Hypergeometric Motives

Let $\alpha=(\alpha_{1},\dots,\alpha_{n})$, and $\beta=(\beta_{1},\dots,\beta_{n})$ be vectors of complex numbers with $\mbox{Re}(\beta_{j})>\mbox{Re}(\alpha_{j})>0$ and $\beta_{n}=1$. For $|t|<1$ define, making use of the standard Gamma function $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}dx$,

Via $\Gamma(1)=1$ and $\Gamma(1/2)=\sqrt{\pi}$, (1.1) is the special case $\alpha=(1/2,1/2)$ and $\beta=(1,1)$.

Expand the denominator of the integrand of (2) via the binomial theorem and use Euler’s beta integral to evaluate the individual terms. Written in terms of Pochhammer symbols $(u)_{k}=u(+1)\cdots(u+k-1)$, the result is

In other words, the integral (2) is an alternative definition of the standard hypergeometric power series (2.2). The case $\alpha=(1/2,1/2)$ and $\beta=(1,1)$ simplifies to (1.2).

An excellent general reference for hypergeometric functions is [BH], and we now give a summary sufficient for this survey. The function $F(\alpha,\beta,t)$ is in the kernel of an $n^{\rm th}$ order differential operator $D(\alpha,\beta)$ with singularities only at $0$, $1$ and $\infty$. This means in particular that $F(\alpha,\beta,t)$, initially defined on the unit disk, extends to a “multivalued function” on the thrice-punctured projective line ${\mathbb{P}}^{1}({\mathbb{C}})-\{0,1,\infty)={\mathbb{C}}-\{0,1\}$. With respect to a given basis, this multivaluedness is codified by a representation $\rho$ of the fundamental group $\pi_{1}({\mathbb{C}}-\{0,1\},1/2)$ into $GL_{n}({\mathbb{C}})$. The fundamental group is free on $g_{0}$ and $g_{1}$, with these elements coming from counterclockwise circular paths of radius $1/2$ about $0$ and $1$ respectively. To emphasize the equal status of $\infty$ and $0$, it is better to present this group as generated by $g_{\infty}$, $g_{1}$, and $g_{0}$, subject to the relation $g_{\infty}g_{1}g_{0}=1$. The assumption $\beta_{n}=1$ was only imposed to present the classical viewpoint (2)-(2.2) cleanly; we henceforth drop it.

A useful fact due to Levelt is the explicit description of the matrices $h_{\tau}=\rho(g_{\tau})\in GL_{n}({\mathbb{C}})$ with respect to a certain well-chosen basis. Define polynomials

Then $h_{\infty}$ and $h_{0}$ are companion matrices of $q_{\infty}$ and $q_{0}$, while $h_{1}$ is determined by $h_{\infty}h_{1}h_{0}=I$. The matrix $h_{1}$ differs minimally from the identity in that $h_{1}-I$ has rank $1$. We will henceforth consider only cases where no $\alpha_{i}-\beta_{j}$ is an integer. This ensures that the $h_{\tau}$ generate an irreducible subgroup $\Gamma$ of $GL_{n}({\mathbb{C}})$. Moreover the representation is rigid, in the following sense: suppose $h_{\infty}^{\prime}$, $h^{\prime}_{1}$, and $h_{0}^{\prime}$ are conjugate to $h_{\infty}$, $h_{1}$, and $h_{0}$ respectively. Then there is a single matrix $c$ such that $h_{\tau}=ch_{\tau}^{\prime}c^{-1}$ for all three $\tau$.

The parameters $(\alpha,\beta)$ contain information which is irrelevant for the sequel. First, the individual $\alpha_{j}$ and $\beta_{j}$ are important only modulo integers. Second, the orderings of the $\alpha_{j}$ and $\beta_{j}$ do not matter. To remove these irrelevancies, we will regard the degree $n$ rational function $Q=q_{\infty}/q_{0}$ as the primary index in the sequel, calling it the family parameter. A bonus of this shift in emphasis is that an important field $E\subset{\mathbb{C}}$ is made evident, the field generated by the coefficients of $q_{\infty}$ and $q_{0}$. By construction, all three $h_{\tau}$ lie in $GL_{n}(E)$.

