Computing periods of hypersurfaces

Let $X\subset\mathbbm{P}^{n+1}$ be an $n$-dimensional smooth hypersurface given as the zero locus of an irreducible homogenous polynomial $f_{X}$. The periods of $X$ are the values of certain integrals and so they are formed by two ingredients: the form to be integrated and the domain of integration. Let us introduce each of these in turn.

The forms to be integrated are based on the holomorphic volume form on projective space, which extends the volume form on its affine charts. On an affine space $\mathbbm{C}^{n+1}$ with coordinates $z_{1},\dots,z_{n+1}$ the standard holomorphic volume form is $\mathrm{d}z_{1}\dots\mathrm{d}z_{n+1}$. On the projective space $\mathbbm{P}^{n+1}$ with homogeneous coordinates $x_{0},\dots,x_{n+1}$ one defines the standard volume form to be

where hat denotes the omission of that factor. Restricting to the standard affine charts, we see that the form $\Omega$ is locally the standard volume form (up to sign):

Notice that the integrand $\frac{p(x)}{f_{X}^{\ell}}\Omega$ is homogeneous of degree 0 with poles over $X$. Since the tube $\tau(\gamma)$ lies in the complement of $X$, the integrals are well defined and independent of the exact construction of the tube $\tau(\gamma)$.

Taken in isolation, a single period does not reveal much of the structure of $X$. Instead, we need the integrals over all $n$-cycles. As the homology groups of a smooth hypersurface are freely generated, we now choose $n$-cycles $\gamma_{1},\dots,\gamma_{m}\subset X$ whose classes generate the middle homology $\mathrm{H}_{n}(X,\mathbbm{Z})$ of $X$.

Most of these integrals are redundant, since any two forms that differ by an exact form will give the same period vector. In fact, one can construct a finite set of forms:

such that every other form $\frac{p(x)}{f_{X}^{\ell}}\Omega$ is a linear combination of $\omega_{i}$ modulo exact forms. In symbols, there are $a_{1},\dots,a_{N}\in\mathbbm{C}$ for which there is an equivalence modulo exact forms

The procedure by which one such basis $\{\omega_{i}\}_{i=1}^{N}$ can be obtained, and the coefficients $a_{i}$ determined, is called the Griffiths–Dwork reduction [griffiths--periods, dwork-62, lairez--periods].

Let us point out that the period matrix is of significantly greater value if the intersection matrix $(\gamma_{i}\cdot\gamma_{j})_{i,j=1}^{m}$ is also known. We are now ready to state the precise nature of the problem that we solve.

The existing algorithms which compute the periods of plane curves attack this problem directly by constructing cycles generating the homology on a given curve and taking the relevant integrals numerically [deconinck01, nils-18]. The 1-cycles generating the homology of a plane curve $X\subset\mathbbm{P}^{2}$ are obtained by projecting $X$ to a coordinate axes $\mathbbm{P}^{1}$, which realizes $X$ as a finite cover of $\mathbbm{P}^{1}$, so that standard tools from topological covering theory may be used to compute the fundamental group $\pi_{1}(X)$ and therefore the middle homology $\mathrm{H}_{1}(X,\mathbbm{Z})$ [tretkoff-84].

A similar technique could in principle be attempted in higher dimensions. For instance in [elsenhans-18], a special case of surfaces is considered: K3 surfaces of rank 16 equipped with a double cover of the plane. There, explicit 2-cycles from the plane are lifted to the K3 surface. The intersection product is then determined numerically by repeated sampling.

Replicating this approach in higher dimensions by giving explicit representations of $n$-cycles generating the homology will likely be very difficult. We solve the problem by giving the cycles implicitly, carrying them over from simpler hypersurfaces in theory, and computing the values of the period integrals by solving ordinary differential equations of Picard–Fuchs type.

