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In mathematics, a '''Fermat quintic threefold''' is a special [[quintic threefold]], in other words a degree 5 dimension 3 [[hypersurface]] in 4-dimensional projective space,  given by 

:<math>V^5+W^5+X^5+Y^5+Z^5=0. \, </math>

It is a  [[Calabi-Yau manifold]].

The [[Hodge diamond]] of a non-singular quintic 3-fold is
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==Rational curves==
{{harvtxt|Clemens|1984}} conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. The Fermat quintic threefold is not generic in this sense, and {{harvtxt|Albano|Katz|1991}} showed that  its lines are contained in 50 1-dimensional families of the form (''x''&nbsp;:&nbsp;&minus;''&zeta;x''&nbsp;:&nbsp;''ay''&nbsp;:&nbsp;''by''&nbsp;:&nbsp;''cy'') for ''&zeta;''<sup>&nbsp;5</sup>&nbsp;=&nbsp;1 and ''a''<sup>5</sup>&nbsp;+&nbsp;''b''<sup>5</sup>&nbsp;+&nbsp;''c''<sup>5</sup>&nbsp;=&nbsp;0.  There 375 lines in more than 1 family, of the form (''x'' : &minus;''&zeta;x''&nbsp;:&nbsp;''y''&nbsp;:&nbsp;&minus;''&eta;y''&nbsp;:&nbsp;0) for 5th roots of&nbsp;1 ''&zeta;''&nbsp;and&nbsp;''&eta;''.

==References==
*{{Citation | last1=Albano | first1=Alberto | last2=Katz | first2=Sheldon | title=Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture | jstor=2001512 | mr=1024767  | year=1991 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=324 | issue=1 | pages=353–368 | doi=10.2307/2001512}}
*{{Citation | last1=Clemens | first1=Herbert | title=Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) | publisher=[[Princeton University Press]] | series=Ann. of Math. Stud. | mr=756858  | year=1984 | volume=106 | chapter=Some results about Abel-Jacobi mappings | pages=289–304}}
*{{Citation | last1=Cox | first1=David A. | last2=Katz | first2=Sheldon | title=Mirror symmetry and algebraic geometry | url=http://www.ams.org/bookstore-getitem/item=surv-68.s | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1059-0 | mr=1677117  | year=1999 | volume=68}}

[[Category:Threefolds]]
[[Category:Complex manifolds]]