# Multiplicative convolution and double shuffle relations: convolution

Nikita Markarian

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Nikita Markarian. Multiplicative convolution and double shuffle relations: convolution. 2024. hal-04855203

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MULTIPLICATIVE CONVOLUTION AND DOUBLE SHUFFLE
RELATIONS: CONVOLUTION

NIKITA MARKARIAN

ABSTRACT. This is the first of two parts of a project devoted to a geometric interpretation of the Deligne–Terasoma approach to regularized double shuf-fle relations. The central fact of this approach is the isomorphism between vanishing cycles of multiplicative convolution of certain perverse sheaves and the tensor product of vanishing cycles, which may be written in two different ways. These isomorphisms depend on a choice of a functorial isomorphism ϕ between vanishing cycles of a perverse sheaf on $C^*$ and cohomology of its cer-tain extension on $P^1$. The isomorphism chosen in the present paper guarantees compatibilities with the isomorphisms.

In the second part of the project, we will study other choices of $\varphi$. We will see that its compatibilities with convolution imply regularized double shuffle relations. In particular, associator relations imply them.

INTRODUCTION

Multiple zeta values are periods of iterated integrals and thereby are coefficients of the Drinfeld associator. As such, they obey the set of relation implied by the relations the associator satisfies. Another set of relations, regularized double shuffle relations, was introduced in [IKZ06], and nicely formalized in [Rac02].

The unfinished project [Del01, Ter06, TD05] of Deligne and Terasoma, developed later in [EF21, EF22, EF23], connects double shuffle relations with convolution of perverse sheaves. The main aim of this project was to show that associator relations imply regularized double shuffle relation. Although since then this fact was proved in different ways, see [Fur11, HS19, Fur22], the geometric approach is still of great interest and has applications we will discuss in the second part of the paper.

The present project grew up from attempts to interpret the Deligne–Terasoma approach. Note that although our approaches have the same main characters, perverse sheaves and convolution, there are some differences. Firstly, the central role in the Deligne–Terasoma approach is played by the quotient category as in Theorem 1 below, we do not use this category at all. Another subject, playing an important role there, Hodge structure and “fake Hodge structure” also do not appear in our approach. It would be very interesting to fill these gaps.

In the first part of the paper presented below, we discuss only interaction between multiplicative convolution of perverse sheaves and vanishing cycles. A paradigmatic example of this relation is the Thom–Sebastiani theorem [ST71, Mas01], which states that vanishing cycles of the direct sum of singularities is equal to tensor product of the ones of summands. The initial proof ([ST71]) is transcendental in nature, essentially they show that the Milnor fiber of the sum is homeomorphic to the join of Milnor fibers of the summands.
The vanishing cycles of singularities are vanishing cycles of the pushforward of the constant sheaf and the direct sum of singularities corresponds essentially to the additive convolution. In contrast, we, following [Del01], consider the category of perverse sheaves on $\mathbb{G}_m$ smooth outside 1 and their multiplicative convolution.

On this category two functors are defined: vanishing cycles at 1 and the 0th cohomology of the extension $H^0(\mathbb{P}^1, j_{\infty!}j_{0*}^-)$. They are isomorphic, we define an isomorphism $\phi_I$ between them. This isomorphism is given by shrinking on the interval $[0,1]$ and is transcendental in nature. It means that it does not respect arithmetic structures such as Hodge structure. This fact could be a link between our approach and the Deligne-Terasoma one.

Both functors are tensor with respect to the multiplicative convolution.

For vanishing cycles, the isomorphism between vanishing cycles of convolution and the product of vanishing cycles of factors is given by the local calculation of the additive convolution of the Verdier specializations of perverse sheaves. Note that Verdier specialization is another possible link between Deligne-Terasoma and our approaches. Indeed, de Rham and Betti fiber functors in [Del89] are defined by means of the Verdier specializations of a vector bundle at infinite points.

The isomorphism between $H^0(\mathbb{P}^1, j_{\infty!}j_{0*}^-)$ of convolution and tensor product of $H^0(\mathbb{P}^1, j_{\infty!}j_{0*}^-)$ of factors is essentially the Künneth formula. Alternatively, one may present this isomorphism in terms of nearby cycles at infinity, this construction reminds the join in the proof of the Thom-Sebastiani theorem mentioned above.

Isomorphism $\phi_I$ connects the two tensor products above. The main result of the text, Theorem 2, states that this isomorphism is compatible with isomorphisms given by $\phi_I$ of factors. This follows from the existence of the pentagon in $\overline{\mathcal{M}}_{0,5}$ restricted by intervals, which are copies of the interval $I$, used to define isomorphism $\phi_I$.

In the second part of the text, we will replace the straight line $I$ with a pro-unipotent path. It is not surprising that the pentagon equation on this path implies analogous compatibilities of the corresponding isomorphism of functor with the convolution product. This compatibility may be reformulated as regularized shuffle relations between the coefficients of the element of the pro-unipotent completion of the fundamental group of $\mathcal{M}_{0,4}$ corresponding to the path.

**Acknowledgments.** I am grateful to B. Enriquez, M. Finkelberg H. Furusho and M. Kapranov for fruitful and deep discussions and support. I would like to thank the Max Planck Institute for Mathematics and IHES for hospitality and perfect work conditions. The project is supported by funding of the program PAUSE and of the ITI IRMIA++.

**Notations.** Everything is over $\mathbb{C}$, although some statement are true in a more general situation.

$\mathbb{P}^1$ is the projective line, $\mathbb{G}_m = \mathbb{P}^1 \setminus \{0, \infty\}$ is the multiplicative group.

$\text{Perv}(\mathbb{G}_m, 1)$ is the category of perverse sheaves on $\mathbb{G}_m$ smooth outside 1.

We omit annotation of derived functors: $j_*$ means $Rj_*$, $i^*$ means $Ri^*$ and so on.

For a map $f: X \to D \subset \mathbb{C}$ from a complex variety X to a disc $D \subset \mathbb{C}$ with the central fiber $X_0 = f^{-1}(0)$, the inclusion map $i: X_0 \to X$, and a complex $\mathcal{F}$ of constructible sheaves $\mathcal{F}^\bullet$ on X, denote by $\phi_{\mathcal{F}^\bullet}$ and $\psi_{\mathcal{F}^\bullet}$ the complex of vanishing and nearby cycles on $X_0$. Functors $\phi_f(-)[-1]$ and $\psi_f(-)[-1]$ conserve the category
of perverse sheaves. They form Milnor triangles:

$$ (1) \qquad \psi_f(\mathcal{F}^\bullet)[-1] \xrightarrow{\text{can}} \phi_f(\mathcal{F}^\bullet)[-1] \longrightarrow i^*\mathcal{F}^\bullet \xrightarrow{+1} $$

$$ (2) \qquad i^!\mathcal{F}^\bullet \longrightarrow \phi_f(\mathcal{F}^\bullet)[-1] \xrightarrow{-\text{var}} \psi_f(\mathcal{F}^\bullet)[-1] \xrightarrow{+1} $$

We denote cohomologies of vanishing and nearby cycles at a point by capital letters:

$$ (3) \qquad \Phi_f(\mathcal{F}) = H^0(\phi_f(\mathcal{F}^\bullet)[-1]) \quad \text{and} \quad \Psi_f(\mathcal{F}) = H^0(\psi_f(\mathcal{F}^\bullet)[-1]) $$

Denote by $i_a: pt \to \mathbb{P}^1$ the embedding of point $a$ in $\mathbb{P}^1$ for $a = 0, \infty, 1$.

