Recent zbMATH articles in MSC 14L30https://zbmath.org/atom/cc/14L302021-06-15T18:09:00+00:00WerkzeugMinimal free resolutions of ideals of minors associated to pairs of matrices.https://zbmath.org/1460.160152021-06-15T18:09:00+00:00"Lőrincz, András Cristian"https://zbmath.org/authors/?q=ai:lorincz.andras-cristianSummary: Consider the affine space consisting of pairs of matrices \((A,B)\) of fixed size, and its closed subvariety given by the rank conditions \(\operatorname{rank} A\leq a\), \(\operatorname{rank} B\leq b\), and \(\operatorname{rank}(A\cdot B)\leq c\), for three non-negative integers \(a,b,c\). These varieties are precisely the orbit closures of representations for the equioriented \(\mathbb{A}_3\) quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest.Hessenberg varieties, Slodowy slices, and integrable systems.https://zbmath.org/1460.141022021-06-15T18:09:00+00:00"Abe, Hiraku"https://zbmath.org/authors/?q=ai:abe.hiraku"Crooks, Peter"https://zbmath.org/authors/?q=ai:crooks.peterLet \(G\) be a simply-connected semisimple complex algebraic group of rank \(r\) with centre \(Z\) and fix a pair of opposite Borel subgroups \(B\) and \(B_{-}\) of \(G\); let \(\mathfrak{g}\), \(\mathfrak{b}\) and \(\mathfrak{b}_{-}\) be their Lie algebras, \(\mathfrak{t} := \mathfrak{b}\cap \mathfrak{b}_{-}\), a Cartan subalgebra of \(\mathfrak{g}\), and let \(\Delta\) be the corresponding set of roots. Now the choice of \(B\) determines subsets of simple, respectively positive, roots \(\Pi\), respectively \(\Delta_+\), so that \(\mathfrak{b} = \oplus_{\alpha \in \Delta_+} \mathfrak{g}_{\alpha}\), \(\mathfrak{b}_{-} = \bigoplus_{\alpha \in \Delta_+} \mathfrak{g}_{-\alpha}\). Set \(\mathfrak{g}_{\alpha}^{\times} = \mathfrak{g}_{\alpha} \backslash 0\) for \(\alpha \in \Delta\). The main character of the paper under review is the quasi-affine variety \[ \mathcal O_{\text{Toda}} := \mathfrak{t}+ \sum_{\alpha \in \Pi} \mathfrak{g}_{-\alpha}^{\times} \] which is isomorphic to a coadjoint orbit of \(B\) and therefore bears a symplectic structure given by the Kostant-Kirillov-Souriau form. Now by the well-known Chevalley restriction theorem, the algebra \(\mathbb C[\mathfrak{g}]^G\) of \(G\)-invariant polynomial functions on \(\mathfrak{g}\) is a polynomial algebra itself. Choose homogeneous algebraically independent generators \(f_1, \dots, f_r\) of \(\mathbb C[\mathfrak{g}]^G\). Pick also \(e_{\alpha} \in \mathfrak{g}_{\alpha}^{\times}\) for \(\alpha \in \Pi\) and set \(\zeta = -\sum_{\alpha \in \Pi} e_{\alpha}\). A celebrated theorem of Kostant states that the restrictions of the translations by \(\zeta\) of \(f_1, \dots, f_r\) form a completely integrable system on \(\mathcal O_{\text{Toda}}\) (with respect to the Kostant-Kirillov-Souriau form) called the \emph{Toda lattice}. See [\textit{B. Kostant}, Sel. Math., New Ser. 2, No. 1, 43--91 (1996; Zbl 0868.14024)]. The purpose of this article is to extend the Toda lattice to a family of Hessenberg varieties. Towards this goal the authors proceed in two steps.
