Holomorphic cohomological convolution and Hadamard product

TL;DR

This paper explores the relationship between a generalized Hadamard product and holomorphic cohomological convolution on complex numbers, introducing new concepts to extend the product to broader function classes.

Contribution

It introduces a generalized Hadamard product applicable to functions not vanishing at infinity and defines strongly convolvable sets, linking these to holomorphic cohomological convolution.

Findings

Established a link between Pohlen's extended Hadamard product and cohomological convolution.

Defined a generalized Hadamard product for broader function classes.

Introduced the concept of strongly convolvable sets.

Abstract

In this article, we explain the link between Pohlen's extended Hadamard product and the holomorphic cohomological convolution on $\mathbb{C}^*$. For this purpose, we introduce a generalized Hadamard product, which is defined even if the holomorphic functions do not vanish at infinity, as well as a notion of strongly convolvable sets.

Equations

$$(A\star B)(z)=\sum _{n=0}^{+\infty }{a}_n{b}_n{z}^n.$$

$$(f_1\star f_2)(z)=\frac{1}{2i\pi }{\int }_{C(0,r{)}^{+}}{f}_1(\zeta )f_2\left(\frac{z}{\zeta }\right)\frac{d\zeta }{\zeta }$$

$$H(\Omega )=\{f\in O(\Omega ):f(\infty )=0\}$$

$$\infty \cdot a=a\cdot \infty =\infty$$

$$A\cdot B=\{a\cdot b:a\in A,b\in B\}.$$

$$S^{-1}=\{z:z^{-1}\in S\}.$$

$${\Omega }_1\star {\Omega }_2=P\backslash ((P\backslash {\Omega }_1)(P\backslash {\Omega }_2)).$$

$$\mathrm{Ind}(c,z)=\{\begin{array}{ll}1& \text{if}z\in K\\ 0& \text{if}z\in P\backslash \Omega \end{array},$$

$$\mathrm{Ind}(c,z)=\{\begin{array}{ll}0& \text{if}z\in K\\ -1& \text{if}z\in P\backslash \Omega \end{array},$$

$$(f_1\star f_2)(z)=\frac{1}{2i\pi }{\int }_{c_z}{f}_1(\zeta )f_2\left(\frac{z}{\zeta }\right)\frac{d\zeta }{\zeta },$$

$${}^{\mathrm{BM}}{H}_k(X):=H^{-k}(X,{\omega }_X),{}^{\mathrm{BM}}{H}_k^c(X):=H_c^{-k}(X,{\omega }_X).$$

$${\alpha }_X\in {}^{\mathrm{BM}}{H}_n(X)\simeq H^{-n}(X,Z_X[n])\simeq H^0(X,Z_X)$$

$$R{\Gamma }_{X\backslash K}(X,{\omega }_X)\to R\Gamma (X,{\omega }_X)\to R\Gamma (K,{\omega }_X)\stackrel{+}{\to }$$

$$R{\Gamma }_c(X\backslash K,{\omega }_X)\to R{\Gamma }_c(X,{\omega }_X)\to R\Gamma (K,{\omega }_X)\stackrel{+}{\to }.$$

$${}^{\mathrm{BM}}{H}_n(X)\to H^{-n}(K,{\omega }_X)\stackrel{\sim }{\to }{H}_n(X,X\backslash K)$$

$$H_1(\Omega )\stackrel{\sim }{\to }{H}_c^0(F,Z_F)$$

$$[c]\mapsto \left(z\mapsto {\mathrm{Ind}}_z(c)\right).$$

$$R{\Gamma }_c(\Omega ,{\omega }_C)\to R{\Gamma }_c(C,{\omega }_C)\to R{\Gamma }_c(F,{\omega }_C)\stackrel{+1}{\to }.$$

$$\delta :H_c^0(F,Z_F)\stackrel{\sim }{\to }{H}_1(\Omega ).$$

$${\delta }_z:Z\simeq H_c^0(\{z\},Z_{\{z\}})\stackrel{\sim }{\to }{H}_1(C\backslash \{z\}).$$

$$H_c^0(F,Z_F)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(P\backslash (z{S}_2^{-1}\cup (\{0,\infty \}\backslash S_1)))\backslash S_1=P\backslash (S_1\cup z{S}_2^{-1}\cup \{0\}\cup \{\infty \}).$$

$$-{\alpha }_{P\backslash (z{S}_2^{-1}\cup (\{0,\infty \}\backslash S_1)),S_1}\in H_2(P\backslash (z{S}_2^{-1}\cup (\{0,\infty \}\backslash S_1)),P\backslash (S_1\cup z{S}_2^{-1}\cup \{0\}\cup \{\infty \}))$$

$$H_2(P\backslash (z{S}_2^{-1}\cup (\{0,\infty \}\backslash S_1)),P\backslash (S_1\cup z{S}_2^{-1}\cup \{0\}\cup \{\infty \}))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$O(P\backslash S_1)\times O(P\backslash S_2)\to O(C^{\ast }\backslash S_1{S}_2)$$

$$(f_1\star f_2)(z)=\frac{1}{2i\pi }{\int }_{c_z}{f}_1(\zeta )f_2\left(\frac{z}{\zeta }\right)\frac{d\zeta }{\zeta },$$

$$(f_1\star f_2)(z)=\frac{1}{2i\pi }{\int }_{c_K}{f}_1(\zeta )f_2\left(\frac{z}{\zeta }\right)\frac{d\zeta }{\zeta },$$

$${\alpha }_{P\backslash (K{S}_2^{-1}\cup (\{0,\infty \}\backslash S_1)),S_1}\in H_2(P\backslash (K{S}_2^{-1}\cup (\{0,\infty \}\backslash S_1)),P\backslash (S_1\cup K{S}_2^{-1}\cup \{0\}\cup \{\infty \}))$$

$$H_2(P\backslash (K{S}_2^{-1}\cup (\{0,\infty \}\backslash S_1)),P\backslash (S_1\cup K{S}_2^{-1}\cup \{0\}\cup \{\infty \}))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_2(P\backslash (K{S}_2^{-1}\cup (\{0,\infty \}\backslash S_1)),P\backslash (S_1\cup K{S}_2^{-1}\cup \{0\}\cup \{\infty \}))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

The first author is supported by a FNRS grant (ASP 111496F)

1. 1 The extended Hadamard product
2. 2 Generalized Hadamard cycles
3. 3 The holomorphic integration map
4. 4 Holomorphic cohomological convolution
5. 5 Multiplicative convolution on $\operatorname{\mathbb{C}}^{*}$
6. 6 The case of strongly convolvable sets

1. The extended Hadamard product

Classically, the Hadamard product of two formal power series $A(z)=\sum_{n=0}^{+\infty}a_{n}z^{n}$ and $B(z)=\sum_{n=0}^{+\infty}b_{n}z^{n}$ is defined by setting

[TABLE]

Using Taylor expansions, one can thus define the Hadamard product $f_{1}\star f_{2}$ of two germs $f_{1}$ and $f_{2}$ of holomorphic functions at the origin. Exploiting the Cauchy’s integral representation, one obtains the formula

[TABLE]

for all $z$ in a neighbordhood of [math], $C(0,r)^{+}$ being a small positively oriented circle centered at the origin (see e.g. [1] and [9] for some applications).