The cases which naturally have underlying motives are exactly the ones with all $\alpha_{j}$ and $\beta_{j}$ rational, so that $E$ is some cyclotomic field. In this survey we will substantially simplify by restricting to cases with $E={\mathbb{Q}}$. With this simplification, there are two natural ways to present $Q$ as follows. Write $\Psi_{m}=T^{m}-1$ and consider its factorization into irreducible polynomials, $\Psi_{m}=\prod_{d|m}\Phi_{d}$. So the factors are cyclotomic polynomials $\Phi_{d}=\prod_{j\in({\mathbb{Z}}/d)^{\times}}(T-e^{2\pi ij/d})$ and thus have degree the Euler totient $\phi(d)=|({\mathbb{Z}}/d)^{\times}|$.

In our introductory example, the ways are

In general, the second way is just the canonical factorization into irreducibles, while the first way is the unique “unreduction” to products of $\Psi_{m}$ in which no factor appears in both a numerator and denominator.

To enter a family parameter $Q$ into Magma, one can use either of these two ways, as in the equivalent commands

In the first method, one inputs just the gamma vector $\gamma=[\gamma_{1},\dots,\gamma_{l}]$ formed by subscripts on the $\Psi$’s, using signs to distinguish between numerator and denominator. In the second method, one inputs just the subscripts of the denominator and then numerator $\Phi$’s, these being called the cyclotomic parameters. When working with underlying varieties, the gamma vectors are so useful that we often simply write $H(\gamma,t)$ rather than $H(Q,t)$. After the transition to motives, the cyclotomic presentation is generally more convenient. To simplify slightly, we henceforth require that $\gcd(\gamma_{1},\dots,\gamma_{l})=1$.

Note that initialization commands like (2.4) do nothing by themselves; in this survey Magma will first start returning useful information in Sections 5 and 11. Note also that Magma requires a semicolon at the end of all commands, as in (2.4). We often omit these semicolons in the sequel.

The number $Q(0)=\det(h_{1})$ is either $-1$ or $1$ under our restriction $E={\mathbb{Q}}$. This dichotomy is strongly felt throughout this survey. It can also be expressed in terms of the fundamental bilinear form $\left(\cdot,\cdot\right)$ on ${\mathbb{Q}}^{n}$ preserved by the monodromy group $\Gamma=\langle h_{\infty},h_{0}\rangle$; see [BH, §4], [RV-bez, §3.5]. In the orthogonal case, $h_{1}$ is conjugate to $\mbox{diag}(-1,1,\dots,1)$ and $\left(\cdot,\cdot\right)$ is symmetric. In the symplectic case, $h_{1}$ is conjugate to $\binom{1\;1}{0\;1}\oplus\mbox{diag}(1,\dots,1)$, and $\left(\cdot,\cdot\right)$ is antisymmetric.

We have already generalized (1.1) to (2) and (1.2) to (2.2). Assuming briefly $\beta_{n}=1$ again, a natural generalization of (1.3) is to

Here $m$ is the least common denominator of the $\alpha_{j}$ and $\beta_{j}$, and the exponents are integers $0\leq a_{j},b_{j}<m$ such that

for $j=1,\ldots,n-1$ and $a_{n}\equiv m\alpha_{n}\bmod m$. The equations (2)-(2.2) show that a specified scalar multiple of ${}_{n}F_{n-1}(\alpha,\beta,t)$ arises as a period of this variety. However the varieties (3.1) depend on how the parameters are paired: $(\alpha_{1},\beta_{1})$, …, $(\alpha_{n},\beta_{n})$. This dependence complicates the arithmetic of these varieties, so we will use an alternative collection of varieties to define hypergeometric motives.