First they consider the Slodowy slice \(S_{\text{reg}} := \xi + \ker \text{ad}_{\eta}\) associated to a suitable (principal) \(\mathfrak{sl}_2\)-triple \((\xi, h, \eta)\). It is known that \(G \times S_{\text{reg}}\) can be identified with a symplectic subvariety of the cotangent bundle of \(G\) and that \(G/Z \times S_{\text{reg}}\) is a symplectic quotient of \(G \times S_{\text{reg}}\). Furthermore the invariant polynomials \(f_1, \dots, f_r\) together with \(\zeta\) above give rise to the so-called Mishchenko-Fomenko polynomials. Pulling back these polynomials along the moment map \(\mu: G /Z \times S_{\text{reg}} \to \mathfrak{g}\) one gets a completely integrable system on \(G /Z \times S_{\text{reg}}\), see [\textit{P. Crooks} and \textit{S. Rayan}, Math. Res. Lett. 26, No. 1, 9--33 (2019; Zbl 1421.32029)]. The main achievement of the first step is the construction of a natural embedding of symplectic varieties \(\kappa : \mathcal{O}_{\text{Toda}} \hookrightarrow G /Z \times S_{\text{reg}}\) relating the Toda lattice with the mentioned completely integrable system.
Second, the \(B\)-submodule \(H_{0} := \mathfrak{b}+ \sum_{\alpha \in \Pi} \mathfrak{g}_{-\alpha}\) of \(\mathfrak{g}\) determines a \(G\)-equivariant vector bundle on \(G/B\) with total space \(X(H_0) := G\times_{B} H_0\). Let \(\mu_0: X(H_{0}) \to \mathfrak{g}\) be the morphism given by \(\mu_0\left([g, x]\right) = \operatorname{Ad}_g (x)\). The fibres \(X(x, H_{0}) := \mu_0^{-1}(x)\) of \(\mu_0\) are the Hessenberg varieties associated to \(H_0\). Set \(H_{0}^{\times} := \mathfrak{b}+ \sum_{\alpha \in \Pi} \mathfrak{g}_{-\alpha}^{\times}\). The authors prove that \(X(H_0)\) carries a natural Poisson structure having a unique open (and dense) symplectic leaf which is \(X(H_0^{\times}) = G\times_{B} H_0^{\times}\). Then they prove that there exists an open immersion \(\varphi: G /Z \times S_{\text{reg}} \hookrightarrow X(H_0)\) such that \(\mu_0\varphi = \mu\). The image of \(\varphi\) is \(X(H_0^{\times})\) which is isomorphic to \( G /Z \times S_{\text{reg}}\) as a symplectic variety via \(\varphi\).
Now pulling back the Mishchenko-Fomenko polynomials along the map \(\mu_0\) one gets a completely integrable system on \(X(H_0)\). In conclusion, the composition \(\varphi\kappa \) is an embedding of completely integrable systems from the Toda lattice on \(\mathcal{O}_{\text{Toda}}\) into that one arising from the Mishchenko-Fomenko polynomials on the family \(X(H_0)\) of Hessenberg varieties. Finally the authors discuss briefly some applications to the geometry of Hessenberg varieties.
