Recent zbMATH articles in MSC 32Shttps://zbmath.org/atom/cc/32S2021-06-15T18:09:00+00:00WerkzeugŁojasiewicz inequality for a pair of semialgebraic functions.https://zbmath.org/1460.141282021-06-15T18:09:00+00:00"Osińska-Ulrych, Beata"https://zbmath.org/authors/?q=ai:osinska-ulrych.beata"Skalski, Grzegorz"https://zbmath.org/authors/?q=ai:skalski.grzegorz"Szlachcińska, Anna"https://zbmath.org/authors/?q=ai:szlachcinska.annaLet \(\Omega =\{x\in \mathbb{R}^{n}:\left\vert x\right\vert <1\}\) be a unit ball in \(\mathbb{R}^{n}\) and \(f,g:\Omega \rightarrow \mathbb{R}\) be two continuous, semialgebraic functions such that \(0\in g^{-1}(0)\subset f^{-1}(0)\neq \Omega\). The greatest lower bound of exponents \(\alpha \) in the Łojasiewicz inequality
\[
\left\vert g(x)\right\vert \geq C\left\vert f(x)\right\vert ^{\alpha}\text{ for }\left\vert x\right\vert <\varepsilon
\]
for some \(C,\varepsilon >0\) is called the Łojasiewicz exponent for the pair of functions \(f\) and \(g\) on \(\Omega \) at \(0.\) The main result is an estimation of \(\alpha \) in terms of the degrees of polynomials describing \(f\) and \(g\) and the degrees of polynomials describing some special semialgebraic decomposition of \(\Omega \) associated to \(f\) and \(g\). The estimation is a version of the Solernó estimation [\textit{P. Solernó}, Appl. Algebra Eng. Commun. Comput. 2, No. 1, 1--14 (1991; Zbl 0754.14036)].
Reviewer: Tadeusz Krasiński (Łódź)\(\mathbb{C}\)-constructible enhanced ind-sheaves.https://zbmath.org/1460.320092021-06-15T18:09:00+00:00"Ito, Yohei"https://zbmath.org/authors/?q=ai:ito.yoheiSummary: \textit{A. D'Agnolo} and \textit{M. Kashiwara} [Publ. Math., Inst. Hautes Étud. Sci. 123, 69--197 (2016; Zbl 1351.32017)] proved that their enhanced solution functor induces a fully faithful embedding of the triangulated category of holonomic \(\mathcal{D}\)-modules into the one of \(\mathbb{R}\)-constructible enhanced ind-sheaves. In this paper, we introduce a notion of \(\mathbb{C}\)-constructible enhanced ind-sheaves and show that the triangulated category of them is equivalent to its essential image. Moreover we show that there exists a t-structure on it whose heart is equivalent to the abelian category of holonomic \(\mathcal{D}\)-modules.Euclidean matchings and minimality of hyperplane arrangements.https://zbmath.org/1460.320642021-06-15T18:09:00+00:00"Lofano, Davide"https://zbmath.org/authors/?q=ai:lofano.davide"Paolini, Giovanni"https://zbmath.org/authors/?q=ai:paolini.giovanniSummary: We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us to prove the isomorphism of Brieskorn's Lemma by a simple bijection of the critical cells. Finally, in the case of line arrangements, we describe the algebraic Morse complex which computes the homology with coefficients in an abelian local system.A global Torelli theorem for singular symplectic varieties.https://zbmath.org/1460.320142021-06-15T18:09:00+00:00"Bakker, Benjamin"https://zbmath.org/authors/?q=ai:bakker.benjamin"Lehn, Christian"https://zbmath.org/authors/?q=ai:lehn.christianSummary: We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of \(K3^{[n]}\)-type varieties.Numerical local irreducible decomposition.https://zbmath.org/1460.141312021-06-15T18:09:00+00:00"Brake, Daniel A."https://zbmath.org/authors/?q=ai:brake.daniel-a"Hauenstein, Jonathan D."https://zbmath.org/authors/?q=ai:hauenstein.jonathan-d"Sommese, Andrew J."https://zbmath.org/authors/?q=ai:sommese.andrew-johnSummary: Globally, the solution set of a system of polynomial equations with complex coefficients can be decomposed into irreducible components. Using numerical algebraic geometry, each irreducible component is represented using a witness set thereby yielding a numerical irreducible decomposition of the solution set. Locally, the irreducible decomposition can be refined to produce a local irreducible decomposition. We define local witness sets and describe a numerical algebraic geometric approach for computing a numerical local irreducible decomposition for polynomial systems. Several examples are presented.