In his thesis [12] (see also [11]), Timo Pohlen introduced the more general notion of Hadamard product for holomorphic functions defined on open subsets of the Riemann sphere $\operatorname{\mathbb{P}}=\operatorname{\mathbb{C}}\cup\{\infty\}$ which do not necessarily contain the origin. This new definition led to interesting applications, (e.g. [8] and [10]). In this introduction, we shall recall the construction and the results of T. Pohlen.

Definition 1.1**.**

Let $\operatorname{\mathbb{P}}$ be the Riemann sphere equipped with its canonical structure of complex manifold. Let $\Omega$ be an open subset of $\operatorname{\mathbb{P}}$. One sets

[TABLE]

if $\infty\in\Omega$ and $\operatorname{\mathcal{H}}(\Omega)=\operatorname{\mathcal{O}}(\Omega)$ otherwise.

Definition 1.2**.**

We set $M=(\operatorname{\mathbb{P}}\times\operatorname{\mathbb{P}})\backslash\{(0,\infty),(\infty,0)\}$ and extend the complex multiplication continuously as a map $\cdot:M\to\operatorname{\mathbb{P}}.$ We then have

[TABLE]

if $a\in\operatorname{\mathbb{P}}$ is not equal to zero. If $A,B$ are subsets of $\operatorname{\mathbb{P}}$ such that $A\times B\subset M$, one sets

[TABLE]

One also extends the inversion $z\mapsto z^{-1}$ continuously from $\operatorname{\mathbb{C}}^{*}$ to $\operatorname{\mathbb{P}}$ by setting $0^{-1}=\infty$ and $\infty^{-1}=0.$ If $S\subset\operatorname{\mathbb{P}},$ one sets

[TABLE]

For the rest of the article, we shall often drop the point and write the multiplication as a concatenation.

Definition 1.3**.**

Two open subsets $\Omega_{1},\Omega_{2}\subset\operatorname{\mathbb{P}}$ are called star-eligible if

1. (1)

$\Omega_{1}$ and $\Omega_{2}$ are proper subsets of $\operatorname{\mathbb{P}},$ 2. (2)

$(\operatorname{\mathbb{P}}\backslash\Omega_{1})\times(\operatorname{\mathbb{P}}\backslash\Omega_{2})\subset M,$ 3. (3)

$(\operatorname{\mathbb{P}}\backslash\Omega_{1})(\operatorname{\mathbb{P}}\backslash\Omega_{2})\neq\operatorname{\mathbb{P}}.$

In this case, the star product of $\Omega_{1}$ and $\Omega_{2}$, noted $\Omega_{1}\star\Omega_{2},$ is defined by

[TABLE]

For the several equivalent definitions of the index/winding number of a cycle $c$ in $\operatorname{\mathbb{C}}$, we refer to [13]. For any cycle $c$ in $\operatorname{\mathbb{C}}$, one sets $\operatorname{Ind}(c,\infty)=0.$

Definition 1.4**.**

Let $\Omega$ be a non-empty open subset of $\operatorname{\mathbb{P}}$, $K$ be a non-empty compact subset of $\Omega$ and $c$ be a cycle in $\Omega\backslash(K\cup\{0\}\cup\{\infty\}).$ If $\infty\notin K$ and

[TABLE]

then $c$ is called a Cauchy cycle for $K$ in $\Omega.$ If $\infty\in\Omega$ and

[TABLE]

then $c$ is called a anti-Cauchy cycle for $K$ in $\Omega.$

In [12], Lemma $2.3.1$, T. Pohlen refers to ad hoc explicit constructions which ensure that Cauchy and anti-Cauchy cycles always exist for any $\Omega$ and any $K$. In the next section, we shall see that this existence can easily be obtained by using singular homology.

Let $\Omega_{1}$ and $\Omega_{2}$ be two star-eligible open subsets of $\operatorname{\mathbb{P}}$. Note that, if $z\in\Omega_{1}\star\Omega_{2}$, then $z(\operatorname{\mathbb{P}}\backslash\Omega_{2})^{-1}$ is a closed subset of $\Omega_{1}.$

Definition 1.5**.**

Let $z\in(\Omega_{1}\star\Omega_{2})\backslash\{0,\infty\}.$ A Hadamard cycle for $z(\operatorname{\mathbb{P}}\backslash\Omega_{2})^{-1}$ in $\Omega_{1}$ is a cycle $c$ in $\Omega_{1}\backslash(z(\operatorname{\mathbb{P}}\backslash\Omega_{2})^{-1}\cup\{0\}\cup\{\infty\})$ which satisfies the condition given in the following table :

[TABLE]

This table should be understood in the following way : The elements in the first row and the first column tell which of these elements are in $\Omega_{1}$ and $\Omega_{2}$ respectively. The abreviation cc (resp. acc) means that $c$ is a Cauchy (resp. anti-Cauchy) cycle for $z(\operatorname{\mathbb{P}}\backslash\Omega_{2})^{-1}$ in $\Omega_{1}$. The abreviation $\text{cc}^{+}$ (resp. $\text{acc}^{-}$) means that $c$ is a Cauchy (resp. anti-Cauchy) cycle with the extra condition $\operatorname{Ind}(c,0)=1$ (resp. $\operatorname{Ind}(c,0)=-1$). A "/" means that this case cannot occur.

One can now extend the standard Hadamard product.

Definition 1.6**.**

Let $f_{1}\in\operatorname{\mathcal{H}}(\Omega_{1})$ and $f_{2}\in\operatorname{\mathcal{H}}(\Omega_{2}).$ For each $z\in(\Omega_{1}\star\Omega_{2})\backslash\{0,\infty\}$ one sets

[TABLE]

where $c_{z}$ is a Hadamard cycle for $z(\operatorname{\mathbb{P}}\backslash\Omega_{2})^{-1}$ in $\Omega_{1}.$ One can check that this integral does not depend on the chosen Hadamard cycle (see Lemma 3.4.2 in [12]). The function $f_{1}\star f_{2}$ is called the Hadamard product of $f_{1}$ and $f_{2}$.

[math]\bullet$$\operatorname{\mathbb{P}}\backslash\Omega_{1}$$>$$>$$z(\operatorname{\mathbb{P}}\backslash\Omega_{2})^{-1}

Proposition 1.7** ([12], Lemma $3.4.5$ and Proposition $3.6.4$).**

The Hadamard product $f_{1}\star f_{2}$ can be continuously extended to $\Omega_{1}\star\Omega_{2}.$ If $0\in\Omega_{1}\star\Omega_{2}$ (resp. $\infty\in\Omega_{1}\star\Omega_{2}$), one has $(f_{1}\star f_{2})(0)=f_{1}(0)f_{2}(0)$ (resp. $(f_{1}\star f_{2})(\infty)=0$). Moreover, $f_{1}\star f_{2}$ is an element of $\operatorname{\mathcal{H}}(\Omega_{1}\star\Omega_{2}).$

Proposition 1.8** ([12], Proposition $3.6.1$).**

The Hadamard product is commutative.

In all this framework, the hypothesis $f(\infty)=0$, when $\infty\in\Omega$, is highly used. In the next section, we shall provide a more general definition of Hadamard cycles and Hadamard product, based on singular homology theory, which does not require the vanishing condition at infinity.