# 1. DELIGNE FIBER FUNCTOR

## 1.1. Vanishing cycles.
Firstly, prove a general useful result. Let $I$ be the real interval $[0, 1]$ in $\mathbb{P}^1$ not containing $\infty$, let $i_I: I \to \mathbb{P}^1$ be the closed embedding. Denote by $j_\infty$ is the inclusion of $\mathbb{A}^1$ into $\mathbb{P}^1$.

**Proposition 1** (Shrinking). For a complex of sheaves $\mathcal{F}^\bullet$ on $\mathbb{A}^1$ constructible with respect to the stratification $(\{0,1\}, \mathbb{A}^1)$, composition of the excision isomorphism and the forgetful map give an isomorphism

$$ (4) \qquad H^*(I, i_I^! \mathcal{F}^\bullet) = H_I^*(\mathbb{P}^1, j_\infty! \mathcal{F}^\bullet) \to H^*(\mathbb{P}^1, j_\infty! \mathcal{F}^\bullet) $$

*Proof.* Let $j_I: \mathbb{P}^1 \setminus I \to \mathbb{P}^1$ be the open embedding. Consider the exact triangle

$$ (5) \qquad i_{I*} i_I^! \mathcal{F}^\bullet = i_{I*} i_I^! j_\infty! \mathcal{F}^\bullet \to j_\infty! \mathcal{F}^\bullet \to j_{I*} j_{I*}^* j_\infty! \mathcal{F}^\bullet \xrightarrow{+1} $$

The pair of topological spaces $(\mathbb{P}^1 \setminus I, \mathbb{P}^1 \setminus \{I, \infty\})$ is homeomorphic to the pair $(\mathbb{A}^1, \mathbb{G}_m)$. The restriction of $\mathcal{F}^\bullet$ on $\mathbb{P}^1 \setminus \{I, \infty\}$ is a complex of local systems. To show that cohomology of its !-extension on the bigger space vanishes, consider the corresponding complex of local systems on $\mathbb{G}_m$. One may see that for a local system $\mathcal{L}$ on $\mathbb{G}_m$, cohomology $H^*(\mathbb{A}^1, j_\infty!\mathcal{L})$ vanishes, which follows, that cohomology of a complex of local systems vanishes as well. It follows that cohomology of the last term in (5) vanish. Thus, cohomologies of first two terms in (5) are isomorphic. $\square$

Let $j: \mathbb{G}_m \to \mathbb{P}^1$ be the embedding of the multiplicative group to the projective line. Factor $j$ as $j_\infty \circ j_0$, where $j_0$ is the inclusion of $\mathbb{G}_m$ into $\mathbb{A}^1$, and $j_\infty$ is the inclusion of $\mathbb{A}^1$ into $\mathbb{P}^1$.

Let $\mathcal{F}$ be a perverse sheaf from Perv($\mathbb{G}_m, 1$). Denote by $(0, 1] \subset \mathbb{G}_m$ the intersection of $I$ and $\mathbb{G}_m$. Combining the canonical isomorphism of local cohomologies

$$ H_{(0,1]}^{\bullet}(\mathbb{G}_m, \mathcal{F}) \simeq H_{(0,1]}^{\bullet}(\mathbb{A}^1, j_{0*} \mathcal{F}) $$

with the excision isomorphism

$$ H_{(0,1]}^{\bullet}(\mathbb{A}^1, j_{0*} \mathcal{F}) \simeq H_{(0,1]}^{\bullet}(\mathbb{P}^1, j_{\infty!*} j_{0*} \mathcal{F}) $$

followed by the forgetful map to $H^\bullet(\mathbb{P}^1, j_\infty!* j_{0*} \mathcal{F})$ gives a map from $H_{(0,1]}^0(\mathbb{G}_m, \mathcal{F})$ to $H^0(\mathbb{P}^1, j_\infty!* j_{0*} \mathcal{F})$.
**Proposition 2.** For a perverse sheaf $\mathcal{F} \in \text{Perv}(\mathbb{G}_m, 1)$ cohomology $H^0_{(0,1]}(\mathbb{G}_m, \mathcal{F})$ is naturally isomorphic to vanishing cycles $\Phi_{1-z}(\mathcal{F})$ of $\mathcal{F}$ at 1 and the map defined above gives an isomorphism

$$ (6) \qquad \varphi_I: \Phi_{1-z}(\mathcal{F}) \to H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F}) $$

For $i \neq 0$ cohomologies $H^i(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F})$ vanish.

*Proof.* Apply Proposition 1 to the perverse sheaf $j_{0!} \mathcal{F}$. As $H^0_{(0,1]}(\mathbb{G}_m, \mathcal{F})$ is equal to $H^0_I(\mathbb{A}^1, j_{0!} \mathcal{F})$, it gives isomorphism (6). An isomorphism between $H^0_{(0,1]}(\mathbb{G}_m, \mathcal{F})$ and $\Phi_{1-z}(\mathcal{F})$ is established in [GGM85] and may be taken as a definition of vanishing cycles. Moreover, it is shown there, that the exact triangle

$$ (7) \qquad H^\bullet(i_1^! \mathcal{F}) \to H^\bullet_{(0,1]}(\mathbb{G}_m, \mathcal{F}) \to H^\bullet_{(0,1)}(\mathbb{G}_m, \mathcal{F}) \xrightarrow{+1} $$

associated with the pair $(0,1) \subset (0,1]$ in $\mathbb{G}_m$ is isomorphic to the cohomology of the Milnor triangle (2) for $\mathcal{F}$, see also [KS16, Remark 9.5]. $\square$

Above we constructed not only the class in $H^1(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F}[-1])$, but the corresponding extension. This extension obeys the natural condition, which essentially specify in which direction interval $I$ meets point 1, see Proposition 5 below.

*Remark 1.* Following [Kat91, Section 3], one may show that the dimension of $\Phi_{1-z}(\mathcal{F})$ coincides with the one of $H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F})$ by means of the exactness properties of push-forwards along affine morphisms of perverse sheaves, see [BBDG82, 4.1]. Indeed, they imply that $H^{>0}(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F}) = H^{>0}(\mathbb{P}^1 \setminus \{0\}, j_1^\infty \mathcal{F}) = 0$ and also $H^{<0}(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F}) = 0$, by the Verdier duality. Thus, dimension of $H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F})$ equals to the Euler characteristic of the cohomology, which can be shown to be equal to the dimension of $\Phi_{1-z}(\mathcal{F})$. This method does not suggest an explicit isomorphism between these spaces.

## 1.2. Nearby cycles.
For a perverse sheaf on $\mathbb{P}^1$, which is an extension of a shifted local system from $\mathbb{P}^1 \setminus \{0, \infty, 1\}$ by $*$, $*$, and $!$ in some order, Proposition 2 implies an isomorphism between vanishing cycles at points, to which the local system is extended by $*$. These vanishing cycles are isomorphic to nearby cycles, the isomorphism is given by *var* from (2). The following proposition reveals the nature of this isomorphism.