For the entire collection see [Zbl 1334.68018].A closedness theorem over Henselian fields with analytic structure and its applications.https://zbmath.org/1460.320472021-06-15T18:09:00+00:00"Nowak, Krzysztof Jan"https://zbmath.org/authors/?q=ai:nowak.krzysztof-janSummary: In this brief note, we present our closedness theorem in geometry over Henselian valued fields with analytic structure. It enables, among others, application of resolution of singularities and of transformation to normal crossings by blowing up in much the same way as over locally compact ground fields. Also given are many applications which, at the same time, provide useful tools in geometry and topology of definable sets and functions. They include several versions of the Łojasiewicz inequality, Hölder continuity of definable functions continuous on closed bounded subsets of the affine space, piecewise continuity of definable functions or curve selection. We also present our most recent research concerning definable retractions and the extension of continuous definable functions. These results were established in several successive papers of ours, and their proofs made, in particular, use of the following fundamental tools: elimination of valued field quantifiers, term structure of definable functions and b-minimal cell decomposition, due to Cluckers-Lipshitz-Robinson, relative quantifier elimination for ordered abelian groups, due to Cluckers-Halupczok, the closedness theorem as well as canonical resolution of singularities and transformation to normal crossings by blowing up due to Bierstone-Milman. As for the last tool, our approach requires its definable version established in our most recent paper within a category of definable, strong analytic manifolds and maps.
For the entire collection see [Zbl 1460.13001].The dimension of the moduli spaces of curves defined by topologically non quasi-homogeneous functions.https://zbmath.org/1460.320672021-06-15T18:09:00+00:00"Loubani, Jinan"https://zbmath.org/authors/?q=ai:loubani.jinanSummary: We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. The 0-level of such a function defines a germ of analytic curve. Proceeding similarly to the homogeneous case [\textit{Y. Genzmer} and \textit{E. Paul}, Mosc. Math. J. 11, No. 1, 41--72 (2011; Zbl 1222.32056)] and the quasi homogeneous case [\textit{Y. Genzmer} and \textit{E. Paul}, J. Singul. 14, 3--33 (2016; Zbl 1338.32028)], we describe an algorithm which computes the dimension of the generic strata of the local moduli space of curves.Topological moduli space for germs of holomorphic foliations.https://zbmath.org/1460.320412021-06-15T18:09:00+00:00"Marín, David"https://zbmath.org/authors/?q=ai:marin-perez.david"Mattei, Jean-François"https://zbmath.org/authors/?q=ai:mattei.jean-francois"Salem, Éliane"https://zbmath.org/authors/?q=ai:salem.elianeSummary: This work deals with the topological classification of germs of singular foliations on \((\mathbb{C}^2,0)\). Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices, and the projective holonomy representations and we compute the moduli space of topological classes in terms of the cohomology of a new algebraic object that we call group-graph. This moduli space may be an infinite-dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological structure.Study of multiple structures on projective subvarieties.https://zbmath.org/1460.140092021-06-15T18:09:00+00:00"Gonzalez-Dorrego, M. R."https://zbmath.org/authors/?q=ai:gonzalez-dorrego.maria-rSummary: Let \(k\) an algebraically closed field, \(\mathrm{char}\, k=0\). We study multiplicity-\(r\) structures on varieties for \(r\in \mathbb{N}, r\ge 2\). Let \(Z\) be a