2. Generalized Hadamard cycles

For classical facts about singular homology, we refer to [5] and [6]. For a general background on sheaf theory and derived functors, we refer to [7]. For a sheaf-theoretic definition of the Borel-Moore homology and the link with singular homology on HLC-spaces, we refer to [3].

Let us recall that on any topological space $X$, there is an orientation complex $\omega_{X}$ which is canonically isomorphic to $\operatorname{\mathbb{Z}}_{X}[n]$ if $X$ is an oriented topological manifold of pure dimension $n$. On a topological space $X$, the Borel-Moore homology (resp. Borel-Moore homology with compact support) of degree $k$ is defined by

[TABLE]

Definition 2.1**.**

Let $X$ be an oriented topological manifold of pure dimension $n$. The orientation class of $X$ is the class

[TABLE]

corresponding to the constant section $1$ of $\operatorname{\mathbb{Z}}_{X}$.

Let $X$ be a topological manifold $X$ of pure dimension $n$. Since $X$ is homologically locally connected, the complex ${\operatorname{R}}\Gamma_{c}(X,\omega_{X})$ is canonically isomorphic to the complex of singular chains on $X$. Hence, ${}^{\operatorname{BM}}\!H_{k}^{c}(X)$ is isomorphic to the usual singular homology group of degree $k$, $H_{k}(X)$. Now, let $K$ be a compact subset of $X$ and consider the two canonical excision distinguished triangles

[TABLE]

and

[TABLE]

The second triangle implies that $H^{-n}(K,\omega_{X})$ is canonically isomorphic to the relative singular homology group $H_{n}(X,X\backslash K)$. Hence, we get a sequence of morphisms

[TABLE]

and $\alpha_{X}\in{}^{\operatorname{BM}}\!H_{n}(X)$ induces a relative orientation class $\alpha_{X,K}\in H_{n}(X,X\backslash K).$

Proposition 2.2**.**

Let $\Omega$ be a proper open subset of $\operatorname{\mathbb{C}}$ and let $F=\operatorname{\mathbb{C}}\backslash\Omega.$ There is a canonical isomorphism

[TABLE]

given by

[TABLE]

Proof.

Let us consider the excision distinguished triangle

[TABLE]

It induces a long exact sequence

{\cdots}$${H_{2}(\Omega)}$${H_{2}(\operatorname{\mathbb{C}})}$${H^{-2}{\operatorname{R}}\Gamma_{c}(F,\omega_{F})}$${H_{1}(\Omega)}$${H_{1}(\operatorname{\mathbb{C}})}$${H^{-1}{\operatorname{R}}\Gamma_{c}(F,\omega_{F})}$${\cdots}

Since $\operatorname{\mathbb{C}}$ is contractible, one has $H_{2}(\operatorname{\mathbb{C}})\simeq H_{1}(\operatorname{\mathbb{C}})\simeq\{0\}$. Therefore, taking into account that $\omega_{F}\simeq\operatorname{\mathbb{Z}}_{F}[2],$ one gets a canonical isomorphism

[TABLE]

Let $z\in F.$ Applying (2.1) with $\operatorname{\mathbb{C}}\backslash\{z\},\operatorname{\mathbb{C}}$ and $\{z\},$ one gets an isomorphism

[TABLE]

Clearly, $\delta_{z}^{-1}([c])=\operatorname{Ind}_{z}(c)$. Moreover, by Proposition $1.3.6$ in [7], there is a commutative diagram

[TABLE]

where $i_{z}(f)=f(z)$ and $j_{z}([c])=[c].$ Hence, one sees that $\delta^{-1}([c])(z)=\operatorname{Ind}_{z}(c).$ Since this argument is valid for all $z\in F$, the conclusion follows. ∎

To introduce our definition of generalized Hadamard cycles, we have to be in the same setting as T. Pohlen. However, looking at Definition 1.3, we find it more natural to start with closed subsets instead of open ones.

Definition 2.3**.**

Two closed subsets $S_{1}$ and $S_{2}$ of $\operatorname{\mathbb{P}}$ are star-eligible if $S_{1},S_{2}$ and $S_{1}S_{2}$ are proper and if $S_{1}\times S_{2}\subset M.$

For the rest of the section we fix $S_{1}$ and $S_{2}$, two star-eligible closed subsets of $\operatorname{\mathbb{P}}$. If $z\in\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}$, $S_{1}$ is a compact subset of $\operatorname{\mathbb{P}}\backslash zS_{2}^{-1}$ and, thus, a compact subset of $\operatorname{\mathbb{P}}\backslash(zS_{2}^{-1}\cup(\{0,\infty\}\backslash S_{1}))$. Moreover, one has

[TABLE]

Let $z\in\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}.$

Definition 2.4**.**

A generalized Hadamard cycle for $S_{1}$ in $\operatorname{\mathbb{P}}\backslash(zS_{2}^{-1}\cup(\{0,\infty\}\backslash S_{1}))$ is a representative $c$ of the class in $H_{1}(\operatorname{\mathbb{P}}\backslash(S_{1}\cup zS_{2}^{-1}\cup\{0\}\cup\{\infty\}))$ which is the image of

[TABLE]

by the canonical map

[TABLE]

Our aim is now to define a product

[TABLE]

which generalizes the extended Hadamard product of T. Pohlen.

Definition 2.5**.**

Let $f_{1}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\backslash S_{1})$ and $f_{2}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\backslash S_{2}).$ For each $z\in\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}$ we set

[TABLE]

where $c_{z}$ is a generalized Hadamard cycle for $S_{1}$ in $\operatorname{\mathbb{P}}\backslash(zS_{2}^{-1}\cup(\{0,\infty\}\backslash S_{1}))$. Since two generalized Hadamard cycles are homologous, the definition does not depend on the chosen generalized Hadamard cycle. The function $f_{1}\star f_{2}$ is called the generalized Hadamard product of $f_{1}$ and $f_{2}.$

[math]\bullet$$S_{1}$$>$$>$$zS_{2}^{-1}

Lemma 2.6**.**

Let $f_{1}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\backslash S_{1})$ and $f_{2}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\backslash S_{2}).$ For each compact subset $K$ of $\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}$, there is a cycle $c_{K}$ in $\operatorname{\mathbb{P}}\backslash(S_{1}\cup KS_{2}^{-1}\cup\{0\}\cup\{\infty\})$ such that

[TABLE]

for all $z\in K.$

Proof.

There is a fundamental class

[TABLE]

We choose $c_{K}$ to be a representative of the class in $H_{1}(\operatorname{\mathbb{P}}\backslash(S_{1}\cup KS_{2}^{-1}\cup\{0\}\cup\{\infty\}))$ which is the image of $-\alpha_{\operatorname{\mathbb{P}}\backslash(KS_{2}^{-1}\cup(\{0,\infty\}\backslash S_{1})),S_{1}}$ by the canonical map

[TABLE]

For each $z\in K$, there is a canonical commutative diagram

[TABLE]

Obviously, $\alpha_{\operatorname{\mathbb{P}}\backslash(zS_{2}^{-1}\cup(\{0,\infty\}\backslash S_{1})),S_{1}}$ is the image of $\alpha_{\operatorname{\mathbb{P}}\backslash(KS_{2}^{-1}\cup(\{0,\infty\}\backslash S_{1})),S_{1}}$ by the left vertical map. Therefore, by the commutativity of the diagram, one can deduce that $c_{K}$ is a generalized Hadamard cycle for $S_{1}$ in $\operatorname{\mathbb{P}}\backslash(zS_{2}^{-1}\cup(\{0,\infty\}\backslash S_{1}))$, for all $z\in K$. Hence the conclusion. ∎

Proposition 2.7**.**

The generalized Hadamard product is a well-defined map

[TABLE]

Proof.