Denote by $j_1: \mathbb{G}_m \setminus \{1\} \to \mathbb{G}_m$ the open embedding and by

$$ (8) \qquad r: z \mapsto 1-z $$

the automorphism of $\mathbb{P}^1$.

**Proposition 3.** Let $\mathcal{L}$ be a local system on $\mathbb{G}_m \setminus \{1\}$. The composition of isomorphisms

$$ (9) \qquad \begin{aligned} \Psi_{1-z}(j_{1*}\mathcal{L}[1]) &\xleftarrow{\text{var}} \Phi_{1-z}(j_{1*}\mathcal{L}[1]) \xrightarrow{\varphi_I} H^0(\mathbb{P}^1, j_{\infty!}j_{0*}j_{1*}\mathcal{L}[1]) \\ \Psi_z(j_{1*}\mathcal{L}[1]) &\xleftarrow{\text{var}} \Phi_z(j_{1*}\mathcal{L}[1]) \xrightarrow{\varphi_I} H^0(\mathbb{P}^1, j_{\infty!}j_{1*}j_{0*}\mathcal{L}[1]) \end{aligned} $$

is the holonomy along $I$.

*Proof.* The statement follows immediately from the proof of Proposition 2: for $j_{1*}\mathcal{L}[1]$ the isomorphism $\phi_I$ identify with sections of $i_I^!\mathcal{L}[1]$ over the set of inner
## 2. CONVOLUTION

**2.1. Moduli spaces.** Denote by $\mathcal{M}_{0,n}$ the moduli space of embeddings of an $n$-element set $\underline{n} \hookrightarrow \mathbb{P}^1$ to the complex projective line considered up to the action of the Möbius group. This is a smooth affine variety, and it has a smooth projective compactification $\overline{\mathcal{M}}_{0,n}$, which is the moduli space of stable curves. The complement $\overline{\mathcal{M}}_{0,n} \setminus \mathcal{M}_{0,n}$ is the union of normal crossing divisors. These divisors are numerated by partitions of $S$ in two subsets with cardinalities more than 2. Denote by $p_i: \mathcal{M}_{0,n} \to \mathcal{M}_{0,n-1}$ and $p_i: \overline{\mathcal{M}}_{0,n} \to \overline{\mathcal{M}}_{0,n-1}$ the projection, forgetting the $i$-th point.

Introduce on $\mathbb{P}^1$ coordinates such that coordinates of first three elements of $\underline{n}$ are 0, $\infty$ and 1 correspondingly. Coordinates of points labeled by the rest of elements of the finite set are called simplicial coordinates on $\mathcal{M}_{0,n}$. Thus, a point of $\mathcal{M}_{0,n+3}$ with simplicial coordinates $(t_1, \dots, t_n)$ is $(0, \infty, 1, t_1, \dots, t_n)$.

Simplicial coordinates identify $\mathcal{M}_{0,4}$ with $\mathbb{P}^1 \setminus \{0, \infty, 1\}$, and $\mathcal{M}_{0,5}$ with $\mathcal{M}_{0,4} \times \mathcal{M}_{0,4} \setminus \Delta$, where $\Delta$ is the diagonal. The projections from $\mathcal{M}_{0,5}$ to $\mathcal{M}_{0,4}$ look in simplicial coordinates as follows:

$$ (13) \qquad \begin{aligned} p_3(0, \infty, 1, t_1, t_2) &= (0, \infty, 1, t_2/t_1) \\ p_4(0, \infty, 1, t_1, t_2) &= (0, \infty, 1, t_2) \\ p_5(0, \infty, 1, t_1, t_2) &= (0, \infty, 1, t_1) \end{aligned} $$

Space $\overline{\mathcal{M}}_{0,4}$ is the obvious compactification of $\mathcal{M}_{0,4}$, which is $\mathbb{P}^1$. $\overline{\mathcal{M}}_{0,5}$ is isomorphic to $\overline{\mathcal{M}}_{0,4} \times \overline{\mathcal{M}}_{0,4}$ with blown-up points $(0,0)$, $(1,1)$ and $(\infty, \infty)$.

Denote by $\overline{\mathcal{M}}'_{0,5}$ the space $\overline{\mathcal{M}}_{0,4} \times \overline{\mathcal{M}}_{0,4}$ with blown-up points $(0,0)$ and $(\infty, \infty)$. In other words, $\overline{\mathcal{M}}'_{0,5}$ is $\overline{\mathcal{M}}_{0,5}$ with the blown-down line, corresponding to singular stable curves of type (12)(345). Denote by

$$ (14) \qquad p'_i: \overline{\mathcal{M}}'_{0,5} \to \overline{\mathcal{M}}_{0,4} $$

the forgetful maps, as in (13).

**2.2. Convolution.** Let $m: \mathbb{G}_m \times \mathbb{G}_m \to \mathbb{G}_m$ be the product map of the multiplicative group. The convolution of two perverse sheaves $\mathcal{E}$ and $\mathcal{F}$ on $\mathbb{G}_m$ is a perverse sheaf on $\mathbb{G}_m$ defined by

$$ (15) \qquad \mathcal{E}^{*'} \mathcal{F} = \tau^{\le 0} \mathbf{R} m_!(\mathcal{E} \boxtimes \mathcal{F}) $$

Note that because map $m$ is affine, the convolution of perverse sheaves is a perverse sheaf due to exactness property of affine morphisms, see [BBDG82, 4.1].

**Remark 4.** Generally, $\mathbf{R} m_!(\mathcal{E} \boxtimes \mathcal{F})$ is a complex of perverse sheaves rather than a perverse sheaf, that is why we add $\tau^{\le 0}$ in the definition. One may avoid it, if, following [Ter06, TD05, Del01], he works in the quotient category as in Theorem 1 because, as it follows from the proof of Proposition 7, the higher terms are smooth at 1.

Let $\iota: \mathbb{P}^1 \to \mathbb{P}^1$ be the inversion map $z \mapsto z^{-1}$. For a sheaf $\mathcal{F}$ on $\mathbb{P}^1$ introduce the notation

$$ \overline{\mathcal{F}} = \iota^* \mathcal{F} $$

Recall, that in Section 1 we denoted by $j_{\infty!}j_{0*}$ the $*$-extension at 0 and $!$-extension at $\infty$ of a perverse sheaf from $\mathbb{G}_m$ to $\mathbb{P}^1$.
**Proposition 6** ([Del01, Kat12]). For perverse sheaves $\mathcal{E}$ and $\mathcal{F}$ on $\mathbb{G}_m$ we have

$$ (16) \qquad j_{\infty!j_0*}(\mathcal{E} *^! \mathcal{F}) = \tau^{\le 0} \mathbf{R} p'_{3*}(p'_{5}*(\overline{j_{\infty!j_0*}\mathcal{E}}) \otimes p'_{4}*(j_{\infty!j_0*}\mathcal{F})) $$

where $p'_i$ are defined in (14).

*Proof.* The proof proceeds in two steps.

