An effective method for computing Grothendieck point residue mappings

The theory of Grothendieck residue and duality is a cornerstone of algebraic geometry and complex analysis (Griffiths and Harris, 1978; Grothendieck, 1957; Hartshorne, 1966; Kunz, 2009). It has been used and applied in diverse problems of several different fields of mathematics (Baum and Bott, 1972; Bykov et al., 1991; Cardinal and Mourrain, 1996; Dickenstein and Sessa, 1991; Griffiths, 1976; Lehmann, 1991; O’Brian, 1975; Perotti, 1998; Suwa, 2005). In the global situation, methods for computing the total sum of Grothendieck residues have been extensively studied and applied by several authors (Bykov et al., 1991; Cattani et al., 1996; Kytmanov, 1988; Yushakov, 1984).

The concept of Grothendieck local residue together with the local duality theory also play quite important roles in complex analysis, especially in singularity theory (Brasselet et al., 2009; Cherveny, 2018; Corrêa et al., 2016; Klehn, 2002; O’Brian, 1975; Suwa, 1988). Computing  Grothendieck local residues is therefore of fundamental importance. However, since the problem is local in nature, it is difficult in general to compute Grothendieck local residues (O’Brian, 1977). In fact, a direct use of the classical transformation law described in (Hartshorne, 1966) only gives algorithms which lack efficiency. Compared to the global situation, despite the importance, much less work has been done on algorithmic aspects of computing Grothendieck local residues (Elkadi and Mourrain, 2007; Mourrain, 1997; Ohara and Tajima, 2019a, b; Tajima and Nakamura, 2005b). Grothendieck local residues with parameters are useful in the study of singularity theory, for example, deformations of singularity and unfoldings of holomorphic foliations (Kulikov, 1998; Saito, 1983; Varchenko, 1986). However, to the best of our knowledge, existing algorithm of computing Grothendieck local residues are not designed to be able to treat parametric cases.

In this paper, we consider methods for computing Grothendieck point residues from the point of view of complex analysis and singularity theory. We propose a new effective method for computing Grothendieck point residues mappings and residues, which can be extended to treat parametric cases.

Let $X\subset{\mathbb{C}}^{n}$ be an open neighborhood of the origin $O\in{\mathbb{C}}^{n}$ and let $f_{1}(z),f_{2}(z),$ $\ldots,f_{n}(z)$ be $n$ holomorphic functions defined on $X$, where $z=(z_{1},z_{2},\ldots,z_{n})\in X$. Assume that their common locus in $X$ is the origin $O$: $\{z\in X\mid f_{1}(z)=f_{2}(z)=\cdots=f_{n}(z)=0\}=\{O\}$.

Then, for a given germ $h(z)$ of holomorphic function at $O$, the Grothendieck point residue at the origin $O$, denoted by

of the differential form $\displaystyle{\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots f_{n}(z)}}$ can be expressed, or defined, as the integral

where $dz=dz_{1}\wedge dz_{2}\wedge\cdots\wedge dz_{n},$ and where $\gamma_{\epsilon}$ is a real $n$-dimensional cycle:

with $0<\epsilon\ll 1.$ (See for instance, (Baum and Bott, 1972; Griffiths and Harris, 1978; Tong, 1973)).

Let

be the Grothendieck point residue mapping that assigns to a holomorphic function $h(z)$ the value of the Grothendieck point residue. We show that, based on the concept of local cohomology, the use of Grothendieck local duality and a transformation law for local cohomology classes given by J. Lipman (Lipman, 1984) allows us to design an effective method for computing Grothendieck local residue mappings and another one for computing Grothendieck local residues. Note that the classical transformation law on Grothendieck residue is of no avail for computing Grothendieck local residue mappings. Since we compute Grothendieck local residue mappings, our method is applicable when the holomorphic function $h(z)$ in the numerator is computable, that is the case when the coefficients of the Taylor expansion of $h(z)$ is computable. This is an advantage of our approach. We also show that the proposed method can be extended to treat parametric cases. This is another advantage of our approach.

In Section 2, we recall the transformation law for local cohomology classes and Grothendieck local duality. In Section 3, we fix our notation and we briefly recall our basic tool, an algorithm for computing Grothendieck local duality. We devise, in the context of exact computation, a new tool which plays a key role in the resulting algorithm. In Section 4, we describe the resulting algorithm for computing Grothendieck point residue mappings and the algorithm for computing Grothendieck point residues. In Section 5, as an application, we generalize the proposed method to treat parametric cases and we show, by using an example, an algorithm for computing Grothendieck point residues associated to a $\mu$-constant deformation of quasi homogeneous isolated hypersurface singularities.

The concept of Grothendieck point residue was introduced by A. Grothendieck in terms of derived categories and local cohomology. In this section, we briefly recall some basics on transformation law for local cohomology classes and Grothendieck local duality.

Let $X\subset{\mathbb{C}}^{n}$ be an open neighborhood of the origin $O\in{\mathbb{C}}^{n}$. Let ${\mathcal{O}}_{X}$ be the sheaf on $X$ of holomorphic functions, and $\Omega_{X}^{n}$ the sheaf of holomorphic $n$-forms. Let ${\mathcal{H}}_{\{O\}}^{n}({\mathcal{O}}_{X})$ (resp. ${\mathcal{H}}_{\{O\}}^{n}(\Omega_{X}^{n})$ ) denote the local cohomology supported at $O$ of ${\mathcal{O}}_{X}$ (resp. $\Omega_{X}^{n}$).