Reviewer: Nicolás Andruskiewitsch (Córdoba)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)Resolvent degree, Hilbert's 13th problem and geometry.https://zbmath.org/1460.141042021-06-15T18:09:00+00:00"Farb, Benson"https://zbmath.org/authors/?q=ai:farb.benson"Wolfson, Jesse"https://zbmath.org/authors/?q=ai:wolfson.jesseSummary: We develop the theory of resolvent degree, introduced by \textit{R. Brauer} [Ann. Mat. Pura Appl. (4) 102, 45--55 (1975; Zbl 0299.12105)] in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.Exotic \({\mathbb{G}}_a\)-quotients of \(\mathrm{SL}_2\times{\mathbb{A}}^1\).https://zbmath.org/1460.141372021-06-15T18:09:00+00:00"Dubouloz, Adrien"https://zbmath.org/authors/?q=ai:dubouloz.adrienSummary: Every deformed Koras-Russell threefold of the first kind \(Y=\{ x^nz=y^m-t^r+xh(x,y,t)\}\) in \({\mathbb{A}}^4\) is the algebraic quotient of proper Zariski locally trivial \({\mathbb{G}}_a\)-action on \(\mathrm{SL}_2\times{\mathbb{A}}^1\).Braid group actions on matrix factorizations.https://zbmath.org/1460.141032021-06-15T18:09:00+00:00"Arkhipov, Sergey"https://zbmath.org/authors/?q=ai:arkhipov.sergey"Kanstrup, Tina"https://zbmath.org/authors/?q=ai:kanstrup.tinaSummary: Let \(X\) be at smooth scheme with an action of a reductive algebraic group \(G\) over an algebraically closed field \(k\) of characteristic zero. We construct an action of the extended affine braid group on the \(G\)-equivariant absolute derived category of matrix factorizations on the Grothendieck variety times \(T^*X\) with potential given by the Grothendieck-Springer resolution times the moment map composed with the natural pairing.
For the entire collection see [Zbl 1458.55002].Covariants, derivation-invariant subsets, and first integrals.https://zbmath.org/1460.141052021-06-15T18:09:00+00:00"Grosshans, Frank"https://zbmath.org/authors/?q=ai:grosshans.frank-d"Kraft, Hanspeter"https://zbmath.org/authors/?q=ai:kraft.hanspeterSummary: Let \(k\) be an algebraically closed field of characteristic 0, and let \(V\) be a finite-dimensional vector space. Let \(\text{End}(V)\) be the semigroup of all polynomial endomorphisms of \(V\). Let \(\mathcal{E}\subseteq \text{End}(V)\) be a linear subspace which is also a subsemigroup. Both \(\text{End}(V)\) and \(\mathcal{E}\) are ind-varieties which act on \(V\) in the obvious way.
In this paper, we study important aspects of such actions. We assign to \(\mathcal{E}\) a linear subspace \(\mathcal{D}_{\mathcal{E}}\) of the vector fields on \(V\). A subvariety \(X\) of \(V\) is said to be \(\mathcal{D}_{\mathcal{E}}\)-invariant if \(\xi(x)\in T_xX\) for all \(\xi\in \mathcal{D}_{\mathcal{E}}\). We show that \(X\) is \(\mathcal{D}_{\mathcal{E}}\)-invariant if and only if it is the union of \(\mathcal{E}\)-orbits. For such \(X\), we define first integrals and show that they are the rational functions on a certain ``quotient'' of \(X\) defined by the action of \(\mathcal{E}\).
An important case occurs when \(G\) is an algebraic subgroup of \(\text{GL}(V)\) and \(\mathcal{E}\) consists of the \(G\)-equivariant polynomial endomorphisms. In this case, the associated \(\mathcal{D}_{\mathcal{E}}\) is the space of \(G\)-invariant vector fields. A significant question here is whether there are non-constant \(G\)-invariant first integrals on \(X\). As examples, we study the adjoint representation, Luna strata, the orbit closures of highest weight vectors, and representations of the additive group. We also look at finite-dimensional irreducible representations of \(\text{SL}_2\) and their nullcones.On the topology of rational \(\mathbb{T}\)-varieties of complexity one.https://zbmath.org/1460.140172021-06-15T18:09:00+00:00"Laface, Antonio"https://zbmath.org/authors/?q=ai:laface.antonio"Liendo, Alvaro"https://zbmath.org/authors/?q=ai:liendo.alvaro"Moraga, Joaquín"https://zbmath.org/authors/?q=ai:moraga.joaquinSummary: We generalize classical results about the topology of toric varieties to the case of projective \(\mathbb{Q}\)-factorial \(\mathbb{T}\)-varieties of complexity one using the language of divisorial fans. We describe the Hodge-Deligne polynomial in the smooth case, the cohomology ring and the Chow ring in the contraction-free case.