The alternative varieties appear under the term “circuits” in [GKZ] and are studied at greater length in [BCM]. For a gamma vector $\gamma$ and a complex number $t$, define $X^{\rm bcm}_{\gamma,t}\subset{\mathbb{P}}^{l-1}$ by two homogeneous equations,

Here and in the sequel, we systematically use the abbreviation $u=t\prod_{j}\gamma_{j}^{\gamma_{j}}$. The canonical variety is by definition the open subvariety $X_{\gamma,t}$ on which all the homogeneous coordinates $y_{j}$ are nonzero. The point $(\gamma_{1}:\cdots:\gamma_{l})$ is an ordinary double point on $X_{\gamma,1}$ and otherwise all the $X_{\gamma,t}$ are smooth. Because of this double point, we exclude the case $t=1$ from consideration until Section 9.

From a dimension-count viewpoint, the BCM equations (3.2) for canonical varieties are inefficient. They start with the $l=\kappa+3$ variables $y_{i}$ and use two equations and projectivization to get the desired $\kappa$-dimensional variety $X_{\gamma,t}$. The toric models from [GKZ] start instead with $d=\kappa+1$ variables $x_{i}$ and present $X_{\gamma,t}$ by just one equation.

To obtain a toric model from a gamma vector $\gamma$, one proceeds as illustrated by Table 3.1. First, for each new variable $x_{i}$ choose a row vector $m_{i*}$ in ${\mathbb{Z}}^{l}$ which is orthogonal to the given $l$-vector $\gamma$. These row vectors are required to be such that ${\mathbb{Z}}^{l}/\langle m_{i*}\rangle$ is torsion-free. Second, choose a row vector $k\in{\mathbb{Z}}^{l}$ which satisfies $\gamma\cdot k=1$. The toric model is then

So in the example of Table 3.1, the resulting equation is

with $u=-2^{6}3^{3}t/5^{5}$. In general, the variety $X_{\gamma,t}$ is the subvariety of the torus ${\mathbb{G}}_{m}^{d}$ given by the equation (3.3).

The relation between the BCM equation for $X_{\gamma,t}$ and a toric model for $X_{\gamma,t}$ is very simple:

When one uses (3.5) to write (3.2) in terms of the $x_{i}$, the second equation is identically satisfied while the first becomes (3.3). Conversely, any point $(y_{1}:\dots:y_{l})$ comes from a unique $(x_{1},\dots,x_{d})$ because of the torsion-free condition.

A toric model gives a polytope $\Delta\subset{\mathbb{R}}^{d}$ which is an aid to understanding the $X_{\gamma,t}$. The case $d=2$ is readily visualizable and Figure 3.1 continues our example. In general, one interprets the column vectors $m_{*j}$ of the chosen matrix as points in ${\mathbb{Z}}^{d}$ and $\Delta$ is their convex hull. Let $\Delta_{j}$ be the convex hull of all the points except the $j^{\rm th}$ one. Normalize volume so that the standard $d$-dimensional simplex has volume $1$, and thus $[0,1]^{d}$ has volume $d!$. Then the volume of $\Delta_{j}$ is $|\gamma_{j}|$. The $\Delta_{j}$ with $\gamma_{j}>0$ form one triangulation of $\Delta$, while the $\Delta_{j}$ with $\gamma_{j}<0$ form another. The total volume of $\Delta$ is the important number $\mbox{vol}(\gamma)=\frac{1}{2}\sum_{j=1}^{l}|\gamma_{j}|$

The common topology of the $X_{\gamma,t}$ with $t\neq 1$ is reflected in the combinatorics of $\Delta$. In the case of $d=2$, the genus $g$ of $X_{\gamma,t}$ is the number of lattice points on the interior, while the number of punctures $k$ is the number of lattice points on the boundary. Pick’s theorem then says that the Euler characteristic $\chi=2-2g-k$ of $X_{\gamma,t}$ is $-\mbox{vol}(\gamma)$. In the example of Figure 3.1, $(g,k,\chi)=(2,5,-7)$. For larger ambient dimension $d$, the situation is of course much more complicated, but always $\chi=(-1)^{d-1}\mbox{vol}(\gamma)$.

