathematical Society, 8 (2): 483-506, doi:10.2307/2152824, JSTOR 2152824, MR 1273413.
- Grothendieck, A. (1969), "Hodge's general conjecture is false for trivial reasons", Topology, 8 (3): 299-303, doi:10.1016/0040-9383(69)90016-0.
- Hodge, W. V. D. (1950), "The topological invariants of algebraic varieties", Proceedings of the International Congress of Mathematicians, Cambridge, MA, 1: 181-192.
- Kollár, János (1992), "Trento examples", i Ballico, E.; Catanese, F.; Ciliberto, C. (red.), Classification of irregular varieties, Lecture Notes in Math., vol. 1515, Springer, s. 134, ISBN 3-540-55295-2.
- Lefschetz, Solomon (1924), L'Analysis situs et la géométrie algébrique, Collection de Monographies publiée sous la Direction de M. Emile Borel (fransk), Paris: Gauthier-Villars Reprinted in Lefschetz, Solomon (1971), Selected papers, New York: Chelsea Publishing Co., ISBN 978-0-8284-0234-7, MR 0299447.
- Moonen, B. J. J.; Zarhin, Yu. G. (1999), "Hodge classes on abelian varieties of low dimension", Mathematische Annalen, 315 (4): 711-733, arXiv:math/9901113, doi:10.1007/s002080050333.
- Mumford, D. (1969), "A Note of Shimura's paper "Discontinuous groups and abelian varieties"", Math. Ann., 181 (4): 345-351, doi:10.1007/BF01350672.
- Totaro, B. (1997), "Torsion algebraic cycles and complex cobordism", Journal of the American Mathematical Society, 10 (2): 467-493, arXiv:alg-geom/9609016, doi:10.1090/S0894-0347-97-00232-4, JSTOR 2152859.
- Voisin, Claire (2002), "A counterexample to the Hodge conjecture extended to Kähler varieties", Int Math Res Notices, 2002 (20): 1057-1075, doi:10.1155/S1073792802111135.
- Weil, A. (1977), "Abelian varieties and the Hodge ring", Collected papers, vol. III, s. 421-429
- Zucker, S. (1977), "The Hodge conjecture for cubic fourfolds", Comp. Math, 34: 199-209
Wikiwand in your browser!
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.