reduced irreducible nonsingular \((N-2)\)-dimensional variety such that \(rZ=X\cap F\), where \(X\) is a normal \((N-1)\)-fold of degree \(n, F\) is a smooth \((N-1)\)-fold of degree \(m\) in \(\mathbb{P}^N\), such that \(r\in \mathbb{N}, r\ge 2, Z\cap \text{Sing} (X)\not =\emptyset \). There are effective divisors \(V\) and \(D_1\) on \(Z\) such that \(O_Z(V-(r-1)D_1)\simeq{\omega_Z}^r(-rm-n+(N+1)r)\), where \(\omega_Z\) is the canonical sheaf of \(Z\). Let \(Z \subset \mathbb{P}^N\) be a reduced irreducible subvariety of codimension 2. Let \(Y\) be an irreducible hypersurface in \(\mathbb{P}^N, Z \subset Y\). Let \({\omega^o}_Z\) be the dualizing sheaf of \(Z\). Then, there exists a hypersurface \(X\) in \(\mathbb{P}^N\) such that \(Z=Y\cap X\) is a scheme-theoretical complete intersection if and only if \(\bullet{\omega^o}_Z\simeq \omega_{\mathbb{P^N}}\otimes{\wedge }^2{\mathcal N }_Z|_{\mathbb{P^N}}. \bullet \text{deg}\,Y\) divides \(\text{deg}\,Z. \bullet{\omega^o}_Z\simeq O_Z (\text{deg}\,Y+(\frac{\text{deg}\,Z}{\text{deg}\,Y})-N-1)\).
For the entire collection see [Zbl 1460.13001].Codimension one holomorphic distributions on the projective three-space.https://zbmath.org/1460.320402021-06-15T18:09:00+00:00"Calvo-Andrade, Omegar"https://zbmath.org/authors/?q=ai:calvo-andrade.omegar"Corrêa, Maurício"https://zbmath.org/authors/?q=ai:correa-barros.mauricio-jun"Jardim, Marcos"https://zbmath.org/authors/?q=ai:jardim.marcosSummary: We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.Singularities, mirror symmetry, and the gauged linear sigma model. Summer school `Crossing the walls in enumerative geometry', Snowbird, UT, USA, May 21 -- June 1, 2018.https://zbmath.org/1460.140022021-06-15T18:09:00+00:00"Jarvis, Tyler J. (ed.)"https://zbmath.org/authors/?q=ai:jarvis.tyler-j"Priddis, Nathan (ed.)"https://zbmath.org/authors/?q=ai:priddis.nathanPublisher's description: This volume contains the proceedings of the workshop `Crossing the walls in enumerative geometry', held in May 2018 at Snowbird, Utah. It features a collection of both expository and research articles about mirror symmetry, quantized singularity theory (FJRW theory), and the gauged linear sigma model.
Most of the expository works are based on introductory lecture series given at the workshop and provide an approachable introduction for graduate students to some fundamental topics in mirror symmetry and singularity theory, including quasimaps, localization, the gauged linear sigma model (GLSM), virtual classes, cosection localization, \(p\)-fields, and Saito's primitive forms. These articles help readers bridge the gap from the standard graduate curriculum in algebraic geometry to exciting cutting-edge research in the field.
The volume also contains several research articles by leading researchers, showcasing new developments in the field.
The articles of this volume will be reviewed individually.Lattices for Landau-Ginzburg orbifolds.https://zbmath.org/1460.320652021-06-15T18:09:00+00:00"Ebeling, Wolfgang"https://zbmath.org/authors/?q=ai:ebeling.wolfgang"Takahashi, Atsushi"https://zbmath.org/authors/?q=ai:takahashi.atsushi.3A Landau-Ginzburg orbifold is a pair \(( f , G)\) where \(f\) is an invertible polynomial (a quasi-homogeneous polynomial with as many terms as variables) and \(G\) a finite abelian group of symmetries of \(f\). For such orbifolds there is a Berglund-Hübsch-Henningson duality. In joint work of the authors with \textit{S. M. Gusein-Zade} [J. Geom. Phys. 106, 184--191 (2016; Zbl 1379.32025)] a symmetry property of the orbifold E-functions of dual pairs has been established. It is used here to show symmetry of the orbifoldized elliptic genera.