Firstly, one needs to build an isomorphism

$$ \mathcal{E} *^! \mathcal{F} \cong \tau^{\le 0} \mathbf{R} p'_{3*}(p'_{5}*(\overline{j_{\infty!j_0*}\mathcal{E}}) \otimes p'_{4}*(j_{\infty!j_0*}\mathcal{F})) $$

on $\mathbb{G}_m \setminus \{1\}$. By (13), on fibers of the projection $p_3$, sheaves $p'_5*(\overline{j_{\infty!j_0*}\mathcal{E}}) \otimes p'_4*(\overline{j_{\infty!j_0*}\mathcal{F}})$ and $\mathcal{E} \boxtimes \mathcal{F}$ are isomorphic out of 0 and $\infty$. Then one may see that the former is the !-extension the latter, it implies that its *-pushforward equals to !-pushforward of the latter.

Secondly, one need to show that $\tau^{\le 0} \mathbf{R} p'_{3*}(p'_{5}*(\overline{j_{\infty!j_0*}\mathcal{E}}) \otimes p'_{4}*(\overline{j_{\infty!j_0*}\mathcal{F}}))$ is *-extension to 0 and !-extension to $\infty$ of $\mathcal{E} *^! \mathcal{F}$. Consider the fiber of $p'_3$ over 0. It consists of two intersecting lines, and $p'_5*(\overline{j_{\infty!j_0*}\mathcal{E}}) \otimes p'_{4}*(\overline{j_{\infty!j_0*}\mathcal{F}})$ is *-extension of $\mathcal{E} \boxtimes \mathcal{F}$ to these lines. Thus, it is true for the pushforward as well. Over $\infty$ the situation is the same. $\square$

**Proposition 7** ([Del01, Kat12]). For perverse sheaves $\mathcal{E}, \mathcal{F} \in \mathrm{Perv}(\mathbb{G}_m, 1)$,

$$ (17) \qquad H^0(\mathbb{P}^1, j_{\infty!j_0*}(\mathcal{E} *^! \mathcal{F})) = H^0(\mathbb{P}^1, j_{\infty!j_0*} \mathcal{E}) \otimes H^0(\mathbb{P}^1, j_{\infty!j_0*} \mathcal{F}) $$

*Proof.* Consider the Leray spectral sequence for $\mathbf{R} p'_{3*}(p'_{5}*(\overline{j_{\infty!j_0*}\mathcal{E}}) \otimes p'_{4}*(\overline{j_{\infty!j_0*}\mathcal{F}}))$. One may see that cohomologies of the derived pushforward outside 1 is *-extension to 0 and !-extension to $\infty$ of a smooth object. By Proposition 2, it follows that only the zeroth row of the spectral sequence does not vanish, thus, it collapses at the second sheet. By the Künneth theorem, the cohomology of the derived pushforward equals to the cohomology of the right side of (17). Because the second sheet of the Leray spectral sequence consists of one row, these cohomologies are presented by 0th cohomology of the pushforward, thus, coincide with the one for $\tau^{\le 0} \mathbf{R} p'_{3*}(p'_{5}*(\overline{j_{\infty!j_0*}\mathcal{E}}) \otimes p'_{4}*(\overline{j_{\infty!j_0*}\mathcal{F}}))$. $\square$

**2.3. Vanishing cycles of the convolution.** Proposition 7 above and its proof shows that $H^0(\mathbb{P}^1, j_{\infty!j_0*} - )$ is a tensor functor on the category of perverse sheaves on $\mathbb{G}_m$ with respect to the tensor structure given by convolution. By Proposition 2, this functor is isomorphic to vanishing cycles at 1. This functor is not faithful, so it is not a fiber functor. It may be fixed by taking an appropriate quotient of $\mathrm{Perv}(\mathbb{G}_m, 1)$.

The following theorem is proved in [Del01, TD05]. We will not need it and cite it for completeness.

**Theorem 1** ([Ter06, TD05, Del01]). Consider the subcategory of $\mathrm{Perv}(\mathbb{G}_m, 1)$ formed by perverse sheaves with nilpotent monodromy and take the Serre quotient of it by the subcategory of perverse sheaves smooth on $\mathbb{G}_m$. This is a Tannakian category with the fiber functor given by vanishing cycles at 1 and the tensor product given by the multiplicative convolution.

*Remark 5.* The category $\mathrm{Perv}_0$ introduced in [Kat91, 12.6] is closely related with the above construction, it gives a splitting of the analogous quotient, but for the category of perverse sheaves on the affine line with the additive convolution as a tensor product.
### 3. SINGULAR FIBERS

**3.1. Sheaves without sections with support at 1.** We say that a perverse sheaf $\mathcal{F} \in \mathrm{Perv}(\mathbb{G}_m, 1)$ has no sections with support at 1 if $H^0(i_1^!(\mathcal{F})) = 0$. Equivalently, sheaf has no sections with support at 1 if the map $\mathrm{var}: \Phi_{1-z}(\mathcal{F}) \to \Psi_{1-z}(\mathcal{F})$ induced by (2) at point 1 is injective. Two main examples are the $*$-extension and the middle extension of a smooth sheaf on $\mathbb{G}_m \setminus \{1\}$. An important feature of such sheaves is that for them (7) defines a two-term complex, representing vanishing cycles.

**Proposition 8.** If one of perverse sheaves $\mathcal{E}, \mathcal{F} \in \mathrm{Perv}(\mathbb{G}_m, 1)$ has no sections with support at 1, then the convolution $\mathcal{E} *! \mathcal{F}$ has no sections with support at 1 as well.

*Proof.* Calculate $i_1^!(\mathcal{E} *! \mathcal{F})$ by means of the base change for the map $p_3'$. Let $f: p_3'^{-1}(1) \to \overline{\mathcal{M}}'_{0,5}$ be the embedding of the fiber. We need to calculate

$$H^0(p_3'^{-1}(1), f^!(p_5'^* (\overline{j_{\infty!}j_{0*} \mathcal{E}}) \otimes p_4'^* (j_{\infty!}j_{0*} \mathcal{F})))$$

The fiber $p_3'^{-1}(1)$ is the projective line. Denote by $s$ the embedding of point $(0, \infty, 1, 1, 1)$ in this line, and by $t$ the embedding of the compliment. Denote $p_5^*(\overline{j_{\infty!}j_{0*} \mathcal{E}}) \otimes p_4^*(j_{\infty!}j_{0*} \mathcal{F})$ by $\mathcal{A}$ and consider the standard triangle

$$s_* s' f' \mathcal{A} \longrightarrow f' \mathcal{A} \longrightarrow t_* t^* f' \mathcal{A} \xrightarrow{+1}$$

Taking cohomology, we get

$$(18) \qquad H^0(s'f'\mathcal{A}) \longrightarrow H^0(\mathbb{P}^1, f'\mathcal{A}) \longrightarrow H^0(\mathbb{P}^1, t_*t^*f'\mathcal{A}) \xrightarrow{+1}$$

where $\mathbb{P}^1$ is $p_3'^{-1}(1)$. The first term is equal to $H^0(i_1^! \mathcal{E}) \otimes H^0(i_1^! \mathcal{F})$ by the Künneth theorem and vanishes by the condition of the proposition. One may see that $t_*t^*f'\mathcal{A}$ is isomorphic to $j_{\infty!}j_{0*}j_1^*j_1^*(\overline{\mathcal{E}} \otimes \mathcal{F})[-2]$, where $j_1$ is $\mathbb{G}_m \setminus \{1\} \hookrightarrow \mathbb{G}_m$. One may see that $i$-th cohomology of the latter sheaf vanishes for $i \neq 1$, applying Proposition 2 to $j_1^*j_1^*(\overline{\mathcal{E}} \otimes \mathcal{F})[-1]$. It implies that the third term in (18) vanishes, thus the middle term vanishes too. $\square$

**3.2. Nearby cycles of convolution at 0 via the Verdier specialization.** For a perverse sheaf from $\mathrm{Perv}(\mathbb{G}_m, 1)$ without sections with support at 1, one may think about vanishing cycles at 1 as a subspace of nearby cycles at 1. For the convolution of sheaves without sections with support at 1, one may define a natural embedding from the tensor product of vanishing cycles of factors $\Phi_{1-z}(\mathcal{E}) \otimes \Phi_{1-z}(\mathcal{F})$ to the nearby cycles at 0 of the convolution $\Psi_z(\mathcal{E} *! \mathcal{F})$ in the following way.