Let $f_{1}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\backslash S_{1})$ and $f_{2}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\backslash S_{2}).$ We have to check that $f_{1}\star f_{2}$ is holomorphic on $\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}$. Since it is a local property, it is enough to prove that $f_{1}\star f_{2}$ is holomorphic on each small open disk $D\subset\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}.$ Let $D$ be such a disk. By Lemma 2.6 there is a cycle $c_{\overline{D}}$ such that

[TABLE]

for all $z\in D.$ We conclude by derivation under the integral sign. ∎

We shall now prove that our product is a good generalization of the extended Hadamard product of T. Pohlen. By doing so, the reader shall see why we chose such a sign convention in Definition 2.4.

Proposition 2.8**.**

Let $f_{1}\in\operatorname{\mathcal{H}}(\operatorname{\mathbb{P}}\backslash S_{1})$ and $f_{2}\in\operatorname{\mathcal{H}}(\operatorname{\mathbb{P}}\backslash S_{2})$. Let $z\in\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}.$ Let $c_{z}$ be a generalized Hadamard cycle for $S_{1}$ in $\operatorname{\mathbb{P}}\backslash(zS_{2}^{-1}\cup(\{0,\infty\}\backslash S_{1}))$ and $d_{z}$ a Hadamard cycle for $zS_{2}^{-1}$ in $\operatorname{\mathbb{P}}\backslash S_{1}.$ Then,

[TABLE]

Proof.

We treat the case where $0,\infty\notin S_{1}$ and $0\in S_{2},\infty\notin S_{2}$ and leave the other ones to the reader. By construction, it is clear that $c_{z}$ verifies

[TABLE]

Let $c^{\prime}_{z}$ be a cycle $\operatorname{\mathbb{P}}\backslash(S_{1}\cup zS_{2}^{-1}\cup\{0\}\cup\{\infty\})$ such that

[TABLE]

Since $d_{z}$ is $\text{acc}^{-}$, by Proposition 2.2, it is clear that $d_{z}$ is homologuous to $c_{z}+c^{\prime}_{z}$ in $\operatorname{\mathbb{P}}\backslash(S_{1}\cup zS_{2}^{-1}\cup\{0\}\cup\{\infty\})$. We then have

[TABLE]

Moreover, by the residue theorem,

[TABLE]

Hence the conclusion. ∎

Remark 2.9**.**

Of course, the generalized Hadamard product is no longer commutative if the functions do not vanish at infinity. For example, let $S_{1}$ and $S_{2}$ be as in the proof of the previous proposition. Let $f_{1}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\backslash S_{1})$ and $f_{2}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\backslash S_{2}).$ By a similar computation, one sees that

[TABLE]

Despite the lack of commutativity, the generalized Hadamard cycles are more symmetric with respect to [math] and $\infty$. In the section $5$, we shall explain how one can define a convolution between $1$-forms which have (not necessarily isolated) singularities at [math] and $\infty$. Generalized Hadamard cycles are key ingredients to compute such a convolution (see also section $6$). Moreover, the commutativity shall eventually be obtained thanks to quotient spaces that naturally occur in this context.

3. The holomorphic integration map

Let $X$ be a complex manifold of complex dimension $d_{X}$ and $r\in\operatorname{\mathbb{Z}}$. Recall that $\operatorname{\mathcal{C}}_{\infty,X}^{r}$ admits a decomposition in bi-types

[TABLE]

which induces a decomposition of the exterior derivative $d$ as

[TABLE]

where

[TABLE]

Similarly, $\operatorname{\mathcal{D}\textit{b}}_{X}^{r}$ admits a decomposition in bi-types

[TABLE]

and an associated decomposition of the distributional exterior derivative. Moreover, for any open subset $U$ of $X$, we have a canonical isomorphism

[TABLE]

between the space of complex distributional $r$-forms and the topological dual of the space of infinitely differentiable complex differential $(2d_{X}-r)$-forms with compact support which induces the similar isomorphism

[TABLE]

In the sequel, we denote by $\Omega_{X}^{p}$ the sheaf of holomorphic differential $p$-forms on $X$ and we set for short $\Omega_{X}=\Omega_{X}^{d_{X}}.$ Of course, $\Omega_{X}^{p}$ is canonically isomorphic to both the kernel of

[TABLE]

and the kernel of

[TABLE]

The double complex \operatorname{\mathcal{C}}_{\infty,X}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{\displaystyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\textstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptscriptstyle\bullet}}}}},\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{\displaystyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\textstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptscriptstyle\bullet}}}}}} (resp. \operatorname{\mathcal{D}\textit{b}}_{X}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{\displaystyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\textstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptscriptstyle\bullet}}}}},\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{\displaystyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\textstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptscriptstyle\bullet}}}}}}) is the infinitely differentiable (resp. distributional) Dolbeault complex of $X$. By construction, the associated simple complex is the infinitely differentiable (resp. distributional) de Rham complex \operatorname{\mathcal{C}}_{\infty,X}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{\displaystyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\textstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptscriptstyle\bullet}}}}}} (resp. \operatorname{\mathcal{D}\textit{b}}_{X}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{\displaystyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\textstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{\scriptscriptstyle\bullet}}}}}}) of $X$. Moreover, we have the following chains of canonical quasi-isomorphisms :

[TABLE]

which are given by de Rham and Dolbeault lemmas.

Let $f:X\to Y$ be a holomorphic map from $X$ to a complex manifold $Y$ of complex dimension $d_{Y}$ and let $V$ be an arbitrary open subset of $Y$. It follows from the holomorphy of $f$ that the pullback

[TABLE]

sends $\operatorname{\mathcal{C}}_{\infty,Y}^{p,q}(V)$ into $\operatorname{\mathcal{C}}_{\infty,X}^{p,q}(f^{-1}(V))$ if $p+q=r$. In particular,

[TABLE]

for all $\omega\in\operatorname{\mathcal{C}}_{\infty,Y}^{p,q}(V)$. By topological duality, it follows that there are canonical pushforward morphisms

[TABLE]

and

[TABLE]

between distributional forms with $f$-proper support on $f^{-1}(V)$ and distributional forms on $V$ and that these morphisms commute with $\partial$ and $\overline{\partial}$. In particular, we get a morphism of double complexes of sheaves of the form

[TABLE]

Moreover, if $f$ is a surjective submersion, one can show that the pushforward of a distibutional form associated to an infinitely differentiable form with $f$-proper support is itself associated to an infinitely differentiable form which can be computed by integration over the fibers of $f$. This shows that, in this case, the preceding morphism factors through a morphism of the form

[TABLE]

Thanks to the quasi-isomorphisms

[TABLE]

this gives us a morphism

[TABLE]

in the derived category for each $p\in\operatorname{\mathbb{Z}}.$ In the particular case where $p=0$, we get the morphism

[TABLE]

which is usually called the holomorphic integration map along the fibers of $f$ (see e.g. [7, p. 129]). Note that, if $g:Y\to Z$ is another holomorphic map between complex manifolds, then the well known relation $(g\circ f)^{*}=f^{*}\circ g^{*}$ entails that $\int_{g\circ f}=\int_{g}\circ\int_{f}$.

