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.
"[1] A quintic threefold is a special class of Calabi–Yau manifolds defined by a degree
Proving that such a polynomial defines a Calabi-Yau requires some more tools, like the Adjunction formula and conditions for smoothness.
Then, using the adjunction formula to compute its canonical bundle, we have
hence in order for the variety to be Calabi-Yau, meaning it has a trivial canonical bundle, its degree must be
It is then a Calabi-Yau manifold if in addition this variety is smooth.
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
Since the only points where they vanish is given by the coordinate axes in
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.
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.
is a single parameter not equal to a 5-th root of unity.
This can be found by computing the partial derivates of
At a point where the partial derivatives are all zero, this gives the relation
But in the first case, these give a smooth sublocus since the varying term in
vanishes, so a singular point must lie in
Computing the number of rational curves of degree
can be computed explicitly using Schubert calculus.
descends to a vector bundle on this projective Grassmannian.
of the bundle corresponds to a linear homogeneous polynomial,
corresponds to a quintic polynomial, a section of
Then, in order to calculate the number of lines on a generic quintic threefold, it suffices to compute the integral[5]
expanding this out in terms of the original Chern classes gives
using relations implied by Pieri's formula, including
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.)
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.