Consider the fiber of $p_3'$ (or, equivalently, $p_3$) over 0. This is the nodal stable curve, consisting of two components. The convolution is the shifted pushforward under $p_3'$ of the sheaf $p_5^*(\overline{j_{\infty!}j_{0*} \mathcal{E}}) \otimes p_4^*(j_{\infty!}j_{0*} \mathcal{F}))$. To calculate nearby cycles of the pushforward of $p_5^*(\overline{j_{\infty!}j_{0*} \mathcal{E}}) \otimes p_4^*(j_{\infty!}j_{0*} \mathcal{F})$, calculate the nearby cycles $\psi_z(p_5^*(\overline{j_{\infty!}j_{0*} \mathcal{E}}) \otimes p_4^*(j_{\infty!}j_{0*} \mathcal{F}))$ on the singular fiber, and then take the pushforward.

Let in

$$(19) \qquad s: * \to p_3^{-1}(0) \leftarrow \mathbb{A}^1 \cup \mathbb{A}^1 : t_{1,2}$$
s be the embedding of the singular point, corresponding to the stable curve of type (15)(3)(24), and $t_i$ be the embedding of the $i$th complement. The first component, corresponding to stables curves of types (15)(234) and the second one — to (135)(24).

Introduce coordinates on the compactification of the first and the second component as follows (see the left part of the picture at the end of the text):

$$ (0, \infty, 1) = ((15)(4)(23), (15)(3)(24), (15)(2)(34)) $$

and:

$$ (0, \infty, 1) = ((15)(3)(24), (13)(5)(24), (35)(1)(24)) $$

Denote by $\mathcal{E}_0$ the shifted vector bundle on $\mathbb{G}_m$ with fiber $\Psi_{az}(\mathcal{E})[1]$ at point $a$ and by $\mathcal{F}_0$ the shifted vector bundle on $\mathbb{G}_m$ with fiber $\Psi_{az}(\mathcal{F})[1]$ at point $a$. One may see that $\mathcal{E}_0$ and $\mathcal{F}_0$ are isomorphic to restriction on $\mathbb{G}_m$ of Verdier specializations ([Ver83]) of $\mathcal{E}$ and $\mathcal{F}$ along 0 after identification of the tangent space to $\mathbb{P}^1$ at 0 with $\mathbb{A}^1 = \mathbb{P}^1 \setminus \infty$ in the natural way.

Denote $p_5^{\prime *}(\overline{j_{\infty!}j_{0*}\mathcal{E}}) \otimes p_4^{\prime *}(j_{\infty!j_{0*}\mathcal{F}})$ by $\mathcal{A}$.

One may see that on the first component in coordinates as above $p_5^{\prime *}(\overline{j_{\infty!}j_{0*}\mathcal{E}})$ is isomorphic to the restriction of $\overline{j_{\infty!}j_{0*}\mathcal{E}}$ and nearby cycles of $p_4^{\prime *}(j_{\infty!j_{0*}\mathcal{F}})$ along $z$ is isomorphic to $j_{\infty!\mathcal{F}}$. Since nearby cycles of tensor product is isomorphic to the tensor product of nearby cycles, we get

$$ (20) \qquad t_{1*}t_1^* \psi_z(p_5^{\prime *}(\overline{j_{\infty!}j_{0*}\mathcal{E}}) \otimes p_4^{\prime *}(j_{\infty!j_{0*}\mathcal{F}})) = \overline{j_{\infty!}j_{0*}(\mathcal{E} \otimes \mathcal{F}_0)} $$

and, analogously,

$$ (21) \qquad t_{2*}t_2^* \psi_z(p_5^{\prime *}(\overline{j_{\infty!}j_{0*}\mathcal{E}}) \otimes p_4^{\prime *}(j_{\infty!j_{0*}\mathcal{F}})) = \overline{j_{\infty!}j_{0*}(\mathcal{E}_0 \otimes \mathcal{F})} $$

where we identify $\mathbb{P}^1$, where right-hand sides of (20) and (21) are defined, with closures of our components.

Consider the standard triangle associated with stratification (19):

$$ s_*s' \psi_z(\mathcal{A}) \longrightarrow \psi_z(\mathcal{A}) \longrightarrow t_*t^* \psi_z(\mathcal{A}) \xrightarrow{+1} $$

Thus, $\psi_z(\mathcal{A})$ may be presented as

$$ (22) \qquad \mathrm{cone}(t_*t^* \psi_z(\mathcal{A}) \to s_*s' \psi_z(\mathcal{A})[1]), $$

where

$$ (23) \qquad t_*t^* \psi_z(\mathcal{A}) = t_{1*}t_1^* \psi_z(\mathcal{A}) \oplus t_{2*}t_2^* \psi_z(\mathcal{A}) $$

Recall that we want to build a map from $\Phi_{1-z}(\mathcal{E}) \otimes \Phi_{1-z}(\mathcal{F})$ to $\Psi_z(\mathcal{E} *^1 \mathcal{F})$. To do it, we will build extensions of $\mathbb{Z}$ by $t_{1*}t_1^* \psi_z(\mathcal{A})$ and $t_{2*}t_2^* \psi_z(\mathcal{A})$ and ensure that their projections on $s_*s' \psi_z(\mathcal{A})[1]$ in (22) coincide.

Combining (20) and (21) with Proposition 2, we get

$$ (24) \qquad \begin{aligned} H^{-1}(\mathbb{P}^1, t_{1*}t_1^* \psi_z(\mathcal{A})) &= \Phi_{1-z}(\mathcal{E} \otimes \mathcal{F}_0) = \Phi_{1-z}(\mathcal{E}) \otimes \Psi_z(\mathcal{F}) \\ H^{-1}(\mathbb{P}^1, t_{2*}t_2^* \psi_z(\mathcal{A})) &= \Phi_{1-z}(\mathcal{E}_0 \otimes \mathcal{F}) = \Psi_z(\mathcal{E}) \otimes \Phi_{1-z}(\mathcal{F}) \end{aligned} $$

where the second equalities are due to the fact that $\mathcal{E}_0$ and $\mathcal{F}_0$ are smooth at 1 and the corresponding nearby cycles are their shifted fibers there.