Let us suppose that we can compute all the periods of another smooth hypersurface $Y=Z(f_{Y})\subset\mathbbm{P}^{n+1}$ of degree $d$ with respect to a basis $\gamma_{1}^{Y},\dots,\gamma_{m}^{Y}$ for the middle homology $\mathrm{H}_{n}(Y,\mathbbm{Z})$ of $Y$. We will compute the periods of $X=Z(f_{X})$ from those of $Y$ by what one might call a period homotopy. That is, we will compute the periods of $X$ from those of $Y$ by deforming $Y$ into $X$ and tracing the value of the periods by ordinary differential equations. We will outline this construction in the remainder of the introduction and delegate some of the details to Section 3.2.

Construct a family of hypersurfaces $\mathcal{X}_{t}=Z(f_{t})\subset\mathbbm{P}^{n+1}$ varying algebraically with respect to a single complex parameter $t\in\mathbbm{C}$ such that $\mathcal{X}_{0}=Y$ and $\mathcal{X}_{1}=X$. For instance, one could take $f_{t}=(1-t)f_{Y}+tf_{X}$ to be the defining equation of $\mathcal{X}_{t}$. Except for finitely many values of $t$ in $\mathbbm{C}$, the hypersurface $\mathcal{X}_{t}$ is smooth.

Using Ehresmann’s fibration theorem we may identify the topological spaces underlying the hypersurfaces $\mathcal{X}_{t}$ along a path in the complex plane. We will choose this path so that it starts from $t=0$ and ends at $t=1$ while avoiding singular values of $t$. In particular, such an identification gives an isomorphism between the homology groups $\mathrm{H}_{n}(Y,\mathbbm{Z})$ and $\mathrm{H}_{n}(\mathcal{X}_{t},\mathbbm{Z})$ for each $t$ along this path. This isomorphism is locally canonical, not depending on the choice of the path or of the identification of the topological spaces.

Let $p(x)\in\mathbbm{C}[x_{0},\dots,x_{n+1}]$ be of degree $d\ell-n-2$ for $\ell\geq 1$. We will now describe how to compute the period vector:

Define the following rational form varying with $t$ and having poles along $\mathcal{X}_{t}$:

We can now define the following period function:

We will compute the value of interest, $\sigma(1)$, by finding an ODE satisfied by $\sigma(t)$ and then determining the initial conditions $\sigma(0),\sigma^{(1)}(0),\sigma^{(2)}(0),\dots$. From these, $\sigma(1)$ is found by numerical integration.

Classically, a differential equation satisfied by a period function is called a Picard–Fuchs equation. In more recent incarnations, it is also referred to as a Gauss–Manin connection. The Picard–Fuchs equations are central to geometry as well as mathematical physics [cox-katz--mirror]. For a perspective from the side of arithmetic geometry, let us refer the interested reader to the landmark paper [katz-70]. From a computational viewpoint, finding differential equations satisfied by integrals of functions is a heavily studied subject [chyzak-00, koutschan-10]. In particular, Lairez [lairez--periods] has dedicated software which can compute the Picard–Fuchs equations of rational integrals.

We will now sketch how the Picard–Fuchs equation for $\sigma(t)$ is found. Denote the differentiation by $t$ operator with $\partial_{t}$. The standard observation here is that we can differentiate under the integral sign:

Indeed, if $t^{\prime}$ is very close to $t$ then $\gamma_{i}(t)$ and $\gamma_{i}(t^{\prime})$ can be represented by cycles that are very close to one another. Then the tube $\tau(\gamma_{i}(t))$ encloses $\gamma_{i}(t^{\prime})$ and is homotopic to the tube $\tau(\gamma_{i}(t^{\prime}))$. Therefore, the domain of integration can be fixed while differentiating $\sigma(t)$.

In particular, suppose $\mathcal{D}\in\mathbbm{C}(t)[\partial_{t}]$ is a differential operator such that

where $\omega(t)$ is defined as in Equation (1.3.3) and equivalence is taken modulo exact forms. Then, the operator $\mathcal{D}$ annihilates the period function $\sigma(t)$.

The construction of an operator $\mathcal{D}$ satisfying (1.4.5) is equivalent to finding a $\mathbbm{C}(t)$-linear relation between the Griffiths–Dwork reductions of the derivatives $\omega^{(k)}(t)$, where the reduction is performed over the field $\mathbbm{C}(t)$. Since these reduced forms must lie in a finite dimensional vector space, after finitely many differentiations and reductions we will find a $\mathbbm{C}(t)$-linearly dependent set. The coefficients of the linear relation give $\mathcal{D}$. We present additional details on the construction of the Picard–Fuchs equations in Section 2.5.

