Topological computation of Stokes matrices of some weighted projective lines

By mirror symmetry, the quantum connection of a weighted projective line is closely related to the localized Fourier–Laplace transform of some Gauß–Manin system. Following an article of D’Agnolo, Hien, Morando, and Sabbah, we compute the Stokes matrices for the latter at ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document} for the cases P(1,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}(1,3)$$\end{document} and P(2,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}(2,2)$$\end{document} by purely topological methods. We compare them to the Gram matrix of the Euler–Poincaré pairing on Db(Coh(P(1,3)))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^b(\mathrm{Coh}({\mathbb {P}}(1,3)))$$\end{document} and Db(Coh(P(2,2)))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^b(\mathrm{Coh}({\mathbb {P}}(2,2)))$$\end{document}, respectively. This article is based on the doctoral thesis of the author.


Introduction
In [6], D'Agnolo et al. describe how to compute the Stokes matrices of the enhanced Fourier-Sato transform of a perverse sheaf on the affine line by purely topological methods. To a regular singular holonomic D-module M ∈ Mod rh D A 1 on the affine line, one associates a perverse sheaf via the regular Riemann-Hilbert correspondence Let ⊂ A 1 denote the set of singularities of M. Following [6,Sect. 4.2], after suitably choosing a total order on , the resulting perverse sheaf F ∈ Perv C A 1 can be described by linear algebra data, namely its quiver where (F) and σ (F) are finite dimensional C-vector spaces and u σ : (F) → σ (F) and v σ : σ (F) → (F) are linear maps such that 1 − u σ v σ is invertible for any σ . The main result in [6] is a determination of the Stokes matrices of the enhanced Fourier-Sato transform of F and therefore of the Fourier-Laplace transform of M in terms of the quiver of F. This result builds on the irregular Riemann-Hilbert correspondence of D'Agnolo and Kashiwara [7], which provides a topological description of holonomic D-modules. As proven by Kashiwara and Schapira [14], this correspondence intertwines the Fourier-Laplace with the (enhanced) Fourier-Sato transform.
Mirror symmetry connects the weighted projective line P(1, 3) with the Landau-Ginzburg model The quantum connection of P(1, 3) is closely related to the Fourier-Laplace transform of the Gauß-Manin system H 0 ( f O) of f . We compute that where denotes the set of singular values of f , is the perverse sheaf associated to H 0 ( f O) by the Riemann-Hilbert correspondence. In Sect. 1, we compute the localized Fourier-Laplace transform of H 0 ( f O). In Sect. 2, analogous to the examples in [6,Sect. 7], we carry out the topological computation of the Stokes matrices of the Fourier-Laplace transform of H 0 ( f O). In Sect. 3, we compare the Stokes matrix S β , that we obtained from our topological computations, to the Gram matrix of the Euler-Poincaré pairing on D b (Coh(P(1, 3))) with respect to a suitable full exceptional collection. Following Dubrovin's conjecture about the Stokes matrix of the quantum connection, proven for the weighted projective space P (ω 0 , . . . , ω n ) by Tanabé and Ueda in [19] and by Cruz Morales and van der Put in [5], they are known to be equivalent after appropriate modifications. We give the explicit braid of the braid group B 4 that deforms the Gram matrix into the Stokes matrix S β . Section 4 tackles the computations for the case of noncoprime parameters. In comparison to the case of coprime parameters, this requires a slightly modified approach. We compute the Stokes matrices of the Fourier-Laplace transform of the Gauß-Manin system of the Landau-Ginzburg model of P(2, 2) and set it into relation with the Gram matrix of the Euler-Poincaré pairing on D b (Coh(P(2, 2))).
This article is based on the doctoral thesis [18] of the author. The figures in Sects. 2 and 4 were mainly produced in SAGE. In the online version of this article, the figures are provided in color.