Motivated by a formula proved here for the signature of the Milnor fibre in terms of the elliptic genus \(Z(\tau, z)\) in the non orbifold case, a definition is given for the orbifoldized signature. This raises the question of the existence of a lattice with this signature. For \(n = 3\), two lattices are defined, one for the A-model when \(G\subset \text{SL}(3;\mathbb{C}) \cap G_f\), the other one for the dual B-model.
The first uses the crepant resolution \(Y\) of the ambient space \(\mathbb{C}^3/G\). Let \(Z\) be the inverse image of the zero set of \(f\). The lattice is the free part of the image of \(H^3(Y , Z;\mathbb{Z})\) in \(H^2(Z;\mathbb{Z})\). For the trivial case \(G=\{\text{id}\}\) this is the usual Milnor lattice. The lattice has the correct rank and signature. It is described in detail for \(14+8\) pairs \(( f , G)\).
The other lattice comes from the numerical Grothendieck group of the stable homotopy category of \(L_{\widehat G}\)-graded matrix factorisations of \(f\), where \(L_{\widehat G}\) is related to the maximal grading of \(f\). The rank of this lattice is equal to the rank of the first lattice for the dual pair. For \(f\) defining an ADE singularity is the root lattice of the corresponding type. The Authors conjecture that these lattices are interchanged under the duality of pairs.
\textit{S. M. Gusein-Zade} and the first author [Manuscr. Math. 155, No. 3--4, 335--353 (2018; Zbl 1393.14056)] also defined an orbifold version of the Milnor lattice for a pair \(( f , G)\). It is not known whether that lattice coincides with one of the lattices here.
Reviewer: Jan Stevens (Göteborg)Symmetries of tilings of Lorentz spaces.https://zbmath.org/1460.520212021-06-15T18:09:00+00:00"Turki, Nasser Bin"https://zbmath.org/authors/?q=ai:turki.nasser-bin"Pratoussevitch, Anna"https://zbmath.org/authors/?q=ai:pratoussevitch.annaSummary: We study tilings of the 3-dimensional simply connected Lorentz manifold of constant curvature. This manifold is modelled on the Lie group
\[
G=\widetilde{\operatorname{SU}(1,1)}\cong\widetilde{\operatorname{SL}(2,\mathbb{R})},
\]
equipped with the Killing form. The tilings are produced by the fundamental domain construction introduced by the second author. The construction gives Lorentz polyhedra as fundamental domains for the action by left multiplication of a discrete co-compact subgroup of finite level. We determine the symmetry groups of these tilings and discuss the connection with the Seifert fibration of the quotient space. We then give an explicit description of the symmetry group of the tiling in the case when the discrete subgroup is a lift of a triangle group.Computing Euler obstruction functions using maximum likelihood degrees.https://zbmath.org/1460.141332021-06-15T18:09:00+00:00"Rodriguez, Jose Israel"https://zbmath.org/authors/?q=ai:rodriguez.jose-israel"Wang, Botong"https://zbmath.org/authors/?q=ai:wang.botongSummary: We give a numerical algorithm computing Euler obstruction functions using maximum likelihood degrees. The maximum likelihood degree is a well-studied property of a variety in algebraic statistics and computational algebraic geometry. In this article we use this degree to give a new way to compute Euler obstruction functions. We define the maximum likelihood obstruction function and show how it coincides with the Euler obstruction function. With this insight, we are able to bring new tools of computational algebraic geometry to study Euler obstruction functions.Convex foliations of degree 4 on the complex projective plane.https://zbmath.org/1460.320392021-06-15T18:09:00+00:00"Bedrouni, Samir"https://zbmath.org/authors/?q=ai:bedrouni.samir"Marín, David"https://zbmath.org/authors/?q=ai:marin-perez.davidA singular holomorphic foliation
\(\mathcal{F}\) on the complex projective space
\(\mathbb{P}^{2}_\mathbb{C}\)
is convex if its leaves other than straight
lines have no inflection points.
In addition, \(\mathcal{F}\) is
homogeneous when it is invariant by homotheties
(of a suitable affine chart).