It remains to find one smooth hypersurface $Y$ for each $n,d\geq 1$ for which we can compute the periods. Fixing $n$ and $d$, let us denote by $Y$ the Fermat hypersurface cut out by the equation $f_{Y}:=x_{0}^{d}+\dots+x_{n}^{d}-x_{n+1}^{d}$.

The Fermat hypersurface has symmetries which we will exploit. In particular, the following automorphisms will be needed.

When $n$ is odd, the set of translates $\{t^{\beta}S\mid\beta\in\mathbbm{Z}^{n+2}\}$ of $S$ generates the homology $\mathrm{H}_{n}(Y,\mathbbm{Z})$. In fact, we can find a finite subset $B\subset\mathbbm{Z}^{n+2}$ such that the set $\{t^{\beta}S\mid\beta\in B\}$ freely generates the homology. The determination of this subset $B$ relies on Corollary 4.8.

When $n$ is even, the Pham cycle and its translates do not generate the homology and we need one other cycle. Let $L$ be the image of the following linear map:

where $\eta:=e^{\frac{\pi\sqrt{-1}}{d}}$ is a $d$-th root of $-1$. The translates of $S$ together with $L$ generate the middle homology of $Y$, see Lemma 2.4. Once again, using Corollary 4.8, a finite subset $B\subset\mathbbm{Z}^{n+2}$ can be found algorithmically so that $\{t^{\beta}S\mid\beta\in B\}\cup\{L\}$ freely generates $\mathrm{H}_{n}(Y,\mathbbm{Z})$.

However, for period computations, this extra class $L$ is redundant. The periods over $t^{\beta}S$ will determine the periods over $L$ by simple linear algebra as we explain in Section 4.5. In light of these statements, the following theorem allows for the computation of all periods on Fermat hypersurfaces.

Now that we have explicit formulas for the periods of the degree $d$ Fermat hypersurface $Y\subset\mathbbm{P}^{n+1}$, the periods of any other degree $d$ hypersurface $X\subset\mathbbm{P}^{n+1}$ can be expressed as initial value problems using the strategy outlined in Sections 1.3 and 1.4. It remains to solve these initial value problems by numerical integration.

The ODEs we obtain from the procedure outlined in Section 1.4 are typically huge. They require dedicated software to be integrated efficiently. We use Marc Mezzarobba’s analytic extension of the ore_algebra package in SageMath which has the added benefit of providing rigorous error bounds for the result [mezzarobba-oa]. As far as we are aware, no other software can handle ODEs that would take many pages to write down.

Currently, the determination of the ODEs and of the initial conditions is fully automated via our implementation in Magma [magma]. A simple script allows for the output to be integrated by Mezzarobba’s solver.

Bernd Sturmfels suggested this problem and provided guidance, support and motivation. Pierre Lairez generously made many expository and technical contributions to this project, in particular, he introduced me to LLL and to Mezzarobba’s work. I’ve also benefited from conversations with numerous other mathematicians. In particular, I would like to thank: Alex Degtyarev for introducing me to Pham cycles and answering related questions. Marc Mezzarobba for helpful comments on how to use his code. Mateusz Michałek for helping with Gröbner bases and combinatorics. Kristin Shaw for finding the paper which got this project off the ground. Lynn Chua for carefully reading through the first draft. In addition, I would like to thank Javier Fresan, Hossein Movasati and Jan Tuitman for valuable comments on the first version of this paper. Finally, heartfelt thanks to the referee for a careful reading of the manuscript and numerous suggestions for improvement.

Understanding the homology and cohomology of $X$ is greatly simplified by passing to its complement $\mathbbm{P}^{n+1}\setminus X$. However, by this passage we lose a small amount of information and describe the primitive (co)homology instead.

Recall that we have a non-degenerate intersection pairing:

which is alternating if $n$ is odd and symmetric if $n$ is even. On cohomology we have the dual cup product.