Given $u \otimes v \in \Phi_{1-z}(\mathcal{E}) \otimes \Phi_{1-z}(\mathcal{F})$, using isomorphisms (24) we get elements

$$ (25) \qquad \begin{aligned} v \otimes (\mathrm{Hol}_I(\mathrm{var}(u))) &\in \Phi_{1-z}(\mathcal{E}) \otimes \Psi_z(\mathcal{F}) = H^{-1}(\mathbb{P}^1, t_{1*}t_1^* \psi_z(\mathcal{A})) \\ (\mathrm{Hol}_I(\mathrm{var}(v))) \otimes u &\in \Psi_z(\mathcal{E}) \otimes \Phi_{1-z}(\mathcal{F}) = H^{-1}(\mathbb{P}^1, t_{2*}t_2^* \psi_z(\mathcal{A})) \end{aligned} $$
where isomorphisms $\text{Hol}_I: \Psi_{1-z}(-) \to \Psi_z(-)$ are as in Proposition 3. Taking the difference between the element given by the first and second lines of (25) and taking into account (23) we get an element of $H^{-1}(t_*t^* \psi_z(\mathcal{A}))$.

To fulfill the construction, we need to check that projections on $s_*s^! \psi_z(\mathcal{A})[1]$ in (22) of extensions given by the first and second lines of (25) coincide. To do it, on need to notice, that these projections are given by the second arrow of the triangle (12) and apply Proposition 5.

**Proposition 9.** The map from $\Phi_{1-z}(\mathcal{E}) \otimes \Phi_{1-z}(\mathcal{F})$ to $\Psi_z(\mathcal{E} *^! \mathcal{F})$ defined above is the composition of maps

$$ (26) \qquad \begin{aligned} & \Phi(\mathcal{E}) \otimes \Phi(\mathcal{F}) \xrightarrow{\varphi_I \otimes \varphi_I} H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F}) \otimes H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{E}) \xrightarrow{\text{Prop.}7} H^0(\mathbb{P}^1, j_{\infty!}j_{0*}(\mathcal{E} *^! \mathcal{F})) \\ & \Psi_z(\mathcal{E} *^! \mathcal{F}) \leftarrow_{\substack{\text{var}_0 \\ \sim}} \Phi_z(j_{1*}j_1^*(\mathcal{E} *^! \mathcal{F})) \leftarrow_{\substack{\sim \\ \sim}} H^0(\mathbb{P}^1, j_{1*}j_1^*j_{\infty!}j_{0*}(\mathcal{E} *^! \mathcal{F})) \end{aligned} $$

where all arrows, except the vertical one, are isomorphisms, and the vertical arrow is the canonical map.

*Proof.* The statement follows directly from definitions and is left to the reader. $\square$

### 3.3. Nearby and vanishing cycles of convolution at 1 via the Verdier specialization.
Analyzing fibers of maps $p_3$ and $p'_3$ over 1 one may identify vanishing cycles of convolution with sections of the products of Verdier specializations of sheaves. Below, we calculate vanishing cycles at 1 and its image in nearby cycles of convolution of perverse sheaves from $\text{Perv}(\mathbb{G}_m, 1)$ without section with support at 1.

The fiber $p_3^{-1}(1)$ is the nodal stable curve consisting of two components. Let in (27) $s: * \to p_3^{-1}(1) \leftarrow \mathbb{A}^1 \cup \mathbb{A}^1: t \cup t'$

$s$ be the embedding of the singular point, corresponding to the stable curve of type (12)(3)(45), $t$ be the embedding of the smooth part of the component (123)(45), and $t'$ be the embedding of the smooth part of the component (12)(345).

Introduce the coordinates on the compactification of the first and the second component as follows (see the right part of the picture at the end of the text):

$$ (0, \infty, 1) = ((13)(2)(45), (23)(1)(45), (12)(3)(45)) $$

and:

$$ (0, \infty, 1) = ((12)(4)(35), (12)(3)(45), (12)(5)(34)) $$

The fiber $p_3'^{-1}(1)$ is the projective line. The natural projection $p_3^{-1}(1) \to p_3'^{-1}(1)$ blows down the component (12)(345) and is an isomorphism on another. Keep notations $s$ and $t$ for the embeddings of point (12)(3)(45) and its complement in $p_3'^{-1}(1)$:

$$ (28) \qquad s: * \to p_3'^{-1}(1) \leftarrow \mathbb{A}^1: t $$

Denote $p_5^*(\overline{j_{\infty!}j_{0*} \mathcal{E}}) \otimes p_4^*(j_{\infty!}j_{0*} \mathcal{F})$ also by $\mathcal{A}$.

As above, for sheaves from $\text{Perv}(\mathbb{G}_m, 1)$ without section with support at 1, we consider vanishing cycles at 1 as a subspace of nearby cycles. Thus, as it follows from the Milnor triangle (2), to build an element from $\Phi_{1-z}(\mathcal{E} *^! \mathcal{F})$ one need to build an element from $\Psi_{1-z}(\mathcal{E} *^! \mathcal{F}) = H^{-1}(p_3'^{-1}(1), \psi_{1-z}(\mathcal{A}))$, which vanishes under the
connecting morphism of the cohomology of the Milnor triangle to $H^1(i'_1(\mathcal{E} *' \mathcal{F}))$. We claim that if $\mathcal{E}$ or $\mathcal{F}$ has no sections with support at 1, for this purpose one can take $v \in H^{-1}(p'_3^{-1}(1), \psi_{1-z}(\mathcal{A}))$ such that

$$ (29) \qquad H^{-1}(p'_3^{-1}(1), t^* \psi_{1-z}(\mathcal{A})) \ni t^* v = 0 $$

Indeed, let $f: p'_3^{-1}(1) \to \overline{M}'_{0,5}$ be the embedding and consider the triangle associated with stratification (28) for $f^!\mathcal{A}$:

$$ s_* s' f^! \mathcal{A} \longrightarrow f^! \mathcal{A} \longrightarrow t_* t^* f^! \mathcal{A} \xrightarrow{+1} $$

Calculate its cohomology:

$$ \begin{array}{ccc} H^0(s^!f^!\mathcal{A}) & \longrightarrow & H^0(p'_3^{-1}(1), f^!\mathcal{A}) \longrightarrow H^0(\mathbb{A}^1, t^!f^!\mathcal{A}) \\ \parallel & & \parallel \\ H^0(i'_1\mathcal{E}) \otimes H^0(i'_1\mathcal{F}) & & H^{-1}(\mathbb{A}^1, t^*\psi_{1-z}(\mathcal{A})) \end{array} $$

Here, the left vertical isomorphism is the Künneth theorem and this term vanishes, because $\mathcal{E}$ or $\mathcal{F}$ has no sections with support at 1. The right isomorphism follows from the fact that $\mathcal{A}$ is smooth in the neighborhood of the image of $t$, hence $\phi_{1-z}(\mathcal{A})$ vanishes there. Consider the image of $v$ under the connecting isomorphism in the middle term of (3.3). Its image under the right arrow vanishes, because it is equal to $t^*v$, which vanishes by (29). The left term is zero, which implies that the image is zero itself.

