In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space . Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
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1 | 101 | 101 | 1 | |||
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Mathematician Robbert Dijkgraaf said "One number which every algebraic geometer knows is the number 2,875 because obviously, that is the number of lines on a quintic."[1]
Definition
A quintic threefold is a special class of Calabi–Yau manifolds defined by a degree projective variety in . Many examples are constructed as hypersurfaces in , or complete intersections lying in , or as a smooth variety resolving the singularities of another variety. As a set, a Calabi-Yau manifold is
Hypersurfaces in P4
Recall that a homogeneous polynomial (where is the Serre-twist of the hyperplane line bundle) defines a projective variety, or projective scheme, , from the algebra
Examples
Fermat Quintic
One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold, which is defined by the vanishing locus of the polynomial
As a Hodge Conjecture testbed
Another application of the quintic threefold is in the study of the infinitesimal generalized Hodge conjecture where this difficult problem can be solved in this case.[2] In fact, all of the lines on this hypersurface can be found explicitly.
Dwork family of quintic three-folds
Another popular class of examples of quintic three-folds, studied in many contexts, is the Dwork family. One popular study of such a family is from Candelas, De La Ossa, Green, and Parkes,[3] when they discovered mirror symmetry. This is given by the family[4] pages 123-125
Other examples
Curves on a quintic threefold
Computing the number of rational curves of degree can be computed explicitly using Schubert calculus. Let be the rank vector bundle on the Grassmannian of -planes in some rank vector space. Projectivizing to gives the projective grassmannian of degree 1 lines in and descends to a vector bundle on this projective Grassmannian. Its total chern class is
Rational curves
Herbert Clemens (1984) conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. (Some smooth but non-generic quintic threefolds have infinite families of lines on them.) This was verified for degrees up to 7 by Sheldon Katz (1986) who also calculated the number 609250 of degree 2 rational curves. Philip Candelas, Xenia C. de la Ossa, and Paul S. Green et al. (1991) conjectured a general formula for the virtual number of rational curves of any degree, which was proved by Givental (1996) (the fact that the virtual number equals the actual number relies on confirmation of Clemens' conjecture, currently known for degree at most 11 Cotterill (2012)). The number of rational curves of various degrees on a generic quintic threefold is given by
Since the generic quintic threefold is a Calabi–Yau threefold and the moduli space of rational curves of a given degree is a discrete, finite set (hence compact), these have well-defined Donaldson–Thomas invariants (the "virtual number of points"); at least for degree 1 and 2, these agree with the actual number of points.
See also
- Mirror symmetry (string theory)
- Gromov–Witten invariant
- Jacobian ideal - gives an explicit basis for the Hodge-decomposition
- Deformation theory
- Hodge structure
- Schubert calculus - techniques for determining the number of lines on a quintic threefold
References
- ^ Robbert Dijkgraaf (29 March 2015). "The Unreasonable Effectiveness of Quantum Physics in Modern Mathematics". youtube.com. Trev M. Archived from the original on 2021-12-21. Retrieved 10 September 2015. see 29 minutes 57 seconds
- ^ Albano, Alberto; Katz, Sheldon (1991). "Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture". Transactions of the American Mathematical Society. 324 (1): 353–368. doi:10.1090/S0002-9947-1991-1024767-6. ISSN 0002-9947.
- ^ Candelas, Philip; De La Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991-07-29). "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory". Nuclear Physics B. 359 (1): 21–74. Bibcode:1991NuPhB.359...21C. doi:10.1016/0550-3213(91)90292-6. ISSN 0550-3213.
- ^ Gross, Mark; Huybrechts, Daniel; Joyce, Dominic (2003). Ellingsrud, Geir; Olson, Loren; Ranestad, Kristian; Stromme, Stein A. (eds.). Calabi-Yau Manifolds and Related Geometries: Lectures at a Summer School in Nordfjordeid, Norway, June 2001. Universitext. Berlin Heidelberg: Springer-Verlag. pp. 123–125. ISBN 978-3-540-44059-8.
- ^ Katz, Sheldon. Enumerative Geometry and String Theory. p. 108.
- Arapura, Donu, "Computing Some Hodge Numbers" (PDF)
- Candelas, Philip; de la Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991), "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory", Nuclear Physics B, 359 (1): 21–74, Bibcode:1991NuPhB.359...21C, doi:10.1016/0550-3213(91)90292-6, MR 1115626
- Clemens, Herbert (1984), "Some results about Abel-Jacobi mappings", Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton University Press, pp. 289–304, MR 0756858
- Cotterill, Ethan (2012), "Rational curves of degree 11 on a general quintic 3-fold", The Quarterly Journal of Mathematics, 63 (3): 539–568, doi:10.1093/qmath/har001, MR 2967162
- Cox, David A.; Katz, Sheldon (1999), Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1059-0, MR 1677117
- Givental, Alexander B. (1996), "Equivariant Gromov-Witten invariants", International Mathematics Research Notices, 1996 (13): 613–663, doi:10.1155/S1073792896000414, MR 1408320
- Katz, Sheldon (1986), "On the finiteness of rational curves on quintic threefolds", Compositio Mathematica, 60 (2): 151–162, MR 0868135
- Pandharipande, Rahul (1998), "Rational curves on hypersurfaces (after A. Givental)", Astérisque, 1997/98 (252): 307–340, arXiv:math/9806133, Bibcode:1998math......6133P, MR 1685628