4. Holomorphic cohomological convolution

Definition 4.1**.**

Let $(G,\mu)$ be a locally compact complex Lie group of complex dimension $n$. Two closed subsets $S_{1}$ and $S_{2}$ of $G$ are said to be convolvable if $S_{1}\times S_{2}$ is $\mu$-proper, i.e. if

[TABLE]

is a compact subset of $G\times G$ for any compact subset $K$ of $G$.

Remark 4.2**.**

A proper map on a locally compact topological space is universally closed, in particular closed (see e.g. [2]). Hence, if $S_{1}$ and $S_{2}$ are convolvable closed subsets of $G$, then $\mu|_{S_{1}\times S_{2}}$ is a proper map and $S_{1}+S_{2}=\mu|_{S_{1}\times S_{2}}(S_{1}\times S_{2})$ is closed.

Definition 4.3**.**

Two distributional $2n$-forms $u_{1}$ and $u_{2}$ of $G$ are convolvable if the support $S_{1}$ of $u_{1}$ and the support $S_{2}$ of $u_{2}$ are convolvable. In that case, the convolution product of $u_{1}$ and $u_{2}$ is a distributional $2n$-form on $G$ defined by

[TABLE]

where $p_{1},p_{2}:G\times G\to G$ are the two canonical projections.

Remark 4.4**.**

By choosing a Haar form $\nu$ on $G$, one can define the convolution product of two distributions by means of the isomorphism $\operatorname{\mathcal{D}\textit{b}}_{G}\simeq\operatorname{\mathcal{D}\textit{b}}_{G}^{2n}$ given by $\nu$ (see e.g. [4]).

Remark 4.5**.**

If we define

[TABLE]

by setting $\phi(g_{1},g_{2})=(g_{1},\mu(g_{1},g_{2}))$ and $\psi(g_{1},g_{2})=(g_{1},\mu(g_{1}^{-1},g_{2}))$, we see that $\phi$ and $\psi$ are reciprocal biholomorphic bijections and that the diagram

[TABLE]

is commutative. This shows in particular that $\mu$ is a surjective submersion and that the preceding procedure allows us also to define the convolution product of infinitely differentiable forms.

Let $S_{1}$ and $S_{2}$ be two convolvable closed subsets of $G$. By construction, the convolution of distributions on $G$ is the composition of the external product of distributions

[TABLE]

and the map

[TABLE]

induced by the integration map along the fibers of $\mu$

[TABLE]

and the fact that $S_{1}$ and $S_{2}$ are convolvable. It is thus natural to define the convolution of cohomology classes of holomorphic forms on $G$ as follows :

Definition 4.6**.**

Let $S_{1},S_{2}$ be two convolvable closed subsets of $G$. Consider the external product morphisms

[TABLE]

and the morphisms

[TABLE]

induced by the holomorphic integration map and the fact that $S_{1}\times S_{2}$ is $\mu$-proper. By composition, these morphisms give derived category morphisms

[TABLE]

that we call the holomorphic convolution morphisms of $G$. Going to cohomology groups, these morphisms give rise to the morphisms

[TABLE]

that we call the holomorphic cohomological convolution morphisms of $G$.

Remark 4.7**.**

Consider the diagram

[TABLE]

where the vertical arrows are given by the Dolbeault complex of $\Omega_{G}$ and the top (resp the bottom) horizontal arrow is given by the holomorphic cohomological morphism of $G$ with $p=q=r=s=0$ (resp. the convolution product of distributions). Obviously, by the definitions, this diagram is commutative. This remark will allow to perform explicit computations in the next section.

5. Multiplicative convolution on $\operatorname{\mathbb{C}}^{*}$

In this section, we will consider the case where the group $G$ is the group $\operatorname{\mathbb{C}}^{*}$ formed by the set of non-zero complex numbers endowed with the complex multiplication (noted as a concatenation). We will assume that $S_{1},S_{2}$ are convolvable proper closed subsets of $\operatorname{\mathbb{C}}^{*}$ (remark that this means that $S_{1}\cap KS_{2}^{-1}$ is compact for any compact subset $K$ of $\operatorname{\mathbb{C}}^{*}$) such that $S_{1}S_{2}$ is also a proper subset of $\operatorname{\mathbb{C}}^{*}$ and we will show how to compute the holomorphic cohomological convolution morphism

[TABLE]

by means of path integral formulas.

Proposition 5.1**.**

Let $S$ be a proper closed subset of $\operatorname{\mathbb{C}}^{*}$, then there is a canonical isomorphism

[TABLE]

Proof.

Any open subset of $\operatorname{\mathbb{C}}$ is a Stein manifold. ∎

Thanks to this proposition, one can see that (5.1) can be interpreted as a bilinear map

[TABLE]

Now, let $\omega_{1}\in\Omega(\operatorname{\mathbb{C}}^{*}\setminus S_{1})$ and $\omega_{2}\in\Omega(\operatorname{\mathbb{C}}^{*}\setminus S_{2})$ be two given holomorphic forms. Ideally, we would like to obtain a formula of the form

[TABLE]

where $\omega$ is a holomorphic form on $\operatorname{\mathbb{C}}^{*}\setminus(S_{1}S_{2})$ which can be computed from $\omega_{1}$ and $\omega_{2}$ by some path integral.

It is in general not possible to find such a nice formula. However, we will show that for any relatively compact open subset $U$ of $\operatorname{\mathbb{C}}^{*}$ and any open neighbourhood $V$ of $S_{1}S_{2}$ in $\operatorname{\mathbb{C}}^{*}$, there is a holomorphic form $\omega$ on $U\setminus\overline{V}$ which can be computed from $\omega_{1}$ and $\omega_{2}$ by some path integral and which is such that

[TABLE]

coincides with the image of $[\omega_{1}]\star[\omega_{2}]$ by the canonical restriction morphism

[TABLE]

Thanks to the following lemma, this is in fact sufficient to completely compute $[\omega_{1}]\star[\omega_{2}]$.

Lemma 5.2**.**

Let $S$ be a closed subset of $\operatorname{\mathbb{C}}^{*}$. Then

[TABLE]

where $\mathcal{U}_{\text{rc}}$ denotes the set of relatively compact open subsets of $\operatorname{\mathbb{C}}^{*}$ ordered by $\subset$ and $\mathcal{V}_{S}$ denotes the set of open neighbourhoods of $S$ in $\operatorname{\mathbb{C}}^{*}$ ordered by $\supset$.

Proof.

This follows from the Mittag-Leffler theorem for projective systems (see e.g. Proposition $2.7.1$ in [7]). ∎

To be able to specify the kind of path integral we need, let us first introduce the following definition :

Definition 5.3**.**

Let $F$ and $G$ be two closed subsets of $\operatorname{\mathbb{C}}^{*}$ which have a compact intersection and let $W$ be an open neighbourhood of $F\cap G$. A relative Hadamard cycle for $F$ with respect to $G$ in $W$ is a relative 1-cycle

[TABLE]

such that its class

[TABLE]

is the image of the fundamental class

[TABLE]

by the Mayer-Vietoris morphism

[TABLE]

associated with the decomposition

[TABLE]

WGF

With this definition at hand, we can now state the main result of this section.