In order to build an element satisfying condition (29), express nearby cycles of $\mathcal{E} *' \mathcal{F}$ in terms of the special fiber of the projection $p_3$ rather than $p'_3$. Denote by $\bar{t}: \mathbb{P}^1 \to p'_3^{-1}$ the closed embedding of the closure of the image of $t$. Consider the stratification

$$ (30) \qquad \bar{t}: \mathbb{P}^1 \to p'_3^{-1}(1) \leftarrow \mathbb{A}^1 : t' $$

and the corresponding triangle for nearby cycles on $p'_3^{-1}(1)$

$$ (31) \qquad t'_1 t'^* \psi_{1-z}(\mathcal{A}) \longrightarrow \psi_{1-z}(\mathcal{A}) \longrightarrow \bar{t}_* \bar{t}^* \psi_z(\mathcal{A}) \xrightarrow{+1} $$

For an element of $H^{-1}(\mathbb{P}^1, t'_1 t'^* \psi_{1-z}(\mathcal{A}))$, where $\mathbb{P}^1$ denotes the closure of image of $t'$, consider its image under the first arrow in (31). Since cohomology of nearby cycles on $p'_3^{-1}(1)$ and $p'_3^{-1}(1)$ coincide and give $\Psi_{1-z}(\mathcal{E} *' \mathcal{F})$, the image gives an element of $H^{-1}(p'_3^{-1}(1), \psi_{1-z}(\mathcal{A}))$. We claim that this element satisfies condition (29). Indeed, one may see that $t^* \psi_{1-z}(\mathcal{A})$ on both $p'_3^{-1}(1)$ and $p'_3^{-1}(1)$ are canonically isomorphic. But $t^*$ on $p'_3^{-1}(1)$ factors through $\bar{t}^*$, which annihilates the constructed element due to the exactness of (31). Thus, an element from $H^{-1}(\mathbb{P}^1, t'_1 t'^* \psi_{1-z}(\mathcal{A}))$ produces an element in nearby cycles of convolution, which may be lifted to vanishing cycles.

**Proposition 10.** For $\mathcal{E}$ and $\mathcal{F}$ from Perv($\mathbb{G}_m, 1$) without section with support at 1, the defined above map to vanishing cycles of $\mathcal{E} *' \mathcal{F}$ lifted from the map to nearby cycles

$$ (32) \qquad H^{-1}(\mathbb{P}^1, t'_1 t'^* \psi_{1-z}(p_5^*(\overline{j_{\infty}!j_{0*}\mathcal{E}}) \otimes p_4^*(j_{\infty}!j_{0*}\mathcal{F}))) \\ \Phi_{1-z}(\mathcal{E} *' \mathcal{F}) \xleftarrow{\text{var}} \Psi_{1-z}(\mathcal{E} *' \mathcal{F}) $$

is an embedding.
**Proof.** The statement follows directly from definitions and is left to the reader. □

Now calculate the domain of the map from the proposition above.

For $\mathcal{E}, \mathcal{F} \in \text{Perv}(\mathbb{G}_m, 1)$ denote by $\mathcal{E}_1$ and $\mathcal{F}_1$ the shifted vector bundles on $\mathbb{A}^1$ which are Verdier specializations ([Ver83]) of $\mathcal{E}$ and $\mathcal{F}$ along point 1 after identification of the tangent space to $\mathbb{P}^1$ at 1 with $\mathbb{A}^1 = \mathbb{P}^1 \setminus \infty$ in the natural way.

One may see that in coordinates as above $t^* \psi_{1-z}(p_5^*(\overline{j_\infty!j_{0*} \mathcal{E}}))$ is isomorphic to $r^* \mathcal{E}_1$ and $t^* \psi_{1-z}(p_4^*(j_\infty!j_{0*} \mathcal{F}))$ is isomorphic to $\mathcal{F}_1$, where $r$ is defined in (8). Thus,

$$ (33) \qquad t^* \psi_{1-z}(\mathcal{A}) = r^* \mathcal{E}_1 \otimes \mathcal{F}_1 $$

**Proposition 11.** For $\mathcal{E}$ and $\mathcal{F}$ from $\text{Perv}(\mathbb{G}_m, 1)$ without section with support at 1,

$$ (34) \qquad H^0(\mathbb{P}^1, j_1^\infty (r^* \mathcal{E}_1 \otimes \mathcal{F}_1)[-1]) = \Phi_{1-z}(\mathcal{E}) \otimes \Phi_{1-z}(\mathcal{F}) $$

**Proof.** Apply Proposition 1 to calculate the left-hand side of (34). The standard triangle for the local cohomology on $I$ associated with the stratification $(\{0,1\}, I)$ presents it as a complex in the derived category, that is a cone of

$$ (35) \qquad \Psi(\mathcal{E}_1) \otimes \Psi(\mathcal{F}_1) \to (H^\bullet(i_0^! \mathcal{E}_1)[2] \otimes \Psi(\mathcal{F}_1)) \oplus (\Psi(\mathcal{E}_1) \otimes H^\bullet(i_1^! \mathcal{F}_1)[2]) $$

Here, we use the identification of the cohomology the Milnor triangle (2) with the complex given by stratification mentioned in the proof of Proposition 2. Analogously, applying the canonical isomorphisms between nearby and vanishing cycles of perverse sheaves and their Verdier specializations, one may present the right-hand side of (34) as a product of complexes

$$ \Psi(\mathcal{E}_1) \to H^\bullet(i_0^! \mathcal{E}_1)[2] \quad \text{and} \quad \Psi(\mathcal{F}_1) \to H^\bullet(i_1^! \mathcal{F}_1)[2] $$

One may see that there is a natural map from this tensor product to (35), and the cone of this map is $H^\bullet(i_0^! \mathcal{E})[2] \otimes H^\bullet(i_1^! \mathcal{F})[2]$. It follows that this map induces an isomorphism on the 0th cohomology, because $\mathcal{E}$ and $\mathcal{F}$ have no section with support at 1. □

Combining Propositions 10 and 11 and (33), we get for perverse sheaves $\mathcal{E}, \mathcal{F} \in \text{Perv}(\mathbb{G}_m, 1)$ without section with support at 1 an isomorphism

$$ (36) \qquad \Phi_{1-z}(\mathcal{E}) \otimes \Phi_{1-z}(\mathcal{F}) = \Phi_{1-z}(\mathcal{E}^*! \mathcal{F}) $$

**Remark 6.** The left-hand side of (34) is the fiber at 1 of the additive convolution of Verdier specializations $\mathcal{E}_1$ and $\mathcal{F}_1$, which is isomorphic to the nearby cycles of the additive convolution, which in turn is isomorphic to vanishing cycles under condition $H^0(i_0^! \mathcal{E}_1) = H^0(i_1^! \mathcal{F}_1) = 0$, compare with the end of the first letter [Del01].

## 4. VANISHING CYCLES AND POLYGONS

### 4.1. Graphical representation.
Isomorphism $\phi_I$ between vanishing cycles $\Phi_{1-z}(\mathcal{F})$ and cohomology $H^0(\mathbb{P}^1, j_\infty!j_{0*} \mathcal{F})$ for $\mathcal{F} \in \text{Perv}(\mathbb{G}_m, 1)$ supplied by Proposition 2 may be realized as a graphical representation of vanishing cycles in the following sense. One associates a cocycle in $H^0(I, i_I^! \mathcal{F})$ with vanishing cycle and then takes the pushforward to $\mathbb{P}^1$ of it. Thus, one may think about interval $I$ and the cocycle
of the local cohomology with support on it as a graphical representation of the vanishing cycle.