Then, ${\mathcal{O}}_{X,O}$, the stalk at $O$ of the sheaf ${\mathcal{O}}_{X}$, and the local cohomology ${\mathcal{H}}_{\{O\}}^{n}(\Omega_{X}^{n})$ are mutually dual as locally convex topological vector spaces (Bănică and Stănăşilă, 1974). The duality is given by the point residue pairing:

Let $F=[f_{1}(z),f_{2}(z),\ldots,f_{n}(z)]$ be an $n$-tuple of $n$ holomorphic functions defined on $X$. Assume that their common locus $\{z\in X\mid f_{1}(z)=f_{2}(z)=\cdots=f_{n}(z)=0\}$ in $X$ is the origin $O.$ Let $I_{F}$ denote the ideal in ${\mathcal{O}}_{X,O}$ generated by $f_{1}(z),f_{2}(z),\ldots,f_{n}(z).$ Let $\omega_{F}$ denote a local cohomology class

in ${\mathcal{H}}_{\{O\}}^{n}(\Omega_{X}^{n})$, where $dz=dz_{1}\wedge dz_{2}\wedge\cdots\wedge dz_{n},$ and $\left[\quad\right]$ stands for Grothendieck symbol (Hartshorne, 1966; Grothendieck, 1967). Residue theory says that, for $h(z)$ in ${\mathcal{O}}_{X,O},$ one has

Let us consider a method for computing Grothendieck point residues in the context of symbolic computation. We start by recalling some basics on an algorithm for computing Grothendieck local duality given in (Tajima and Nakamura, 2009; Tajima et al., 2009).

Let $K={\mathbb{Q}}$ be the field of rational numbers and let $z=(z_{1},z_{2},\ldots,z_{n})\in{\mathbb{C}}^{n}.$ Let $H^{n}_{[O]}(K[z])$ denote the algebraic local cohomology defined to be

where ${\mathfrak{m}}$ is the maximal ideal ${\mathfrak{m}}=\langle z_{1},z_{2},\ldots,z_{n}\rangle$ in $K[z]=K[z_{1},z_{2},\ldots,z_{n}].$

We adopt the notation used in (Nabeshima and Tajima, 2015a, b, 2016a, 2016b) to handle local cohomology classes. For instance, a polynomial $\sum_{\lambda}c_{\lambda}\xi^{\lambda}$ in $K[\xi]=K[\xi_{1},\xi_{2},\ldots,\xi_{n}]$ represents the local cohomology class of the form $\displaystyle\sum_{\lambda=(\ell_{1},\ell_{2},\ldots,\ell_{n})}c_{\lambda}\left[\begin{array}[]{c}1\\ z_{1}^{\ell_{1}+1}z_{2}^{\ell_{2}+1}\cdots z_{n}^{\ell_{n}+1}\end{array}\right].$ Note that a multiplication on $\xi^{\beta}$ by $z^{\alpha}$ is

Let $\succ$ be a term ordering on $K[\xi]$. For a local cohomology class $\displaystyle\psi=c_{\alpha}\xi^{\alpha}+\sum_{\xi^{\alpha}\succ\xi^{\gamma}}c_{\gamma}\xi^{\gamma}$, we call $\xi^{\alpha}$ the head monomial of $\psi$, and $\alpha\in{\mathbb{N}}^{n}$ the head exponent of $\psi$.

Let $F=[f_{1}(z),f_{2}(z),\ldots,f_{n}(z)]$ be a list of $n$ polynomials $f_{1},\dots,f_{n}$ in $K[z]$. We also assume as in the previous section that there exists an open neighborhood $X$ of the origin $O$ such that their common locus is the origin: $\{z\in X\mid f_{1}(z)=f_{2}(z)=\cdots=f_{n}(z)=0\}=\{O\}.$

We set

Let $\tau_{F}$ denote the local cohomology class in $H_{F}$ defined to be

Since $\omega_{F}=\tau_{F}dz,$ the local cohomology class $\tau_{F}$ is the kernel function of the point residue mapping.

Let

$q(z)z_{i}^{m_{i}}=p_{i,1}(z)f_{1}(z)+p_{i,2}(z)f_{2}(z)+\cdots+p_{i,n}(z)f_{n}(z),\ i=1,2,\ldots,n,$

and set ${\rm Det}(z)={\rm det}(p_{i,j}(z))_{1\leq i,j\leq n}.$

Let $I_{M}$ be the ideal in $K[z]$ generated by $z_{1}^{m_{1}},z_{2}^{m_{2}},\ldots,z_{n}^{m_{n}}.$ Let $u(z)\in K[z]$ be a polynomial such that $u(z)q(z)-1\in I_{M}$.

Since $A(z)=\det(p_{i,j}(z)/q(z))_{1\leq i,j\leq n}$ is equal to $\frac{1}{q(z)^{n}}{\rm Det}(z),$ the transformation law implies the following

$\displaystyle\tau_{F}=\left[\begin{array}[]{c}u(z)^{n}{\rm Det}(z)\\ z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots z_{n}^{m_{n}}\end{array}\right].$

Let $\lambda_{F}=(\ell_{F,1},\ell_{F,2},\ldots,\ell_{F,n}).$ Since $m_{i}=\ell_{F,i}+1,$ the formula above can be rewritten as $\tau_{F}=u(z)^{n}{\rm Det}(z)\ast\xi^{\lambda_{F}}.$

Note that, according to an algorithm in (Sato and Suzuki, 2009) discovered by Y. Sato and A. Suzuki, the inverse $u(z)$ of $q(z)$ in $K[z]/I_{M}$ can be obtained by using Gröbner basis computation.

The following algorithm computes a representation of the local cohomology class $\tau_{F}$, the kernel function of the point residue mapping.