Theorem 5.4**.**

Let $S_{1}$ and $S_{2}$ be two convolvable proper closed subsets of $\operatorname{\mathbb{C}}^{*}$ such that $S_{1}S_{2}\neq\operatorname{\mathbb{C}}^{*}$ and let us assume that $\omega_{1}=f_{1}dz$ and $\omega_{2}=f_{2}dz$ with $f_{1}\in\mathcal{O}(\operatorname{\mathbb{C}}^{*}\setminus S_{1}),f_{2}\in\mathcal{O}(\operatorname{\mathbb{C}}^{*}\setminus S_{2})$. Fix a relatively compact open subset $U$ of $\operatorname{\mathbb{C}}^{*}$ and an open neighbourhood $V$ of $S_{1}S_{2}$ in $\operatorname{\mathbb{C}}^{*}$. Then, the image of

[TABLE]

in

[TABLE]

is the class of the form $\omega=fdz\in\Omega(U\setminus\overline{V})$ where

[TABLE]

and $c$ is a relative Hadamard cycle for $S_{1}$ with respect to $\overline{U}S_{2}^{-1}$ in $\operatorname{\mathbb{C}}^{*}\setminus(\overline{U}\setminus V)S_{2}^{-1}$.

Lemma 5.5**.**

Let $S_{1}$ and $S_{2}$ be convolvable closed subsets of $\operatorname{\mathbb{C}}^{*}$ and let $\mathcal{W}$ be a fundamental system of compact neighbourhoods of $1$ in $\operatorname{\mathbb{C}}^{*}$. Then

1. (1)

The set $S_{1}^{W}=WS_{1}$ (resp. $S_{2}^{W}=WS_{2}$, $S_{1}^{W}S_{2}^{W}=W^{2}S_{1}S_{2}$) is a closed neighbourhood of $S_{1}$ (resp. $S_{2}$, $S_{1}S_{2}$) in $\operatorname{\mathbb{C}}^{*}$ for any $W\in\mathcal{W}$. 2. (2)

The closed subsets $S_{1}^{W}$ et $S_{2}^{W}$ are convolvable in $\operatorname{\mathbb{C}}^{*}$ for any $W\in\mathcal{W}$. 3. (3)

One has $\bigcap_{W\in\mathcal{W}}S_{1}^{W}=S_{1}$, $\bigcap_{W\in\mathcal{W}}S_{2}^{W}=S_{2}$, and $\bigcap_{W\in\mathcal{W}}S_{1}^{W}S_{2}^{W}=S_{1}S_{2}$. 4. (4)

In particular, if $S_{1}$ and $S_{2}$ are proper convolvable closed subsets of $\operatorname{\mathbb{C}}^{*}$ such that $S_{1}S_{2}\neq\operatorname{\mathbb{C}}^{*}$, if $U$ is a relatively compact open subset of $\operatorname{\mathbb{C}}^{*}$ and if $V$ is an open neighbourhood of $S_{1}S_{2}$ in $\operatorname{\mathbb{C}}^{*}$, then there is $W\in\mathcal{W}$ such that $S_{1}^{W}$ and $S_{2}^{W}$ are convolvable proper closed subsets of $\operatorname{\mathbb{C}}^{*}$ such that $S_{1}^{W}S_{2}^{W}\neq\operatorname{\mathbb{C}}^{*}$ and $S_{1}^{W}S_{2}^{W}\cap\overline{U}\subset V$.

Proof.

(1) This follows from the fact that $FK$ is closed in $\operatorname{\mathbb{C}}^{*}$ if $F$ (resp. $K$) is closed (resp. compact) in $\operatorname{\mathbb{C}}^{*}$ and from the fact that $(zW)_{W\in\mathcal{W}}$ is a fundamental system of neighbourhoods of $z\in\operatorname{\mathbb{C}}^{*}$.

(2) This follows from the inclusion

[TABLE]

which is satisfied for any compact subset $K$ of $\operatorname{\mathbb{C}}^{*}$.

(3) This is clear since for any closed subset $F$ of $\operatorname{\mathbb{C}}^{*}$ and any $z\not\in F$ there is $W\in\mathcal{W}$ such that $zW^{-1}\cap F=\emptyset$.

(4) By contradiction, assume that

[TABLE]

for all $W\in\mathcal{W}.$ Then, by compactness,

[TABLE]

but this contradicts the fact that $S_{1}^{W}S_{2}^{W}\cap\overline{U}\subset V$. ∎

Lemma 5.6**.**

Let $S$ be a proper closed subset of $\operatorname{\mathbb{C}}^{*}$ and let $\omega\in\Omega(\operatorname{\mathbb{C}}^{*}\setminus S)$. Assume that $\omega$ admits an infinitely differentiable extension to $\operatorname{\mathbb{C}}^{*}$ and denote by $\underline{\omega}$ such an extension. Then $[\omega]$, seen as an element of $H^{1}_{S}(\operatorname{\mathbb{C}}^{*},\Omega_{\operatorname{\mathbb{C}}^{*}})$, is the image of

[TABLE]

by the canonical morphism obtained by applying $H^{1}$ to the composition in the derived category of the canonical morphism

[TABLE]

and the inverse of the canonical isomorphism

[TABLE]

Proof.

It follows from the distinguished triangle

[TABLE]

that ${\operatorname{R}}\Gamma_{S}(\operatorname{\mathbb{C}}^{*},\Omega_{\operatorname{\mathbb{C}}^{*}})$ is canonically isomorphic to the mapping cone $M(\rho_{S})$ of the restriction morphism

[TABLE]

shifted by $-1$. We know that $M[\rho_{S}][-1]$ is a complex concentrated in degrees [math], $1$ and $2$ of the form

[TABLE]

where the differentials in degree [math] and $1$ are given by the matrices

[TABLE]

What we have to show is that

[TABLE]

are two 1-cycles of this complex which are in the same cohomology class. This is clear since

[TABLE]

∎

Proof of Theorem 5.4.

Let $U$ and $V$ be as in the statement of the theorem. Thanks to Lemma 5.5, we know that it is possible to find a closed neighbourhood $\underline{S}_{1}$ of $S_{1}$ and a closed neighbourhood $\underline{S}_{2}$ of $S_{2}$ in $\operatorname{\mathbb{C}}^{*}$ such that $\underline{S}_{1}$ and $\underline{S}_{2}$ are convolvable and

[TABLE]

Let $\underline{f}_{1}$ (resp. $\underline{f}_{2}$) be an infinitely differentiable function on $\operatorname{\mathbb{C}}^{*}$ which coincides with $f_{1}$ (resp. $f_{2}$) on $\operatorname{\mathbb{C}}^{*}\setminus\underline{S}_{1}$ (resp. $\operatorname{\mathbb{C}}^{*}\setminus\underline{S}_{2}$) and set

[TABLE]

It follows from Lemma 5.6 that the image of

[TABLE]

by the canonical morphism

[TABLE]

is the same as the image of

[TABLE]

by the canonical morphism

[TABLE]

considered in this lemma. A similar conclusion is true for the image of

[TABLE]

in $H^{1}_{\underline{S}_{2}}(\operatorname{\mathbb{C}}^{*},\Omega_{\operatorname{\mathbb{C}}^{*}})$. Therefore, the image of