Gauß-Manin system and its Fourier-Laplace transform
Let X be affine and f a regular function f : X → A 1 on X . Denote by f (•) the direct image in the category of D-modules and by M := H 0 ( f O X ) ∈ Mod rh D A 1 the zeroth cohomology of the Gauß-Manin system of f . Following [9, Sect. 2.c], it is given by Denote by G := M[τ −1 ] the Fourier-Laplace transform of M, localized at τ = 0. It is given by G is endowed with a flat connection given as follows. For γ = k∈Z ω k θ k ∈ G, where n (X ) ω k = 0 for almost all k, the connection is given by (cf. [12, Definition 2.3.1]): It is known that (G, ∇) has a regular singularity at θ = ∞ and possibly an irregular one at θ = 0.
We now consider the Laurent polynomial f = x + x −3 ∈ C x, x −1 , being a regular function on the multiplicative group G m . For our computations we pass to the variable θ = τ −1 . We compute that for the given f , G is given by the free C θ, θ −1 -module In this basis, the connection is given by Via the cyclic vector m = (1, 0, 0, 0) t , we compute the relation and therefore associate the differential operator As it is well known, one can read the type of the singularities at 0 and ∞ from the Newton polygon in the sense of Ramis (cf. [15,Chapter V]). The Newton polygon in Fig. 1 confirms that P-and therefore system (1)-has the nonzero slope 1 and therefore is irregular singular at θ = 0 and regular singular at θ = ∞.

Topological computation of the Stokes matrices
We consider the Laurent polynomial f = x + x −3 : G m → A 1 . Its critical points are given by The critical values of f are given by The preimages of Since f is proper, we compute by the adjunction formula that This induces the following order on (cf. [6, Sect. 4]): In Fig. 4, the σ i are depicted in the following colors: • σ 1 : green, • σ 2 : red, • σ 3 : purple, • σ 4 : orange.
The blue area in Fig. 2 shows where f has real (resp. imaginary) part greater than or equal to 0. In Fig. 3, the preimage of the imaginary (resp. real) axis under f is plotted in blue (resp. red) color. We consider lines passing through the singular values with phase π 8 , as depicted in Fig. 4. The preimages of these lines are plotted in Fig. 5. We fix a base point e with (e) > (σ i ) for all i and denote its preimages by e 1 , e 2 , e 3 , e 4 , as depicted in Fig. 6. In the following, we adopt the notation of [6,Sect. 4]. The nearby and global nearby cycles of F are given by The exponential components at ∞ of the Fourier-Laplace transform of H 0 ( f O) are known to be of linear type with coefficients given by the σ i ∈ . The Stokes rays are therefore given by We consider loops γ σ i , starting at e and running around the singular value σ i in counterclockwise orientation, 1 as depicted in Fig. 6. We denote by γ j σ i the preimage of γ σ i starting at e j , j = 1, 2, 3, 4. The figure constitutes a rough drawing of the preimages of γ σ i . By taking into account the preimages of the different segments of the axes and the intersections of γ σ i with them, one recovers the γ j σ i as depicted in 1 Counterclockwise orientation since the imaginary part of α, β is positive.
In order to obtain the maps b σ i , we consider the half-lines σ i := σ i + αR ≥0 . We denote their preimages under f by { j σ i } j=1,2,3,4 , depending on which γ j σ i they intersect. We label the preimages of σ i by σ 1 i , σ 2 i , σ 3 i , as depicted in Fig. 7. By the derivation of the short exact sequence of quivers [6, (7.1.3)] and passing to Borel-Moore homology as described in [6, Lemma 5.3.1.(i)], b σ i is induced from the corresponding boundary value map from σ i to its origin σ i . Therefore, b σ i encodes which lift of σ i starts at which preimage of σ i . Namely, from Fig. 7 we read the following: Therefore, b σ 2 is the transpose of We obtain, in the ordered bases as the cokernels of the following diagrams: We identify the cokernels of b σ i in the following way:

Quantum connection
The quantum connection of a Fano variety (resp. an orbifold) X is a connection on the trivial vector bundle over P 1 with fiber H * (X, C) (resp. H * orb (X, C)), where z denotes the standard inhomogeneous coordinate at ∞. By [11, (2.2.1)], the quantum connection is the connection given by where the first term on the right hand side is ordinary differentiation, the second one is pointwise quantum multiplication by (−K X ), and the third one is the grading operator The quantum connection is regular singular at z = ∞ and irregular singular at z = 0. For the weighted projective line P (a, b), the orbifold cohomology ring is given by (cf. [16,Example 3.20]) H * orb (P(a, b), gcd(a, b) and m, n ∈ Z such that am + bn = d. The grading is given as For gcd(a, b) = 1, −K P(a,b) is given by the element [x a + y b ] ∈ H 1 orb (P(a, b), C). Taking into account that the grading is scaled by 2, the grading operator is defined by μ(a) = i − dim X 2 a for a ∈ H i orb (X, C). We obtain the quantum connection of P(1, 3) as follows. Therefore, the quantum connection of P(1, 3) is given by Observation. By the gauge transformation h = diag(θ − 1 2 , θ − 1 2 , θ − 1 2 , θ − 1 2 ), which subtracts 1 2 on the diagonal entries, and passing to −θ , connection (1) arising from the Landau-Ginzburg model is exactly the quantum connection (2) of P (1, 3), as predicted by mirror symmetry.