A foliation of degree \(d\geq 1\) cannot have more than \(3d\)
(distinct) invariant lines;
if this bound is reached by some \(\mathcal{F}\),
then it is reduced convex.
The authors show that up to automorphism of \(\mathbb{P}^{2}_\mathbb{C}\)
there are five homogeneous convex foliations of degree
four on \(\mathbb{P}^{2}_\mathbb{C}\).
The 1-forms defining these foliations are provided.
Using this result,
a partial answer to Problem 9.1 posed in
[\textit{D. Marín} et al.,
Asian J. Math. 17, No. 1, 163--192 (2013; Zbl 1330.53020)]
about the classification of reduced convex foliations
is provided.
Roughly speaking:
up to automorphism of \(\mathbb{P}^{2}_\mathbb{C}\),
the Fermat foliation and the Hesse pencil are
the only reduced convex foliations of degree four.
Reviewer: Jesus Muciño Raymundo (Morelia)Palais leaf-space manifolds and surfaces carrying holomorphic flows.https://zbmath.org/1460.320662021-06-15T18:09:00+00:00"Ferreira, Ana Cristina"https://zbmath.org/authors/?q=ai:ferreira.ana-cristina"Rebelo, Julio C."https://zbmath.org/authors/?q=ai:rebelo.julio-c"Reis, Helena"https://zbmath.org/authors/?q=ai:reis.helenaSummary: Given a pair of commuting holomorphic vector fields defined on a neighborhood of \((0,0)\in\mathbb{C}^2\), we discuss the problem of globalizing them as an action of \(\mathbb{C}^2\) on a suitable complex surfaces along with some related questions. A review of Palais' theory about globalization of local transformation groups is also included in our discussion.The global sections of chiral de Rham complexes on compact Ricci-flat Kähler manifolds.https://zbmath.org/1460.810402021-06-15T18:09:00+00:00"Song, Bailin"https://zbmath.org/authors/?q=ai:song.bailinSummary: The space of the global sections of the chiral de Rham complex on a compact Ricci-flat Kähler manifold is calculated and it is expressed as the subspace of invariant elements in a \(\beta \gamma - bc\) system under the action of certain Lie algebra of Cartan type.On the growth behaviour of Hironaka quotients.https://zbmath.org/1460.320632021-06-15T18:09:00+00:00"Maugendre, H."https://zbmath.org/authors/?q=ai:maugendre.helene"Michel, F."https://zbmath.org/authors/?q=ai:michel.francoiseLet $\phi=(f,g): (X,p)\to (\mathbb C^2,0)$ be a finite analytic morphism which is defined on a complex analytic surface germ $(X,p)$ by
two complex analytic function germs $f$ and $g$. A resolution $\pi: (Y,E_Y)\to (X,p)$ is said to be a \textit{good} resolution of $\phi$ if it is a resolution of the singularity $(X,p)$ in which the total transform $E_Y^+=((f,g)\circ \pi)^{-1}(0)$ is a normal crossing divisor and such that the irreducible components of the exceptional divisor $E_Y=\pi^{-1}(p)$ are non-singular. An irreducible component $E_i$ of $E_Y$ is called a \textit{rupture} component if it is not a rational curve or if it intersects at least three other components of the exceptional divisor. A \textit{curvetta} $c_i$ of an irreducible component $E_i$ of $E_Y$ is a smooth curve in $Y$ that intersects transversally $E_i$ at a smooth point of the total transform $E^+_Y$. To each irreducible component $E_i$ of $E$ the authors associate the rational number $q_{E_i}:=V_{f\circ \pi}(c_i)/V_{g\circ \pi}(c_i)$ where $V_{f\circ\pi}(c_i)$ resp. $V_{g\circ\pi}(c_i)$ is the order of $f\circ\pi$ resp. $g\circ\pi$ on the curvetta $c_i$. This quotient is called the Hironaka quotient of $(f,g)$ on $E_i$. This set of rational numbers associated to a complex analytic normal surface germ was introduced by \textit{D. T. Lê} et al. [J. Lond. Math. Soc., II. Ser. 63, No. 3, 533--552 (2001; Zbl 1018.32027)].