[TABLE]

in $H^{1}_{\underline{S}_{1}\underline{S}_{2}}(\operatorname{\mathbb{C}}^{*},\Omega_{\operatorname{\mathbb{C}}^{*}})$ is the same as the image of $[\overline{\partial}\underline{\omega}_{1}\star\overline{\partial}\underline{\omega}_{2}]$ by the canonical morphism

[TABLE]

Let us note $p_{1},p_{2}:\operatorname{\mathbb{C}}^{*}\times\operatorname{\mathbb{C}}^{*}\to\operatorname{\mathbb{C}}^{*}$ the two canonical projections and consider the commutative diagram

[TABLE]

where $\phi(z_{1},z_{2})=(z_{1},z_{1}z_{2})$ and $\psi(\zeta,z)=(\zeta,z/\zeta)$. Since $\phi\circ\psi=\text{id}=\psi\circ\phi$, we have

[TABLE]

Therefore,

[TABLE]

where $h(\zeta,z)=z/\zeta$. Since

[TABLE]

we have

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

Since $\underline{f}_{1}$ coincides with $f_{1}$ on $\operatorname{\mathbb{C}}^{*}\setminus\underline{S}_{1}$, one has

[TABLE]

Similarly, one has

[TABLE]

Hence,

[TABLE]

is an infinitely differentiable function on $\operatorname{\mathbb{C}}^{*}$ supported by $\underline{S}_{1}\cap z\underline{S}_{2}^{-1}$ which is a compact subset of $\operatorname{\mathbb{C}}^{*}$.

Since $U$ is a relatively compact open subset of $\operatorname{\mathbb{C}}^{*}$ and $\underline{S}_{1}$ and $\underline{S}_{2}$ are convolvable closed subsets of $\operatorname{\mathbb{C}}^{*}$,

[TABLE]

is a compact subset of $\operatorname{\mathbb{C}}^{*}$. Let $c$ be a singular infinitely differentiable $2$-chain of $\operatorname{\mathbb{C}}^{*}$ such that

[TABLE]

is the relative orientation class $\alpha_{\operatorname{\mathbb{C}}^{*},K}$, which is the image of the orientation class $\alpha_{\operatorname{\mathbb{C}}^{*}}$ by the canonical morphism

[TABLE]

Then, on $U$, one has

[TABLE]

since the integrated form is supported by $\underline{S}_{1}\cap z\underline{S}_{2}^{-1}\subset K$ for any $z\in U$. Moreover, the function $\underline{f}_{2}$ is infinitely differentiable on $\operatorname{\mathbb{C}}^{*}$ and the chain $c$ is supported by a compact subset of $\operatorname{\mathbb{C}}^{*}$. Thus, the function

[TABLE]

is infinitely differentiable on $\operatorname{\mathbb{C}}^{*}$ and

[TABLE]

Therefore, on $U$, one has

[TABLE]

where $\omega=f(z)dz$. Since $\operatorname{supp}(\overline{\partial}\underline{\omega}_{1}\star\overline{\partial}\underline{\omega}_{2})\subset\underline{S}_{1}\underline{S}_{2}$, the function $f$ is holomorphic on $U\setminus\underline{S}_{1}\underline{S}_{2}$ and it follows from what precedes that

[TABLE]

in

[TABLE]

Let us now show how to compute $[\omega|_{U}]$ in $\Omega(U\setminus\overline{V})/\Omega(U)$ by means of $f_{1}$ and $f_{2}$ alone. Since $V$ is an open neighbourhood of $\underline{S}_{1}\underline{S}_{2}$,

[TABLE]

Therefore,

[TABLE]

and, replacing if necessary $c$ by a barycentric subdivision, we may assume that $c=c_{1}+c_{2}$ where

[TABLE]

Since $\operatorname{supp}\frac{\partial\underline{f}_{1}}{\partial\overline{z}}\subset\underline{S}_{1}$, it is then clear that

[TABLE]

Moreover, for any $z\in\overline{U}\setminus V$ one has

[TABLE]

and since the function $\zeta\mapsto\underline{f}_{2}(z/\zeta)$ is holomorphic on $\operatorname{\mathbb{C}}^{*}\setminus z\underline{S}_{2}^{-1}$, it follows that

[TABLE]

By construction,

[TABLE]

Up to replacing $c$ by a one of its barycentric subdivisions, we may thus assume that $\partial c=c^{\prime}_{1}+c^{\prime}_{2}$ where $\operatorname{supp}c^{\prime}_{1}\subset\operatorname{\mathbb{C}}^{*}\setminus\underline{S}_{1}$ and $\operatorname{supp}c^{\prime}_{2}\subset\operatorname{\mathbb{C}}^{*}\setminus\overline{U}\underline{S}_{2}^{-1}$. Since

[TABLE]

there is a chain $c_{3}$ such that

[TABLE]

Since $\operatorname{supp}c^{\prime}_{2}\subset\operatorname{\mathbb{C}}^{*}\setminus\overline{U}\underline{S}_{2}^{-1}$, the function

[TABLE]

is clearly holomorphic on $U$. Hence, the image of $[\omega|_{U}]$ in $\Omega(U\backslash\overline{V})/\Omega(U)$ is $[g(z)dz]$ where $g$ is the holomorphic function on $U\setminus\overline{V}$ defined by setting

[TABLE]

Since

[TABLE]

and

[TABLE]

it is clear that

[TABLE]

Therefore, we have in fact

[TABLE]

for any $z\in U\setminus\overline{V}$. Moreover, since $\partial c_{3}=\partial c^{\prime}_{1}=-\partial c^{\prime}_{2}$, it is clear that

[TABLE]

So,

[TABLE]

and it follows by the construction of the chain $c_{3}$ that its class in

[TABLE]

coïncides with the image of orientation class of $\operatorname{\mathbb{C}}^{*}$ by the following sequence of canonical morphisms

[TABLE]

This clearly shows that $c_{3}$ is a relative Hadamard cycle for $\underline{S}_{1}$ with respect to $\overline{U}\underline{S}_{2}^{-1}$ in $\operatorname{\mathbb{C}}^{*}\setminus(\overline{U}\setminus V)\underline{S}_{2}^{-1}$. Thus, $c_{3}$ is also a relative Hadamard cycle for $S_{1}$ with respect to $\overline{U}S_{2}^{-1}$ in $\operatorname{\mathbb{C}}^{*}\setminus(\overline{U}\setminus V)S_{2}^{-1}$.