Dubrovin's conjecture
Let X be a Fano variety (or an orbifold), such that the bounded derived category D b (Coh(X )) of coherent sheaves on X admits a full exceptional collection E 1 , . . . , E n , where the collection E 1 , . . . , E n is called containing E 1 , . . . , E n .
In [10], Dubrovin conjectured that, under appropriate choices, the Stokes matrix of the quantum connection of X equals the Gram matrix of the Euler-Poincaré pairing with respect to some full exceptional collection-modulo some action of the braid group, sign changes and permutations (cf. [4,Sect. 2.3]). Then the second Stokes matrix is the transpose of the first one. The Euler-Poincaré pairing is given by the bilinear form The Gram matrix of χ with respect to a full exceptional collection is upper triangular with ones on the diagonal.  (Coh(P(a, b))) (cf. [2, Theorem 2.12]). Following [3, Theorem 4.1], the cohomology of the twisting sheaves for k ∈ Z is given by (P(a, b), O(k)) = (m,n)∈I 0 Cx m y n , where • H 1 (P(a, b), O(k)) = (m,n)∈I 1 Cx m y n , where • H i (P(a, b), O(k)) = 0 for all i ≥ 2.
We only need to compute Ext k (O(i), O( j)) for i < j, which is given by H k (O ( j − i)) (cf. [17,Lemma 4.5]). Therefore, the zeroth cohomologies of the twisting sheaves O ( j − i) are the only ones that contribute to the Gram matrix of χ . For P(1, 3) we obtain the cohomology groups  (Coh(P(1, 3))) with respect to the full exceptional collection E :

Comparison of the Gram and Stokes matrix
Mirror symmetry relates the Laurent polynomial f = x + x −3 to the weighted projective line P (1, 3).  P(1, 3))). Note that there is a natural action of the braid group on the Stokes matrix reflecting variations in the choices involved to determine the Stokes matrix (cf. [13]). In our case, we have to consider the braid group on four strands, namely

Proposition. S Gram and S β correspond to each other under the action of the elementary braid
Proof. We computed that the Gram matrix of χ with respect to the full exceptional collection E is given by (3). Following [13,Sect. 6], the braid β 1 acts on the Gram matrix as where A β 1 (S Gram ) is given by We obtain that

Non-coprime parameters
In this section, we consider the weighted projective line P(2, 2) as an example for the case of non-coprime parameters. The topological computation of the Stokes matrices of the quantum connection at ∞ requires some adaptions.
A Landau-Ginzburg model of P(2, 2) is given by the curve together with the potential f = x + y. This splits into two disjoint components where we identified y = −x −1 and y = x −1 , respectively. The blue area in Fig. 8 shows where f 1 has real (resp. imaginary) part greater than or equal to 0. The blue area in Fig. 9 shows where f 2 has real (resp. imaginary) part greater than or equal to 0. In Fig. 10, the preimages of the real (resp. imaginary) axis under f 1 and f 2 are plotted. f has singular fibers at := {±2i, ±2}. For our topological computations, we consider the perverse sheaf F = R f * C[1] ∈ Perv A 1 . The exponential components at ∞ of the Fourier-Sato transform of F are of linear type, with coefficients given by the σ i ∈ . The Stokes rays are therefore given by 0, ± π 4 , ± π 2 , ± 3π 4 , π .
Denote by σ i = σ i + R ≥0 α. Their preimages are depicted in Figs. 11 and 12. As in the previous example, only the lifts of γ σ i and σ i around the double preimages of σ i , which we denote by σ 1 i , contribute to the monodromy and the cokernel of b σ i . Therefore, in our figures, we restricted to this information. From Fig. 13 we read the monodromies in the ordered basis e 1 , e 2 , e 3 , e 4 to be acting on the Gram matrix S Gram as P · S Gram · P −1 (cf. [13,Sect. 6.c]), we find that the Gram matrix S Gram (5) is transformed into the topologically computed Stokes matrix S β (4).