In algebraic geometry, one normally wants to compactify a given open variety such as $X_{\gamma,t}$ and there are typically many natural ways of doing it.

We already saw the compactification $X_{\gamma,t}^{\rm bcm}$. It is a hypersurface of degree $\mbox{vol}(\gamma)$ in the projective space ${\mathbb{P}}^{d}$ defined by the first equation of (3.2). On the other hand, for any choice of matrix $m$ with all entries nonnegative, homogenization of (3.3) gives a alternative compactification $\overline{X}_{\gamma,t}\subset{\mathbb{P}}^{d}$.

In our continuing example $\gamma=[-5,-2,3,4]$, the plane curve $X_{\gamma,t}^{\rm bcm}$ has degree seven. In contrast, the plane curve $\overline{X}_{\gamma,t}$ has degree just four, this number arising as the maximum column sum of the matrix $m$ in Table 3.1. Smooth curves in these degrees have genera $15$ and $3$ respectively. For $t\neq 1$, $X_{\gamma,t}$ has genus $2$ so $X_{\gamma,t}^{\rm bcm}$ must have bad singularities while $\overline{X}_{\gamma,t}$ has just a single node.

Another compactification $X_{\gamma,t}^{\rm BCM}$ is a major focus of [BCM]. It is typically not smooth, but only has quotient singularities. These singularities are mild in the sense that $X_{\gamma,t}^{\rm BCM}$ looks smooth from the viewpoint of rational cohomology, and may be ignored when discussing motives as in the next section.

Let $K$ and $E$ be subfields of ${\mathbb{C}}$; the case of principal interest to us is $K=E={\mathbb{Q}}$. Minimally modifying Grothendieck’s original conditional definitions, André unconditionally defined a category $\mathcal{M}(K,E)$ of pure motives over $K$ with coefficients in $E$ [And]. The formal structures of this category can best be understood in terms of a huge proreductive algebraic group ${\mathbb{G}}_{K}$ over ${\mathbb{Q}}$, the absolute motivic Galois group of $K$. Then $\mathcal{M}(K,E)$ is exactly the category of representations of ${\mathbb{G}}_{K}$ on finite-dimensional $E$ vector spaces.

When taking cohomology, we are always implicitly working with the complex points of a variety. For a smooth projective variety $X$ over $K$ and an integer $w$, the singular cohomology space $M=H^{w}(X,E)$ is an object of $\mathcal{M}(K,E)$. The image $G_{M}$ of ${\mathbb{G}}_{K}$ in the general linear group of $M$ is by definition the motivic Galois group of $M$. The purpose of $G_{M}$, as the rest of this survey will make clear, is to group-theoretically coordinate very concrete structures on the vector space $H^{w}(X,E)$.

Two copies of the multiplicative group ${\mathbb{G}}_{m}=GL_{1}$ play important roles in the formalism of motives. One is a normal subgroup and the other a quotient: ${\mathbb{G}}_{m}\subset{\mathbb{G}}_{K}\twoheadrightarrow{\mathbb{G}}_{m}$. A rank $n$ motive $M$ is said to be of weight $w$ if the representation restricted to the subgroup ${\mathbb{G}}_{m}$ consists of $n$ copies the representation $r\mapsto r^{w}$. The motives $H^{w}(X,E)$ all have weight $w$. The representation of ${\mathbb{G}}_{K}$ on the rank one motive $E(-1):=H^{2}({\mathbb{P}}^{1},E)$ corresponds to the representation $t\mapsto t$ of the quotient group ${\mathbb{G}}_{m}$. The motive corresponding to the representation $t\mapsto t^{j}$ is denoted $E(-j)$. The motives $M(j):=M\otimes E(j)$ are called the Tate twists of $M$.