To conclude, it remains to show that if $c^{\prime}_{3}$ is another relative Hadamard cycle for $S_{1}$ with respect to $\overline{U}S_{2}^{-1}$ in $\operatorname{\mathbb{C}}^{*}\setminus(\overline{U}\setminus V)S_{2}^{-1}$ and if

[TABLE]

for any $z\in U\setminus\overline{V}$, then $[g(z)dz]=[\check{g}(z)dz]$ in $\Omega(U\backslash\overline{V})/\Omega(U)$. For such a $c^{\prime}_{3}$, we have $[c_{3}]=[c^{\prime}_{3}]$ in

[TABLE]

Therefore, $c^{\prime}_{3}=c_{3}+c_{4}+\partial c_{5}$ where $c_{4}$ is a $1$-chain of $(\operatorname{\mathbb{C}}^{*}\setminus\underline{S}_{1})\cap(\operatorname{\mathbb{C}}^{*}\setminus\overline{U}\underline{S}_{2}^{-1})$ and $c_{5}$ is a $2$-chain of $(\operatorname{\mathbb{C}}^{*}\setminus\underline{S}_{1})\cap(\operatorname{\mathbb{C}}^{*}\setminus(\overline{U}\setminus V)\underline{S}_{2}^{-1})$. It follows that the function

[TABLE]

is a holomorphic function on $U\setminus\overline{V}$ and that

[TABLE]

on $U\setminus\overline{V}$. Since

[TABLE]

is clearly holomorphic on $U$, we have $[g(z)dz]=[\check{g}(z)dz]$ in $\Omega(U\backslash\overline{V})/\Omega(U)$ as expected. ∎

6. The case of strongly convolvable sets

It is natural to ask if one can compute the holomorphic cohomological multiplicative convolution on $\operatorname{\mathbb{C}}^{*}$ thanks to a global formula, by adding extra-conditions on $S_{1}$ and $S_{2}$. Recalling Definition 2.3, we are led to introduce the following one :

Definition 6.1**.**

Let $S_{1}$ and $S_{2}$ be two convolvable proper closed subsets of $\operatorname{\mathbb{C}}^{*}$ such that $S_{1}S_{2}\neq\operatorname{\mathbb{C}}^{*}$. These two closed sets are said to be strongly convolvable if, furthermore, $\overline{S}_{1}$ and $\overline{S}_{2}$ are star-eligible, that is to say, if $\overline{S}_{1}\times\overline{S}_{2}\subset M.$ (Here $\overline{(.)}$ denotes the closure in $\operatorname{\mathbb{P}}.$)

Remark 6.2**.**

One can find convolvable subsets of $\operatorname{\mathbb{C}}^{*}$ which are not strongly convolvable. For example, consider

[TABLE]

We shall now highlight the link with the generalized Hadamard product. Recall Definitions 2.4 and 2.5.

Proposition 6.3**.**

Let $S_{1}$ and $S_{2}$ be two strongly convolvable proper closed subsets of $\operatorname{\mathbb{C}}^{*}.$ Assume that $\omega_{1}=f_{1}dz$ and $\omega_{2}=f_{2}dz$ with $f_{1}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*}\backslash S_{1})$ and $f_{2}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*}\backslash S_{2})$. For all $z\in\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}$, let $c_{z}$ be a generalized Hadamard cycle for $\overline{S}_{1}$ in $\operatorname{\mathbb{P}}\backslash(z\overline{S}_{2}^{-1}\cup(\{0,\infty\}\backslash\overline{S}_{1})).$ Then

[TABLE]

where

[TABLE]

for all $z\in\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2}.$

Proof.

Let $U$ be a relatively compact open subset of $\operatorname{\mathbb{C}}^{*}$ and $V$ an open neighbourhood of $S_{1}S_{2}$ in $\operatorname{\mathbb{C}}^{*}$. Let $c$ be a relative Hadamard cycle for $S_{1}$ with respect to $\overline{U}S_{2}^{-1}$ in $\operatorname{\mathbb{C}}^{*}\backslash(\overline{U}\backslash V)S_{2}^{-1}$. Then, by a similar argument as in the proof of Lemma 2.6, it is clear that the image of $[c_{z}]$ by the sequence of canonical maps

[TABLE]

is $[-c]$ for all $z\in\overline{U}\backslash V.$ Hence

[TABLE]

Since this argument is valid for all $U$ and all $V$, the conclusion follows from Theorem 5.4. ∎

In this context, we set $(f_{1}\star f_{2})(z)=\int_{c_{z}}f_{1}(\zeta)f_{2}\left(\frac{z}{\zeta}\right)\frac{d\zeta}{\zeta}.$ If $f_{1}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\setminus\overline{S}_{1})$ and $f_{2}\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{P}}\setminus\overline{S}_{2}),$ this really coincides with the generalized Hadamard product.

Remark 6.4**.**

Let $S_{1}$ and $S_{2}$ be two strongly convolvable proper closed subsets of $\operatorname{\mathbb{C}}^{*}.$ Let us make an identification $fdz\leftrightarrow-2i\pi f$ between holomorphic $1$-forms and holomorphic functions. Then, by the previous proposition, the holomorphic cohomological convolution morphism

[TABLE]

can be seen as a bilinear map

[TABLE]

which can be computed by

[TABLE]

For the following example, we use the notation $D(0,R)=\{z\in\operatorname{\mathbb{C}}:|z|<R\}$ with $R>0.$

Example 6.5**.**

Let $S=\operatorname{\mathbb{C}}^{*}\backslash D(0,s)$ and $T=\operatorname{\mathbb{C}}^{*}\backslash D(0,t)$ with $s>0$, $t>0$ and let

[TABLE]

be two holomorphic functions. Then, $S$ and $T$ are strongly convolvable proper closed subsets of $\operatorname{\mathbb{C}}^{*}$ and we can write $f(z)=\sum_{n=-\infty}^{+\infty}a_{n}z^{n}$, $g(z)=\sum_{n=-\infty}^{+\infty}b_{n}z^{n}$. Since the polar part of $f$ (resp. $g$) is holomorphic on $\operatorname{\mathbb{C}}^{*}$, we have $[f]=\left[\sum_{n=0}^{+\infty}a_{n}z^{n}\right]$ in $\operatorname{\mathcal{O}}(D(0,s)\backslash\{0\})/\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*})$ and $[g]=\left[\sum_{n=0}^{+\infty}b_{n}z^{n}\right]$ in $\operatorname{\mathcal{O}}(D(0,t)\backslash\{0\})/\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*}).$ Using the preceding remark, we see that the holomorphic cohomological convolution $[f]\star[g]$ is given by

[TABLE]

since the generalized Hadamard product coincides with the usual one in this case.

Let us now state a trivial proposition :

Proposition 6.6**.**

Let $S_{1}$ and $S_{2}$ be two convolvable closed subsets of $\operatorname{\mathbb{C}}^{*}$ and $S^{\prime}_{1}\subset S_{1}$, $S^{\prime}_{2}\subset S_{2}$ two closed subsets. Then, $S^{\prime}_{1}$ and $S^{\prime}_{2}$ are convolvable and the diagram

[TABLE]

where the horizontal arrows are given by the holomorphic cohomological convolution morphisms, is commutative.

Example 6.5 combined with Proposition 6.6 allows to compute several other examples.

Example 6.7**.**

Let $S_{1}=S_{2}=(-\infty,-1].$ The principal determination of the function $z\mapsto\ln(1+z)$ is holomorphic on $\operatorname{\mathbb{C}}^{*}\backslash S_{1}.$ Moreover, $S_{1}$ and $S_{2}$ are strongly convolvable and thus, there is $g\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*}\backslash[1,+\infty))$ such that

[TABLE]

Using the previous results, one has

[TABLE]

where $\text{Li}_{2}$ is the principal dilogarithm function, holomorphic on $\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*}\backslash[1,+\infty))$. Hence, there is $h\in\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*})$ such that

[TABLE]

By the uniqueness of the analytic continuation, one deduces that $g-\text{Li}_{2}=h$ on $\operatorname{\mathbb{C}}^{*}\backslash[1,+\infty)$ and, thus, that

[TABLE]

in $\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*}\backslash S_{1}S_{2})/\operatorname{\mathcal{O}}(\operatorname{\mathbb{C}}^{*}).$