Quintic threefold

A quintic threefold is a special class of Calabi–Yau manifolds defined by a degree $5$ projective variety in $\backslash mathbb\{P\}^4$. Many examples are constructed as hypersurfaces in $\backslash mathbb\{P\}^4$, or complete intersections lying in $\backslash mathbb\{P\}^4$, or as a smooth variety resolving the singularities of another variety. As a set, a Calabi-Yau manifold is$$X\; =\; \backslash \{x\; =\; [x\_0:x\_1:x\_2:x\_3:x\_4]\; \backslash in\; \backslash mathbb\{CP\}^4\; :\; p(x)\; =\; 0\; \backslash \}$$where $p(x)$ is a degree $5$ homogeneous polynomial. One of the most studied examples is from the polynomial$$p(x)\; =\; x\_0^5\; +\; x\_1^5\; +\; x\_2^5\; +\; x\_3^5\; +\; x\_4^5$$called a Fermat polynomial. Proving that such a polynomial defines a Calabi-Yau requires some more tools, like the Adjunction formula and conditions for smoothness.

Recall that a homogeneous polynomial $f\; \backslash in\; \backslash Gamma(\backslash mathbb\{P\}^4,\backslash mathcal\{O\}(d))$ (where $\backslash mathcal\{O\}(d)$ is the Serre-twist of the hyperplane line bundle) defines a projective variety, or projective scheme, $X$, from the algebra$$\backslash frac\{k[x\_0,\backslash ldots,\; x\_4]\}\{(f)\}$$where $k$ is a field, such as $\backslash mathbb\{C\}$. Then, using the adjunction formula to compute its canonical bundle, we have$$\backslash begin\{align\}$$

\Omega_X^3 &= \omega_X \\

&= \omega_{\mathbb{P}^4}\otimes \mathcal{O}(d) \\

&\cong \mathcal{O}(-(4+1))\otimes\mathcal{O}(d) \\

&\cong \mathcal{O}(d-5)

\end{align}hence in order for the variety to be Calabi-Yau, meaning it has a trivial canonical bundle, its degree must be $5$. It is then a Calabi-Yau manifold if in addition this variety is smooth. This can be checked by looking at the zeros of the polynomials$$\backslash partial\_0f,\backslash ldots,\; \backslash partial\_4f$$and making sure the set$$\backslash \{\; x\; =\; [x\_0:\backslash cdots:x\_4]\; |\; f(x)\; =\; \backslash partial\_0f(x)\; =\; \backslash cdots\; =\; \backslash partial\_4f(x)\; =\; 0\; \backslash \}$$is empty.

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$$f\; =\; x\_0^5\; +\; x\_1^5\; +\; x\_2^5\; +\; x\_3^5\; +\; x\_4^5$$Computing the partial derivatives of $f$ gives the four polynomials$$\backslash begin\{align\}$$

\partial_0f = 5x_0^4\\

\partial_1f = 5x_1^4 \\

\partial_2f = 5x_2^4 \\

\partial_3f = 5x_3^4 \\

\partial_4f = 5x_4^4 \\

\end{align}Since the only points where they vanish is given by the coordinate axes in $\backslash mathbb\{P\}^4$, the vanishing locus is empty since $[0:0:0:0:0]$ is not a point in $\backslash mathbb\{P\}^4$.

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.{{Cite journal|last1=Albano|first1=Alberto|last2=Katz|first2=Sheldon|date=1991|title=Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture|url=https://www.ams.org/tran/1991-324-01/S0002-9947-1991-1024767-6/|journal=Transactions of the American Mathematical Society|language=en|volume=324|issue=1|pages=353–368|doi=10.1090/S0002-9947-1991-1024767-6|issn=0002-9947|doi-access=free}} 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,{{Cite journal|last1=Candelas|first1=Philip|last2=De La Ossa|first2=Xenia C.|last3=Green|first3=Paul S.|last4=Parkes|first4=Linda|date=1991-07-29|title=A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory|url=https://dx.doi.org/10.1016/0550-3213%2891%2990292-6|journal=Nuclear Physics B|language=en|volume=359|issue=1|pages=21–74|doi=10.1016/0550-3213(91)90292-6|bibcode=1991NuPhB.359...21C|issn=0550-3213}} when they discovered mirror symmetry. This is given by the family{{Cite book|last1=Gross|first1=Mark|url=https://www.springer.com/gp/book/9783540440598|title=Calabi-Yau Manifolds and Related Geometries: Lectures at a Summer School in Nordfjordeid, Norway, June 2001|last2=Huybrechts|first2=Daniel|last3=Joyce|first3=Dominic|date=2003|publisher=Springer-Verlag|isbn=978-3-540-44059-8|editor-last=Ellingsrud|editor-first=Geir|series=Universitext|location=Berlin Heidelberg|pages=123–125|language=en|editor-last2=Olson|editor-first2=Loren|editor-last3=Ranestad|editor-first3=Kristian|editor-last4=Stromme|editor-first4=Stein A.}} ^{pages 123-125}$$f\_\backslash psi\; =\; x\_0^5\; +\; x\_1^5\; +\; x\_2^5\; +\; x\_3^5\; +\; x\_4^5\; -\; 5\backslash psi\; x\_0x\_1x\_2x\_3x\_4$$where $\backslash psi$ is a single parameter not equal to a 5-th root of unity. This can be found by computing the partial derivates of $f\_\backslash psi$ and evaluating their zeros. The partial derivates are given by$$\backslash begin\{align\}$$