When $n$ is odd, primitive (co)homology coincides with (co)homology. When $n$ is even, $h\cdot h=d$ and thus $h\notin\mathrm{PH}_{n}(X,\mathbbm{Z})$. However, we need to add more than just $h$ into the primitive homology in order to recover homology.

Let $U:=\mathbbm{P}^{n+1}\setminus X$ be the complement of $X$. Recall that for any $n$-cycle $\gamma\subset X$ we defined by $\tau(\gamma)\subset U$ the $(n+1)$-cycle obtained by forming a thin tube, i.e., $S^{1}$-bundle, over $\gamma$. This induces the following map between homology groups:

In cohomology, the corresponding map is:

Let $V$ be the space of all holomorphic top forms on $U=\mathbbm{P}^{n+1}\setminus X$, i.e., $V=\mathrm{H}^{0}(U,\Omega^{n+1}_{U/\mathbbm{C}})$. Let us denote by $V_{\ell}\subset V$ the subspace of forms which extend to $\mathbbm{P}^{n+1}$ with a pole order at most $\ell$ on $X$. In symbols, $V_{\ell}=\mathrm{H}^{0}(\mathbbm{P}^{n+1},\mathcal{O}_{\mathbbm{P}^{n+1}}(\ell[X]))$. Then we have $V=\bigoplus_{\ell\geq 1}V_{\ell}$.

The cohomology of $X$ admits the Hodge decomposition $\mathrm{H}^{n}(X,\mathbbm{C})=\bigoplus_{k=0}^{n}\mathrm{H}^{n-k,k}(X,\mathbbm{C})$. The corresponding Hodge filtration is denoted by:

This filtration on cohomology induces a filtration on the primitive cohomology by restriction:

There is a finite procedure to maximally reduce an element in $V$ modulo the kernel of the residue map. This reduction procedure is based on the following two theorems. The first theorem states that we do not need infinitely many pole orders to describe the primitive cohomology of $X$.

The second theorem makes the kernel of the residue map more explicit and is a strengthening of Proposition 2.8 above. Let $W_{\ell}=\mathrm{H}^{0}(\mathbbm{P}^{n+1},\Omega^{n}_{\mathbbm{P}^{n}/\mathbbm{C}}(\ell[X]))$ be the space of $n$-forms on $\mathbbm{P}^{n+1}$ with pole order at most $\ell$ on $X$. Once again, these can be viewed as holomorphic $n$-forms on $U$. We have the algebraic derivation map $\mathrm{d}:W\to V:\eta\mapsto\mathrm{d}\eta$ which allows us to write:

Given a form $\omega\in V$ we will describe the Griffiths–Dwork reduction on $\omega$ which modifies $\omega$ modulo the kernel of the residue map and puts it into a reduced form. The primary motivation for this reduction is Lemma 2.18.

Recall the nature of the elements in $V$ from Remark 2.7. We will now describe the elements in $W$, the holomorphic $n$-forms on $\mathbbm{P}^{n+1}\setminus X$. Let us use $\mathrm{d}x^{i,j}$ to refer to the $n$-form $\mathrm{d}x_{0}\dots\widehat{\mathrm{d}x_{i}}\dots\widehat{\mathrm{d}x_{j}}\dots\mathrm{d}x_{n+1}$, where hat denotes omission. A form in $\varphi\in W_{\ell}$ can be written as:

where $s_{i}\in\mathbbm{C}[x_{0},\dots,x_{n+1}]$ is homogeneous of degree $(\ell d-n-1)$ (see [griffiths--periods] for a derivation of this fact).

Let us say that a form $\omega\in V$ has pole order $\ell$ if $\omega\in V_{\ell}$ but $\omega\notin V_{\ell-1}$. Pole order induces a natural grading on $V$. If $\omega\in V$ we will write $\omega=\sum_{i=1}^{r}\omega_{j}$ where $\omega_{j}$ is zero or has pole order $j$, and the highest term $\omega_{r}$ is non-zero. Let us refer to $r$ as the pole order of $\omega$. Similarly, we can grade $W$ by pole order.