Each category of pure motives $\mathcal{M}(K,E)$ is contained in a larger category $\mathcal{M}\mathcal{M}(K,E)$ of mixed motives, where an irreducible motive has a canonical weight filtration with subquotients in $\mathcal{M}(K,E)$. We will mention mixed motives at several junctures, but our focus is sharply on pure motives.

Let $\gamma$ be a gamma vector of length $\kappa+3$ with $r$ negative entries and let $t\in{\mathbb{Q}}^{\times}-\{1\}$. The hypergeometric motive $H(\gamma,t)$ is defined from the cohomology of the affine variety $X_{\gamma,t}$ [RV-Mixed]. We start with the compactly supported middle cohomology space $H_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ and begin by cutting out a subquotient $H^{\prime}(\gamma,t)$ in two steps.

First, we eliminate the contribution of the ambient $d$-dimensional torus to obtain the primitive subspace $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$. Second, we take any smooth compactification $\bar{X}$ of $X_{\gamma,t}$, or one with at worst mild singularities as mentioned above, and consider the image $H^{\prime}(\gamma,t)$ of $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ under the natural map to $H^{\kappa}(\bar{X},{\mathbb{Q}})$. As a quotient of $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$, the space $H^{\prime}(\gamma,t)$ is independent of the choice of compactification.

For example, for the compactification $X^{BCM}_{\gamma,t}$ there is a decomposition of its middle cohomology, $H^{\kappa}(X^{BCM}_{\gamma,t},{\mathbb{Q}})=H^{\prime}(\gamma,t)\oplus T$. It is described at the level of point counts in [BCM, Thm 1.5]. Here $T$ is zero if $\kappa$ is odd and the sum of $\binom{\kappa+1}{r-1}$ copies of ${\mathbb{Q}}(-\kappa/2)$ if $\kappa$ is even.

Finally, we define the hypergeometric motive $H(\gamma,t)\in\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ as the Hodge-normalized Tate twist $H^{\prime}(\gamma,t)(j)$, as discussed in the next section. So $H(\gamma,t)$ has weight $w=\kappa-2j$ with $j\in{\mathbb{Z}}_{\geq 0}$ specified there.

More conceptually, $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ is a mixed motive of rank $\operatorname{vol}(\gamma)-1$ and the pure motive $H^{\prime}(\gamma,t)$ is its weight $\kappa$ quotient. The passage from $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ to $H^{\prime}(\gamma,t)$ is closely related to the reduction of fractions as in (2.3). In particular, $H^{\prime}(\gamma,t)$ has rank $n=\deg(Q)$.

It is worth stressing that the full mixed motive $PH_{c}^{\kappa}(X_{\gamma,t},{\mathbb{Q}})$ is itself of great interest, with its lower weight parts playing an important role in deeper studies of hypergeometric motives.

For $t\in{\mathbb{C}}^{\times}-\{1\}$, one likewise gets a motive $M=H(\gamma,t)\in\mathcal{M}({\mathbb{Q}}(t),{\mathbb{Q}})$. lf $t$ is transcendental then the motivic Galois group of $M$ can be cleanly expressed in terms of the monodromy group $\Gamma$ of Section 2, as follows. If $w=0$, then $\Gamma$ is finite and $G_{M}=\Gamma$. If $w>0$, then $\Gamma$ is infinite and $G_{M}$ is the smallest algebraic group containing both $\Gamma$ and scalars; more explicitly, $G_{M}$ is the conformal symplectic group $\mbox{CSp}_{n}$ in the symplectic case of odd $w$, and a conformal orthogonal group $\mbox{CO}_{n}$ in the orthogonal case of even $w$. In the case that $t$ is algebraic, including our main case that $t$ is rational, the same identification of $G_{M}$ holds almost always.