Let us construct the graphical presentation of vanishing cycles of the convolution. To be more precise, we will construct its image in nearby cycles.

For two perverse sheaves $\mathcal{E}, \mathcal{F} \in \mathrm{Perv}(\mathbb{G}_m, 1)$ take the external product of the corresponding cocycles in $H_I^0(\mathbb{P}^1, -)$. It gives a cocycle with support on the square in $\mathbb{G}_m \times \mathbb{G}_m$. The projection of this square under multiplication is $I$. The preimage of this projection over an inner point of $I$ is the interval $[\lambda_1, \lambda_2]$ for $0 < \lambda_1 < \lambda_2 < 1$. The cocycle representing the cohomology with support on this interval, which we identify with $I$, is presented by the section of the restriction of $\mathcal{E} \boxtimes \mathcal{F}$ on this interval, which is the product of restrictions of factors. Note that restrictions of this product at the ends of this interval vanish.

The picture at the end of the text could be helpful at this point, one should imagine that the right vertical side of the pentagon-like gray figure is shrunk, making the pentagon in a quadrupole.

Thus, we get a section of the local system $\mathbb{R}^1 m!(\mathcal{E}[-1] \boxtimes \mathcal{F}[-1])$ over inner points of $I$, which is isomorphic to $\mathcal{E} *^! \mathcal{F}$ there. After the pushforward, we get a class in $H^0(\mathbb{P}^1, j_{1*}j_1^*(\mathcal{E} *^! \mathcal{F}))$. Combining this morphism with $\phi_I$, we get a class in $\Phi_{1-z}(j_{1*}j_1^*(\mathcal{E} *^! \mathcal{F})) = \Psi_{1-z}(\mathcal{E} *^! \mathcal{F})$.

**Proposition 12.** For $\mathcal{E}$ and $\mathcal{F}$ from $\mathrm{Perv}(\mathbb{G}_m, 1)$ without section with support at 1, the defined above map

$$ (37) \qquad H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{E}) \otimes H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F}) \to \Psi_{1-z}(\mathcal{E} *^! \mathcal{F}) $$

factors through

$$ (38) \qquad \mathrm{var}: \Phi_{1-z}(\mathcal{E} *^! \mathcal{F}) \to \Psi_{1-z}(\mathcal{E} *^! \mathcal{F}) $$

and is an isomorphism on the image, which is $\Phi_{1-z}(\mathcal{E} *^! \mathcal{F})$.

*Proof.* By Proposition 7, the external product gives an isomorphism

$$ H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{E}) \otimes H^0(\mathbb{P}^1, j_{\infty!}j_{0*} \mathcal{F}) \xrightarrow{\sim} H^0(\mathbb{P}^1, j_{\infty!}j_{0*}(\mathcal{E} *^! \mathcal{F})) $$

By Proposition 2, map $\phi_I$ establishes an isomorphism between $H^0(\mathbb{P}^1, j_{\infty!}j_{0*}(\mathcal{E} *^! \mathcal{F}))$ and $\Phi_{1-z}(\mathcal{E} *^! \mathcal{F})$. Compose it with the composition

$$ (39) \qquad \Phi_{1-z}(\mathcal{E} *^! \mathcal{F}) \to \Phi_{1-z}(j_{1*}j_1^*(\mathcal{E} *^! \mathcal{F})) = \Psi_{1-z}(\mathcal{E} *^! \mathcal{F}) $$

where the arrow is induced by the canonical map.

$$ (40) \qquad \mathcal{E} *^! \mathcal{F} \to j_{1*}j_1^*(\mathcal{E} *^! \mathcal{F}) $$

By the very definition of $\phi_I$, this is the map constructed before the proposition. By Proposition 21, the composite map (39) is equal to $\mathrm{var}$. By Proposition 8, $\mathcal{E} *^! \mathcal{F}$ has no section with support at 1, thus $\mathrm{var}$ is injective. $\square$

**4.2. Compatibility.** Summarizing constructions from the previous section with the graphical presentation introduced above, we get the following result.
**Theorem 2.** For $\mathcal{E}, \mathcal{F} \in \text{Perv}(\mathbb{G}_m, 1)$ without sections with support at 1 the following diagram commutes

<figure><img src="image_3.png" /></figure>

where we denote by $\Phi_1(-)$, $\Phi_0(-)$ and $\Psi_0(-)$ vanishing cycles at 1 along $1-z$ and vanishing and nearby cycles at 0 along $z$ correspondingly.

*Proof.* Let us start with the right triangle of the diagram. The right isomorphism $\phi_I$ may be factored through

$$ \Phi_{1-z}(\mathcal{E}^{*'} \mathcal{F}) \to \Phi_{1-z}(j_1^*j_1^*(\mathcal{E}^{*'} \mathcal{F})) = \Psi_{1-z}(\mathcal{E}^{*'} \mathcal{F}) $$

as in Proposition 12. In that proposition, we present elements of as cohomology with support on $I$, that is as a section of $\mathcal{E}^{*'} \mathcal{F}$ over $I$. By the very definition of $\phi_I$ in Proposition 2 via [GGM85], the image of such a class in $\Psi_1(\mathcal{E}^{*'} \mathcal{F})$ is the limit of this section. If we use the blown-up fiber over 1 as in Subsection 3.3, on the picture at the end of the text, this section is presented by the right vertical side of the gray pentagon-like figure. The limit of it is the bold line connecting 4 and 5 drawn on the figure presented, the blown-up fiber over 1. Thus, the class of nearby cycles where $\phi_I$ landed is exactly the class, constructed in Proposition 11 by means of the shrinking arguments from Proposition 1.

The left part of the diagram may be treated likewise. But this fact is essentially the content of Proposition 9. $\square$

The theorem gives a map from vanishing cycles of the convolution at 1 to its nearby cycles at 0. The following proposition interprets this maps in terms of holonomy along $I$.

**Proposition 13.** In notations of Theorem 2 the following diagram commutes

$$ \begin{array}{ccc} (42) & \left\| \right\| & \Psi_0(\mathcal{E}^{*'} \mathcal{F}) \xrightarrow{\substack{(26) \\ \text{Hol}_I}} \Psi_1(\mathcal{E}^{*'} \mathcal{F}) \end{array} $$

where the top row is the bottom row of (41) and the isomorphism downstairs is the holonomy along $I$.

*Proof.* By the very construction preceding Proposition 12, images of maps from $\Phi_1(\mathcal{E}) \otimes \Phi_1(\mathcal{F})$ to nearby cycles coincide with the ones of $H^0(\mathbb{P}^1, j_1^*j_1^*(\mathcal{E}^{*'} \mathcal{F}))$ under isomorphisms $\phi_I$. Then, the statement is a corollary of Proposition 3. $\square$

The following picture illustrates the proposition above. The gray pentagon-like figure is the graphical representation of the cohomology of the convolution. It may be thought of as well as a trace of a graphical representation of a cohomology class in the fiber when transporting along $I$.
<figure><img src="image_4.png" /></figure>

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Email address: nikita.markarian@gmail.com

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