Given $\omega=\sum_{i=1}^{r}\omega_{i}\in V$, for each $i=1,\dots,r$ write $\omega_{i}=p_{i}(x)\frac{\Omega}{f_{X}^{i}}$ where $p_{i}$ is a polynomial of degree $id-n-2$ not divisible by $f_{X}$. Let $q_{r}$ be the remainder of $p_{r}$ modulo $J(f_{X})$ and find polynomials $s_{0},\dots,s_{n+1}$ satisfying:

Define the polynomial $q_{r-1}=\frac{1}{r-1}\sum_{j=0}^{n+1}\frac{\partial s_{j}}{\partial x_{j}}$. Then by Equation 2.4.10 we may conclude that $\omega$ is equivalent modulo exact forms to

The choice of the “coordinates” $s_{i}$ for the difference $p_{r}-q_{r}$ is not canonical and therefore the partially reduced form $\tilde{\omega}$ is not canonical. However, an application of Theorem 2.14 implies that any other choice of coordinates would yield a reduction that is equivalent to $\tilde{\omega}$ modulo exact forms of pole order $r-1$. This difference will vanish in the next step of the reduction.

Writing $\tilde{\omega}=\sum_{i=1}^{r}\tilde{\omega}_{i}$, and $\tilde{\omega}_{i}=\frac{\tilde{p}_{i}\Omega}{f_{X}^{i}}$, we can apply the same reduction method to the form $\tilde{\omega}_{r-1}$. After repeated application, all terms in the pole decomposition have been put into a normal form with regards to $J(f_{X})$.

Let $R$ be the polynomial ring $\mathbbm{C}[x_{0},\dots,x_{n+1}]$. We will denote by $R_{s}\subset R$ the subspace of homogeneous degree $s$ polynomials in $R$. For a homogeneous ideal $I\subset R$, we denote by $I_{s}$ the intersection $I\cap R_{s}$. Define the following vector space:

Observe that $S\subset R/J(f_{X})$ is finite dimensional since $Z(f_{X})$ is smooth.

Let $p_{1},\dots,p_{N}\in\mathbbm{C}[x_{0},\dots,x_{n+1}]$ be homogeneous polynomials which descend to a basis on $S$. For each $i$, let $\ell_{i}$ be the positive integer satisfying $\deg p_{i}=d\ell_{i}-n-2$. Define the forms

Fix an integer $n>0$ and let $f\in\mathbbm{C}[s][x_{0},\dots,x_{n+1}]$ be a homogeneous polynomial of degree $d$ with coefficients which are polynomial in the variable $s$, further assume that the coefficients of $f$ are relatively prime. For $t\in\mathbbm{C}$ we will write $f_{t}:=f|_{s=t}$, $\mathcal{X}_{t}:=Z(f_{t})\subset\mathbbm{P}^{n+1}$ and $U_{t}:=\mathbbm{P}^{n+1}\setminus\mathcal{X}_{t}$. Let us assume that $\mathcal{X}_{t}$ is smooth for generic $t$.

Fix a non-zero polynomial $p\in\mathbbm{C}[x_{0},\dots,x_{n+1}]$ of degree $d\ell-n-2$ where $\ell\geq 1$ is an integer. Define the following holomorphic form on $U_{t}$:

We will now describe how to find the Picard–Fuchs equation corresponding to $\omega(t)$.

Recall that we denote by $\partial_{t}$ the differentiation by $t$ operator. Let $\omega^{(k)}(t):=\partial^{k}_{t}\omega(t)$ be the $k$-th derivative of $\omega(t)$ with respect to $t$. For any $k=0,1,\dots$ the Griffiths–Dwork reduction $[\omega^{(k)}(t)]_{\operatorname{GD}}$ of the $k$-th derivative has pole order bounded by $n+1$ due to Theorem 2.13. In particular, the reductions of these infinitely many derivatives live in a finite dimensional $\mathbbm{C}(t)$-vector space.

Let $\delta\geq 1$ be the smallest integer for which there is a linear linear relation:

where $a_{i}(t)\in\mathbbm{C}(t)$ are rational functions in $t$.