The toric model viewpoint is part of the program in [GKZ] to approach algebraic geometry by emphasizing the number $\kappa+\epsilon$ of terms in polynomials defining $\kappa$-dimensional varieties. By scaling to normalize coefficients, such varieties come in $(\epsilon-2)$-dimensional families.

For $\epsilon=2$, the Newton polytope $\Delta$ is a simplex. An abelian group $A$ of order $\mbox{vol}(\Delta)$ and some exponent $m$ acts on the single associated complex variety $X$. The essential cases here are the Fermat varieties in ${\mathbb{P}}^{\kappa+1}$, defined by

The group $A$ comes from scaling the variables by $m^{\rm th}$ roots of unity and has order $m^{\kappa+1}$ and exponent $m$. Writing $K={\mathbb{Q}}(e^{2\pi i/m})$, the action decomposes $H^{\kappa}(X,K)$ into one-dimensional motives in $\mathcal{M}(K,K)$. This setting of $\epsilon=2$ was the focus of several influential papers of Weil from around 1950, and the rank one motives appearing are Jacobi motives.

The case $\epsilon=3$ corresponds to general hypergeometric motives where the $\alpha_{j}$ and $\beta_{j}$ can be arbitrary rational numbers. The group $A$ now has order a divisor of $\mbox{vol}(\Delta)$ and some exponent $m$. For example, for $m=\kappa+2\geq 3$ one can add the term $ux_{1}\cdots x_{m}$ to (4.1); then $A$ is reduced to having order $m^{\kappa}$ but still has exponent $m$. The action of $A$ again decomposes $H^{\kappa}(X,K)$ in $\mathcal{M}(K,K)$ and the summands include general hypergeometric motives. Our torsion-free requirement for exponent matrices is equivalent to requiring that the column vectors affine span ${\mathbb{Z}}^{d}$; in turn, this means that our HGMs constitute exactly the case $|A|=1$.

Much of what we are describing in this article both has simpler analogs for Jacobi motives and extends to general hypergeometric motives. Indeed [BH] and [Katz-ESDE] are in the latter setting. However the associated $L$-functions correspond to motives that have been descended to $\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ and have rank $\phi(m)n$. Because of the factor $\phi(m)$, inclusion of these other settings would only modestly increase the collection of computationally accessible $L$-functions. Also the resulting motives in $\mathcal{M}({\mathbb{Q}},{\mathbb{Q}})$ have motivic Galois groups which are more complicated than the $\mbox{CSp}_{n}$ and $\mbox{CO}_{n}$ arising ubiquitously in our setting of $H(\gamma,t)$.

For a smooth projective variety $X$ over $K\subseteq{\mathbb{C}}$, there is a decomposition of complex vector spaces $H^{w}(X,{\mathbb{C}})=\bigoplus_{p=0}^{w}H^{p,w-p}$, with complex conjugation on coefficients switching $H^{p,q}$ and $H^{q,p}$. The Hodge numbers $h^{p,q}:=\dim(H^{p,q})$ therefore satisfy Hodge symmetry $h^{p,q}=h^{q,p}$ and sum to the Betti number $b_{w}:=\dim(H^{w}(X,{\mathbb{C}}))$. Classical examples are given in (6.1)-(6.2) below.

Likewise, the rank of a weight $w$ motive $M\in\mathcal{M}(K,E)$ is decomposed into Hodge numbers $h^{p,w-p}$. The decomposition has a simple group-theoretic reformulation: ${\mathbb{G}}_{K}({\mathbb{R}})$ contains a subgroup ${\mathbb{C}}^{\times}$ which acts on $H^{p,q}$ by $z^{p}\overline{z}^{q}$. If either $K$ or $E$ is in ${\mathbb{R}}$, as will generally be the case for us, then Hodge symmetry continues to hold.