Reviewer: Karl-Heinz Kiyek (Paderborn)Bernstein-Sato varieties and annihilation of powers.https://zbmath.org/1460.140482021-06-15T18:09:00+00:00"Bath, Daniel"https://zbmath.org/authors/?q=ai:bath.danielLet \(X\) be a smooth scheme over \(\mathbb{C}\) or analytic variety of dimension \(n\). Take \(r\) different global functions \(f_1,\ldots,f_r\in\mathcal{O}_X\) and let \(f=f_1\cdots f_r\). Associated with that family of functions, we can consider the \(\mathcal{O}_X[S]\)-module \(\mathcal{O}_X[f^{-1},S]\cdot F^S\), where \(F\) and \(S\) are multi-index notation for \((f_1,\ldots,f_r)\) and \((s_1,\ldots,s_r)\) and \(F^S\) is the symbol \(\prod_if_i^{s_i}\). Such an object can be endowed with a natural structure of \(\mathcal{D}_X[S]\)-module, extending the one over \(\mathcal{O}_X[S]\).
Analogously as in the case \(r=1\), we can consider the Bernstein-Sato ideal \(B_F=\mathbb{C}[S]\cap\langle\mathcal{D}_X[S]F+\operatorname{ann}_{\mathcal{D}_X[S]} F^S\rangle\). The paper under review is a generalization to the multivariable case of some of the results of [\textit{L. Narváez Macarro}, Adv. Math. 281, 1242--1273 (2015; Zbl 1327.14090)] and [\textit{U. Walther}, Invent. Math. 207, No. 3, 1239--1287 (2017; Zbl 1370.14022)] under certain conditions on \(f\).
We say that \(f\) is strongly Euler-homogeneous at some point \(x\in X\) if there is a logarithmic vector field \(E_x\in\operatorname{Der}_{X,x}(-\log f)\) such that \(E_x\bullet f=f\) and \(E_x\) vanishes at \(x\). We also say that \(f\) is tame if the projective dimension of the logarithmic \(k\)-forms \(\Omega_X^k(\log f)\) is at most \(k\) at each \(x\in X\). Last, the author defines \(f\) to be Saito-holonomic if certain stratification that we can consider on \(X\), arising from logarithmic derivations, is locally finite.
Under those three conditions, the author proves the first main result of the paper: the annihilator \(\operatorname{ann}_{\mathcal{D}_X[S]} F^S\) is generated by derivations, namely by the elements \(\delta-\sum_is_i(\delta\bullet f_i)/f_i\), where \(\delta\in\operatorname{Der}_{X}(-\log f)\). Using such result, the author goes on proving the generalized \(-\frac{n}{d}\) conjecture as stated by \textit{Budur} [Ann. Inst. Fourier 65, No. 2, 549--603 (2015; Zbl 1332.32038)] for central, essential, indecomposable, tame hyperplane arrangements in \(\mathbb{C}^n\).
Let now \(A=(a_1,\ldots,a_r)\in\mathbb{C}^r\) be an \(r\)-uple and let \(x\in X\) be a point and consider the \(\mathcal{D}_{X,x}\)-module map \[\nabla_A:\frac{\mathcal{D}_{X,x}[S]F^S}{(S-A)\mathcal{D}_{X,x}[S]F^S}\rightarrow \frac{\mathcal{D}_{X,x}[S]F^S}{(S-(A-1))\mathcal{D}_{X,x}[S]F^S},\] induced by sending \(F^S\) to \(F^{S+1}\). The other main result of the paper is that if \(\nabla_A\) is injective, then it is surjective too, whenever \(f\) is strongly Euler-homogeneous, Saito-holonomic and tame. Moreover, if \(f\) is also reduced, and free instead of just tame, the converse statement holds. This is used to study the dual of the \(\mathcal{D}_{X,x}[S]\)-modules from above and the cohomology support loci of \(f\) (defined in the text).
Reviewer: Alberto Castaño Domínguez (Sevilla)