# zbMATH — the first resource for mathematics

Duality of Gauß-Manin systems associated to linear free divisors. (English) Zbl 1272.32026
This article is a continuation of the author’s work on certain Gauß-Manin connections associated with linear free divisors, as defined in [I. de Gregorio, D. Mond and C. Sevenheck, Compos. Math. 145, No. 5, 1305–1350 (2000; Zbl 1238.32022)]. Let $$h$$ be a reduced, homogeneous polynomial on $$V := \mathbb{C}^n$$ and $$D = V(h)$$ the associated divisor. When the sheaf of logarithmic vector fields $$\Theta(- \log D)$$ has a free $$\mathcal{O}_V$$-basis given by homogeneous linear vector fields, $$D$$ is said to be a linear free divisor. Moreover, it is said to be reductive if $$G_D := \{ g \in \mathrm{GL}(V) \;| \;g(D) \subset D \}$$ is a reductive group. To a reductive linear free divisor one can associate the top cohomology group $G( * ) = \mathbb{H}^{n-1}\big(\Omega_{V/T}^{\bullet}(* D)[\theta,\theta^{-1}], \theta d - df \wedge \big),$ where $$h : V \rightarrow T = \mathrm{Spec\,} \mathbb C[t]$$, $$f$$ is a generic linear form on $$\mathbb{C}^n$$ and $$\big(\Omega_{V/T}^{\bullet}(* D)[\theta,\theta^{-1}], \theta d - df \wedge\big )$$ is a certain complex of relative differential forms. The finite rank, free $$\mathbb{C}[\theta^{\pm 1}, t^{\pm 1}]$$-module $$G( * D)$$ is equipped with an integrable connection, the “Gauß-Manin connection”, thus making it a $$\mathcal{D}$$-module, where $$\mathcal{D} = \mathbb{C}[\theta^{\pm 1}, t^{\pm 1}] \langle \partial_t, \partial_{\theta} \rangle$$. This $$\mathcal{D}$$-module was defined and extensively studied in [loc. cit.].
The main result of the article under review is the identification $\mathrm{Hom}_{\mathbb{C}[\theta^{\pm 1}, t^{\pm 1}]} \left( G(*D),\mathbb{C}[\theta^{\pm 1}, t^{\pm 1}] \right) \simeq \iota^* G(*D)$ of $$\mathcal{D}$$-modules, where $$\iota$$ is a certain involution on $$\mathrm{Spec} \;\mathbb{C}[\theta^{\pm 1}, t^{\pm 1}]$$. This implies that there exists a non-degenerate pairing $$G(*D) \otimes \iota^* G(* D) \rightarrow \mathbb{C}[\theta^{\pm 1}, t^{\pm 1}]$$. The key to proving this result is an explicit presentation of the $$\mathcal{D}$$-module $$G(*D)$$. This allows the author to construct a free resolution of $$G(*D)$$ and hence to calculate the Verdier dual of $$G(*D)$$.
It is shown that the duality theorem allows one to sharpen several of the main results from [loc. cit.]. In particular, it gives a proof of Conjecture 5.5 from [loc. cit.].

##### MSC:
 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
Full Text:
##### References:
  Buchweitz, R.-O., Mond, D.: Linear free divisors and quiver representations, Singularities and computer algebra (Cambridge). In: Lossen, C., Pfister, G. (eds.) London Math. Soc. Lecture Note Ser., vol. 324. Cambridge University Press, Cambridge; 2006. Papers from the conference held at the University of Kaiserslautern, Kaiserslautern, pp. 41-77 (2004)  Bridgeland, T.: Spaces of stability conditions. Algebraic geometry—Seattle 2005. Part 1. In: Abramovich, D., Bertram, A., Katzarkov, L., Pandharipande, R., Thaddeus, M. (eds.) Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence; 2009. Papers from the AMS Summer Research Institute held at the University of Washington, Seattle, pp. 1-21 (2005)  Antoine, D., Etienne, M.: The small quantum cohomology of a weighted projective space, a mirror $${\mathcal{D}}$$ -module and their classical limits. Geom. Dedicata (2012). doi:10.1007/s10711-012-9768-3 · Zbl 1221.34237  Douai, A., A canonical Frobenius structure, Math. Z., 261, 625-648, (2009) · Zbl 1208.53091  Douai, A.; Sabbah, C., Gauss-Manin systems, Brieskorn lattices and Frobenius structures I, Ann. Inst. Fourier (Grenoble), 53, 1055-1116, (2003) · Zbl 1079.32016  Michel, G.; Mond, D.; Nieto, A.; Schulze, M., Free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble), 59, 811-850, (2009) · Zbl 1163.32014  Gregorio, I.; Mond, D.; Sevenheck, C., Linear free divisors and Frobenius manifolds, Compos. Math., 145, 1305-1350, (2009) · Zbl 1238.32022  Granger, M.; Schulze, M., On the symmetry of $$b$$-functions of linear free divisors, Publ. Res. Inst. Math. Sci., 46, 479-506, (2010) · Zbl 1202.14046  Reichelt, T., A construction of Frobenius manifolds with logarithmic poles and applications, Commun. Math. Phys., 287, 1145-1187, (2009) · Zbl 1197.53117  Saito, M., Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., 24, 849-995, (1989) · Zbl 0691.14007  Saito, M., On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble), 39, 27-72, (1989) · Zbl 0644.32005  Schapira P.: Microdifferential systems in the complex domain, Grundlehren der Mathematischen Wissenschaften. Fundamental Principles of Mathematical Sciences, vol. 269.. Springer, Berlin (1985) · Zbl 0554.32022  Sevenheck, C., Bernstein polynomials and spectral numbers for linear free divisors, Ann. Inst. Fourier (Grenoble), 61, 379-400, (2011) · Zbl 1221.34237  Sato, M., Kawai, T., Kashiwara Masaki.: Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations. In: Komatsu, H. (ed.) Proc. Conf., Katata, 1971. Lecture Notes in Mathematics, vol. 287. Springer, Berlin; 1973 (dedicated to the memory of André Martineau) · Zbl 0277.46039  Saito M., Sturmfels B., Takayama N.: Gröbner deformations of hypergeometric differential equations. Algorithms and Computation in Mathematics, vol. 6. Springer, Berlin (2000) · Zbl 0946.13021  Takahashi, A.: Matrix factorizations and representations of quivers I. math.AG/0506347 (2005, preprint) · Zbl 1079.32016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.