If a motive $M$ has Hodge numbers $\underline{h}^{p,q}$ then the Hodge numbers of its Tate twist $M(j)$ are ${h}^{p-j,q-j}=\underline{h}^{p,q}$. The Hodge-normalization of a pure weight motive is the Tate twist for which all the nonzero Hodge numbers are in the vector $h=(h^{w,0},\dots,h^{0,w})$ and at least one of the outer ones is nonzero.

The procedure we are about to describe is equivalent to a formula conjectured by Corti and Golyshev [CG] and proved by different methods in Fedorov [Fed] and [RV-Mixed]. The procedure is completely combinatorial and only depends on the interlacing pattern of the roots of $q_{\infty}$ and $q_{0}$ in the unit circle.

To pass from a family parameter $Q=q_{\infty}/q_{0}$ to its Hodge vector $h$ one proceeds as illustrated by Figure 5.1. One orders the parameters $\alpha_{j}$ and $\beta_{j}$, viewed as elements in say $(0,1]$; for more immediate readability, we associate the colors red and blue to $\infty$ and $0$ respectively. One draws a point at $(0,0)$ corresponding to the smallest parameter in a Cartesian plane. One then proceeds in uniform steps from left to right, drawing a point for each parameter and then moving diagonally upwards after red points and diagonally downwards after blue points. One focuses on one color or the other, counting the number of points on horizontal lines. The numbers obtained form the Hodge vector $h$. The red and blue dots yield the same Hodge vector but contain more information. They may be used to describe the limiting mixed Hodge structure at $t=\infty$ and $0$ respectively.

Complete intertwining of the $\alpha_{j}$ and $\beta_{j}$ gives the extreme where the resulting Hodge vector is just $(n)$. Beukers and Heckman [BH] proved that complete intertwining is exactly the condition one needs for the monodromy group $\langle h_{\infty},h_{0}\rangle$ to be finite. They also established the complete list of such pairs $(\alpha,\beta)$. Actually they, like Schwarz who famously treated the $n=2$ case more than a century earlier, worked without our standing assumption $E={\mathbb{Q}}$. Then one needs to require complete intertwining of all the natural conjugates of $(\alpha,\beta)$ and the list obtained is longer.

In our setting of $E={\mathbb{Q}}$, the corresponding $\gamma$-vectors are of odd lengths $3$ to $9$. There are infinite collections of length $3$ and $5$ given by coprime positive integers $a,b$:

Here and always when discussing classification, we omit consideration of $-\gamma$ whenever $\gamma$ is listed. In case $(i)$, the canonical variety consists of just $a+b$ points. Removing a variable, the BCM presentation takes the form

For $b=1$ this presentation is already trinomial; in general, one has to make a non-trivial change of variables to pass to the trinomial presentation of $X_{a,b,t}$ given by a toric model.

Beyond the closely related collections $(i)$ - $(iii)$, there are only finitely many further $\gamma$, all related to Weyl groups. [BH, Table 8.3] says that, modulo the quadratic twisting operation $Q(T)\mapsto Q(-T)$, there are just one, five, five, and fifteen respectively for the groups $W(F_{4})$, $W(E_{6})$, $W(E_{7})$, and $W(E_{8})$. One of the $W(E_{6})$ cases is discussed in Section 7 and the remaining $W(E_{n})$ cases are similarly treated in [Rob-polys].

Complete separation of the $\alpha_{j}$ and $\beta_{j}$ gives the extreme where the resulting Hodge vector is $(1,1,\dots,1,1)$. The subcase where $q_{0}=(T-1)^{n}$ has the simplifying feature that $h_{0}$ consists of a single Jordan block. Families in this subcase have received particular attention in the literature; the condition is sometimes verbalized as MUM, for maximal unipotent monodromy.