\partial_0f_\psi = 5x_0^4 - 5\psi x_1x_2x_3x_4 \\

\partial_1f_\psi = 5x_1^4 - 5\psi x_0x_2x_3x_4 \\

\partial_2f_\psi = 5x_2^4 - 5\psi x_0x_1x_3x_4 \\

\partial_3f_\psi = 5x_3^4 - 5\psi x_0x_1x_2x_4\\

\partial_4f_\psi = 5x_4^4 - 5\psi x_0x_1x_2x_3\\

\end{align}At a point where the partial derivatives are all zero, this gives the relation $x\_i^5\; =\; \backslash psi\; x\_0x\_1x\_2x\_3x\_4$. For example, in $\backslash partial\_0f\_\backslash psi$ we get$$\backslash begin\{align\}$$

5x_0^4 &= 5\psi x_1x_2x_3x_4 \\

x_0^4 &= \psi x_1x_2x_3x_4 \\

x_0^5 &= \psi x_0x_1x_2x_3x_4

\end{align}by dividing out the $5$ and multiplying each side by $x\_0$. From multiplying these families of equations $x\_i^5\; =\; \backslash psi\; x\_0x\_1x\_2x\_3x\_4$ together we have the relation$$\backslash prod\; x\_i^5\; =\; \backslash psi^5\; \backslash prod\; x\_i^5$$showing a solution is either given by an $x\_i\; =\; 0$ or $\backslash psi^5\; =\; 1$. But in the first case, these give a smooth sublocus since the varying term in $f\_\backslash psi$ vanishes, so a singular point must lie in $\backslash psi^5\; =\; 1$. Given such a $\backslash psi$, the singular points are then of the form$$[\backslash mu\_5^\{a\_0\}:\backslash cdots:\backslash mu\_5^\{a\_4\}]$$ such that $$\backslash mu\_5^\{\backslash sum\; a\_i\}=\backslash psi^\{-1\}$$where $\backslash mu\_5\; =\; e^\{2\; \backslash pi\; i\; /\; 5\}$. For example, the point$$[\backslash mu\_5^4:\backslash mu\_5^\{-1\}:\backslash mu\_5^\{-1\}:\backslash mu\_5^\{-1\}:\backslash mu\_5^\{-1\}]$$is a solution of both $f\_1$ and its partial derivatives since $(\backslash mu\_5^i)^5\; =\; (\backslash mu\_5^5)^i\; =\; 1^i\; =\; 1$, and $\backslash psi\; =\; 1$.

• Barth–Nieto quintic
• Consani–Scholten quintic

Computing the number of rational curves of degree $1$ can be computed explicitly using Schubert calculus. Let $T^*$ be the rank $2$ vector bundle on the Grassmannian $G(2,5)$ of $2$-planes in some rank $5$ vector space. Projectivizing $G(2,5)$ to $\backslash mathbb\{G\}(1,4)$ gives the projective Grassmannian of degree $1$ lines in $\backslash mathbb\{P\}^4$ and $T^*$ descends to a vector bundle on this projective Grassmannian. Its total Chern class is$$c(T^*)\; =\; 1\; +\; \backslash sigma\_1\; +\; \backslash sigma\_\{1,1\}$$in the Chow ring $A^\backslash bullet(\backslash mathbb\{G\}(1,4))$. Now, a section $l\; \backslash in\; \backslash Gamma(\backslash mathbb\{G\}(1,4),T^*)$ of the bundle corresponds to a linear homogeneous polynomial, $\backslash tilde\{l\}\; \backslash in\; \backslash Gamma(\backslash mathbb\{P\}^4,\backslash mathcal\{O\}(1))$, so a section of $\backslash text\{Sym\}^5(T^*)$ corresponds to a quintic polynomial, a section of $\backslash Gamma(\backslash mathbb\{P\}^4,\backslash mathcal\{O\}(5))$. Then, in order to calculate the number of lines on a generic quintic threefold, it suffices to compute the integral{{Cite book|last=Katz|first=Sheldon|title=Enumerative Geometry and String Theory|pages=108}}$$\backslash int\_\{\backslash mathbb\{G\}(1,4)\}\; c(\backslash text\{Sym\}^5(T^*))\; =\; 2875$$This can be done by using the splitting principle. Since$$\backslash begin\{align\}$$

c(T^*) &= (1+\alpha)(1+\beta) \\

&= 1 + (\alpha + \beta) + \alpha\beta

\end{align}and for a dimension $2$ vector space, $V\; =\; V\_1\backslash oplus\; V\_2$,$$\backslash text\{Sym\}^5(V)\; =\; \backslash bigoplus\_\{i=0\}^5\; (V\_1^\{\backslash otimes\; 5-i\}\backslash otimes\; V\_2^\{\backslash otimes\; i\})$$so the total Chern class of $\backslash text\{Sym\}^5(T^*)$ is given by the product$$c(\backslash text\{Sym\}^5(T^*))\; =\; \backslash prod\_\{i=0\}^5\; (1\; +\; (5-i)\backslash alpha\; +\; i\backslash beta)$$Then, the Euler class, or the top class is$$5\backslash alpha(4\backslash alpha\; +\; \backslash beta)(3\backslash alpha\; +\; 2\backslash beta)(2\backslash alpha\; +\; 3\backslash beta)(\backslash alpha\; +\; 4\backslash beta)5\backslash beta$$expanding this out in terms of the original Chern classes gives$$\backslash begin\{align\}$$

c_6(\text{Sym}^5(T^*)) &= 25\sigma_{1,1}(4\sigma_1^2 + 9\sigma_{1,1})(6\sigma_1^2 + \sigma_{1,1}) \\

&= (100 \sigma_{3,1} + 100\sigma_{2,2} + 225\sigma_{2,2})(6\sigma_1^2 + \sigma_{1,1}) \\

&= (100 \sigma_{3,1} + 325\sigma_{2,2})(6\sigma_1^2 + \sigma_{1,1})\\

&= 600 \sigma_{3,3} + 2275 \sigma_{3,3}\\

&= 2875 \sigma_{3,3}

\end{align}using relations implied by Pieri's formula, including $\backslash sigma\_1^2\; =\; \backslash sigma\_2\; +\; \backslash sigma\_\{1,1\}$, $\backslash sigma\_\{1,1\}\backslash cdot\; \backslash sigma\_1^2\; =\; \backslash sigma\_\{3,1\}\; +\; \backslash sigma\_\{2,2\}$, $\backslash sigma\_\{1,1\}^2\; =\; \backslash sigma\_\{2,2\}$.

{{harvs|txt|last=Clemens | first=Herbert | authorlink=Herbert Clemens | year=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 {{harvs|txt| last=Katz | first=Sheldon |authorlink=Sheldon Katz|year=1986}} who also calculated the number 609250 of degree 2 rational curves.

{{harvs|txt| last1=Candelas | first1=Philip | author1-link=Philip Candelas | last2=de la Ossa | authorlink2=Xenia de la Ossa | first2=Xenia C. | last3=Green | first3=Paul S. | last4=Parkes | first4=Linda | title=A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory | doi=10.1016/0550-3213(91)90292-6 | mr=1115626 | year=1991 | journal=Nuclear Physics B | volume=359 | issue=1 | pages=21–74}}

conjectured a general formula for the virtual number of rational curves of any degree, which was proved by {{harvtxt|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 {{harvtxt|Cotterill|2012}}).

The number of rational curves of various degrees on a generic quintic threefold is given by

:2875, 609250, 317206375, 242467530000, ...{{OEIS|A076912}}.

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.

• Enumerative geometry

• 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

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Category:Algebraic varieties

Category:3-folds

Category:Complex manifolds