Classification of families in the completely separated case is easier than in the completely intertwined case. It becomes trivial in the MUM subcase because $q_{\infty}$ is arbitrary except for the fact that it contains no factors of $(T-1)$. Accordingly, the number $c_{n}$ of rank $n$ families in the MUM subcase is given by a generating function

Always $c_{2j}=c_{2j+1}$ as under the MUM restriction multiplying by $(T+1)/(T-1)$ gives a bijection on parameters. Arithmetic information about the list underlying $c_{4}=14$ is in [RV]. For general $n$, the case $q_{\infty}=1+T+\cdots+T^{n}$ is the “mirror dual” of the Dwork case discussed after (4.1), and so motives of this family have been given special attention in the physics literature.

A motive defined over a subfield of ${\mathbb{R}}$ has a signature $\sigma$, which is the trace of complex conjugation. For odd weight motives, it is always zero. For even weight HGMs $H(Q,t)$, it depends only on $Q$ and the interval $(-\infty,0)$, $(0,1)$, or $(1,\infty)$ in which $t$ lies. Magma’s command HodgeStructure returns both the Hodge vector and the signature in coded form. To see just the Hodge vector clearly, one can implement Q as in (2.4) and get the Hodge vector from say

HodgeVector(HodgeStructure(Q,2));

Let $X\subset{\mathbb{P}}^{\kappa+1}$ be a smooth hypersurface of degree $\delta$. Let $PH^{\kappa}(X,{\mathbb{Q}})$ be the primitive part of its middle cohomology, meaning the part that does not come from the ambient projective space. If $\kappa$ is odd, then this primitive part is all of $H^{\kappa}(X,{\mathbb{Q}})$. If $\kappa$ is even, then the complementary piece that we are discarding is ${\mathbb{Q}}(-\kappa/2)$.

Hirzebruch gave a formula for the Hodge numbers of $PH^{\kappa}(X,{\mathbb{Q}})$ as a function of $\kappa$ and $\delta$. For example, the sum of the Hodge numbers and first Hodge number are respectively

These special cases and Hodge symmetry are sufficient to give Hodge vectors when $\kappa\leq 3$:

For $\kappa=1$, either part of (6.1) reduces to the genus formula for smooth plane curves, $g=(\delta-1)(\delta-2)/2$.

Let $\delta=e+1\geq 3$ be a desired degree and let $\kappa$ be a desired dimension. Define

The toric procedure illustrated by Table 3.1 yields the completed canonical variety:

The necessary orthogonality relations on each variable’s exponents are illustrated by the case of cubic fourfolds where $\gamma=[1,-2,4,-8,16,-33,22]$. Then (6.4) becomes

For $x_{5}$, the relation is that $m_{5*}=(1,0,0,0,2,1,0)$ is orthogonal to $\gamma$. In general, partial derivatives of (6.4) are very simple since the row vectors $m_{i*}$ have just two nonzero entries, except for the case $i=\kappa+1$ and its three nonzero entries. It is then a pleasant exercise to check via the Jacobian criterion that $X_{t}$ is smooth for $t\in{\mathbb{C}}^{\times}-\{1\}$.

The degree of the rational function $Q$ determined by $\gamma$ can be computed uniformly in $(\delta,\kappa)$ as the cancellations to be analyzed are very structured. This degree agrees with the Betti number $b_{\kappa}$ from (6.1). Thus $H(\gamma,t)$ is the full primitive middle cohomology of $X_{t}$, while a priori it might have been a proper subspace. The zigzag procedure for computing Hodge numbers must agree in the end with the Hirzebruch formula. The reader might want to check the above case of cubic fourfolds, where Hirzebruch’s full formula gives $(0,1,20,1,0)$.

An interesting problem is to find all $\gamma$ which give projective smooth $\kappa$-folds of degree $\delta$. For small parameters, this problem can be solved by direct computation. For example, consider $(\delta,\kappa)=(3,4)$, thus cubic fourfolds. In this case, one has the standardization $[-33,-8,-2,1,4,16,22]$ of the above example, and then exactly ten more: