Open Embedding Research Articles

Normal Completions of Toric Varieties Over Rank One Valuation Rings and Completions of Γ-Admissible Fans

Abstract We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $\Gamma $ is discrete or divisible then no base-change is needed. We give explicit examples that show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $\Gamma $-admissible fans admit $\Gamma $-admissible completions. In order to show this we prove a combinatorial analog of Noetherian reduction, which we believe will be of independent interest.

Reconstruction of a surface from the category of reflexive sheaves

We define a normal surface X to be codim-2-saturated if any open embedding of X into a normal surface with the complement of codimension 2 is an isomorphism. We show that any normal surface X allows a codim-2-saturated model Xˆ together with the canonical open embedding X→Xˆ.Any normal surface which is proper over its affinisation is codim-2-saturated, but the converse does not hold. We give a criterion for a surface to be codim-2-saturated in terms of its Nagata compactification and the boundary divisor.We reconstruct the codim-2-saturated model of a normal surface X from the additive category of reflexive sheaves on X. We show that the category of reflexive sheaves on X is quasi-abelian and we use its canonical exact structure for the reconstruction.In order to deal with categorical issues, we introduce a class of weakly localising Serre subcategories in abelian categories. These are Serre subcategories whose categories of closed objects are quasi-abelian. This general technique might be of independent interest.

Comparing two Proj-like constructions on toric varieties

The paper explores the relation between Perling's toric Proj of a multigraded ring A associated with a toric variety (tProj A) and Brenner and Schröer's homogeneous spectrum ProjMHA of the same ring. We show that there is always a canonical open embedding and study a criterion for them to be isomorphic.

Open, Closed, and Non-Degenerate Embedding Dimensions of Neural Codes

We study the open, closed, and non-degenerate embedding dimensions of neural codes, which are the smallest respective dimensions in which one can find a realization of a code consisting of convex sets that are open, closed, or non-degenerate in a sense defined by Cruz, Giusti, Itskov, and Kronholm. For a given code C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}$$\\end{document} we define the embedding dimension vector to be the triple (a, b, c) consisting of these embedding dimensions. Existing results guarantee that max{a,b}≤c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\max {\\{a,b\\}}\\le c$$\\end{document}, and we show that when any of these dimensions is at least 2 this is the only restriction on such vectors. Specifically, for every triple (a, b, c) with 2≤min{a,b}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\le \\min {\\{a,b\\}}$$\\end{document} and max{a,b}≤c≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\max {\\{a,b\\}}\\le c\\le \\infty $$\\end{document} we construct a code C(a,b,c)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_{(a,b,c)}$$\\end{document} whose embedding dimension vector is exactly (a, b, c) (where an embedding dimension is ∞\\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} if there is no realization of the corresponding type). Our constructions combine two existing tools in the convex neural codes literature: sunflowers of convex open sets, and rigid structures, the latter of which was recently defined in work of Chan, Johnston, Lent, Ruys de Perez, and Shiu. Our constructions provide the first examples of codes whose closed embedding dimension is larger than their open embedding dimension, but still finite.

Hodge modules and singular hermitian metrics

The purpose of this paper is to study certain notions of metric positivity called “minimal extension property” for the lowest nonzero piece in the Hodge filtration of a Hodge module. Let X be a complex manifold and let \(\mathcal {M}\) be a polarized pure Hodge module on X with strict support X. Let \(F_p\mathcal {M}\) be the smallest nonzero piece in the Hodge filtration. Assume that \(\mathcal {M}\) is smooth outside a closed analytic subset Z and let \(j:X\setminus Z \hookrightarrow X\) be the open embedding. Let h be the smooth hermitian metric on \(F_p\mathcal {M}|_{X\setminus Z}\) induced by the polarization. We show that the canonical morphism of \(\mathcal {O}_X\)-modules $$\begin{aligned} F_p\mathcal {M}\rightarrow j_{*}(F_p\mathcal {M}|_{X\setminus Z}) \end{aligned}$$induces an isomorphism between \(F_p\mathcal {M}\) and the subsheaf of \(j_{*}(F_p\mathcal {M}|_{X\setminus Z})\) consisting of sections which are locally \(L^2\) near Z with respect to h and the standard Lebesgue measure on X. In particular, h extends to a singular hermitian metric on \(F_p\mathcal {M}\) with minimal extension property.

Lefschetz Open Book Embeddings of 4-Manifolds

In this article, we define the notion of a generalized open book of a n-manifold over the k−sphere Sk , k < n. We discuss Lefschetz open book embeddings of Lefschetz open books of closed oriented 4-manifolds into the trivial open book over S2 of the 7−sphere S7 . If X is the double of a bounded achiral Lefschetz fibration over D2 , then X naturally admits a Lefschetz open book given by the bounded achiral Lefschetz fibration. We show that this natural Lefschetz open book of X admits a Lefschetz open book embedding into the trivial open book over S2 of the 7−sphere S7 .

On the Image of the Period Map for Polarized Hyperkähler Manifolds

Abstract The moduli space for polarized hyperkähler manifolds of ${\textrm {K3}^{[m]}}$-type or ${\textrm {Kum}}_m$-type with a given polarization type is not necessarily connected, which is a phenomenon that only happens for $m$ large. The period map restricted to each connected component gives an open embedding into the period domain, and the complement of the image is a finite union of Heegner divisors. We give a simplified formula for the number of connected components, as well as a simplified criterion to enumerate the Heegner divisors in the complement. In particular, we show that the image of the period map may be different when restricted to different components of the moduli space.

Non-Monotonicity of Closed Convexity in Neural Codes

Neural codes are lists of subsets of neurons that fire together. Of particular interest are neurons called place cells, which fire when an animal is in specific, usually convex regions in space. A fundamental question, therefore, is to determine which neural codes arise from the regions of some collection of open convex sets or closed convex sets in Euclidean space. This work focuses on how these two classes of codes – open convex and closed convex codes – are related. As a starting point, open convex codes have a desirable monotonicity property, namely, adding non-maximal codewords preserves open convexity; but here we show that this property fails to hold for closed convex codes. Additionally, while adding non-maximal codewords can only increase the open embedding dimension by 1, here we demonstrate that adding a single such codeword can increase the closed embedding dimension by an arbitrarily large amount. Finally, we disprove a conjecture of Goldrup and Phillipson, and also present an example of a code that is neither open convex nor closed convex.

Moduli spaces of Hecke modifications for rational and elliptic curves

We propose definitions of complex manifolds $\mathcal{P}_M(X,m,n)$ that could potentially be used to construct the symplectic Khovanov homology of $n$-stranded links in lens spaces. The manifolds $\mathcal{P}_M(X,m,n)$ are defined as moduli spaces of Hecke modifications of rank 2 parabolic bundles over an elliptic curve $X$. To characterize these spaces, we describe all possible Hecke modifications of all possible rank 2 vector bundles over $X$, and we use these results to define a canonical open embedding of $\mathcal{P}_M(X,m,n)$ into $M^s(X,m+n)$, the moduli space of stable rank 2 parabolic bundles over $X$ with trivial determinant bundle and $m+n$ marked points. We explicitly compute $\mathcal{P}_M(X,1,n)$ for $n=0,1,2$. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold $\mathcal{P}_M(\mathbb{CP}^1,3,n)$ is isomorphic for $n$ even to a space $\mathcal{Y}(S^2,n)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$-stranded links in $S^3$.

The endpoint theorem * *This paper is dedicated to the memory of Christopher J S Clarke.

The endpoint theorem links the existence of a sequence (curve), without accumulation points, in a manifold to the existence of an open embedding of that manifold so that the image of the given sequence (curve) has a unique endpoint. It plays a fundamental role in the theory of the abstract boundary as it implies that there is always an abstract boundary point to represent the endpoint of such sequences and curves. The endpoint theorem will be of interest to researchers analysing specific spacetimes as it shows how to construct a chart in the original manifold which contains the sequence (curve). In particular, it has application to the study of singularities predicted by the singularity theorems.

Kostant–Toda lattices and the universal centralizer

To each complex semisimple Lie algebra g decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant–Toda lattice, while the second is an integrable system defined on the universal centralizer Zg of g. These systems are similar in that each exploits and closely reflects the invariant theory of g, as developed by Chevalley, Kostant, and others. One also has Kostant’s description of level sets in the Kostant–Toda lattice, which turns out to suggest deeper similarities between the two integrable systems in question. Some more recent works allude to a strong connection between the two systems, e.g. Bezrukavnikov et al. (2005), Teleman (2014, 2018).We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant–Toda lattice into Zg. Secondary results include some qualitative features of the integrable system on Zg.

Absolute sets and the decomposition theorem

We give a framework to produce constructible functions from natural functors between categories, without need of a morphism of moduli spaces to model the functor. We show using the Riemann-Hilbert correspondence that any natural (derived) functor on constructible sheaves on smooth complex algebraic varieties can be used to construct a kind of constructible sets, called absolute sets, generalizing a notion introduced by Simpson in presence of moduli. We conjecture that the absolute sets of local systems satisfy a special varieties package, among which is an analog of the Manin-Mumford, Mordell-Lang, and Andre-Oort conjectures. The conjecture gives a simple proof of the Decomposition Theorem for all semi-simple perverse sheaves, assuming the Decomposition Theorem for the geometric ones. We prove the conjecture in the rank one case by showing that the closed absolute sets in this case are finite unions of torsion-translated affine tori. This extends a structure result of the authors for cohomology jump loci to any other natural jump loci. For example, to jump loci of intersection cohomology and Leray filtrations. We also show that the Leray spectral sequence for the open embedding in a good compactification degenerates for all rank one local systems at the usual page, not just for unitary local systems.

Fourier–Mukai and autoduality for compactified Jacobians, II

To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) Calabi-Yau projective varieties with locally complete intersection singularities. We define a Poincare' sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the integral transform with kernel the Poincare' sheaf is an equivalence of their derived categories. In particular, any two fine compactified Jacobians are derived equivalent. When applied to the same fine compactified Jacobian, one gets a Fourier-Mukai autoequivalence, which generalizes the classical result of Mukai for Jacobians of smooth curves (or more generally abelian varieties) and of Arinkin for compactified Jacobians of integral curves, thus providing further evidence for the classical limit of the geometric Langlands conjecture (as formulated by R. Donagi and T. Pantev). As a corollary of our main result, we prove an autoduality result for fine compactified Jacobians: there is a natural equivariant open embedding of the connected component of the scheme parametrizing rank-1 torsion-free sheaves on X into the connected component of the algebraic space parametrizing rank-1 torsion-free sheaves on a given fine compactified Jacobian of X.

Approximation forte pour les variétés avec une action d’un groupe linéaire

Let$G$be a connected linear algebraic group over a number field$k$. Let$U{\hookrightarrow}X$be a$G$-equivariant open embedding of a$G$-homogeneous space$U$with connected stabilizers into a smooth$G$-variety$X$. We prove that$X$satisfies strong approximation with Brauer–Manin condition off a set$S$of places of$k$under either of the following hypotheses:(i)$S$is the set of archimedean places;(ii)$S$is a non-empty finite set and$\bar{k}^{\times }=\bar{k}[X]^{\times }$.The proof builds upon the case$X=U$, which has been the object of several works.

Luna's fundamental lemma for diagonalizable groups

We study relatively affine actions of a diagonalizable group $G$ on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a $G$-equivariant morphism $f: X'\to X$ to be $strongly equivariant$, namely the base change of the morphism $f/\!/G$ of quotient schemes, and establish descent criteria for $f/\!/G$ to be an open embedding, etale, smooth, regular, syntomic, or lci.

Luna's fundamental lemma for diagonalizable groups

We study relatively affine actions of a diagonalizable group $G$ on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a $G$-equivariant morphism $f: X'\to X$ to be $strongly\ equivariant$, namely the base change of the morphism $f/\!/G$ of quotient schemes, and establish descent criteria for $f/\!/G$ to be an open embedding, \'etale, smooth, regular, syntomic, or lci.

Cohomology support loci of local systems

The support S of Sabbah's specialization complex is a simultaneous generalization of the set of eigenvalues of the monodromy on Deligne's nearby cycles complex, of the support of the Alexander modules of an algebraic knot, and of certain cohomology support loci. Moreover, it equals conjecturally the image under the exponential map of the zero locus of the Bernstein-Sato ideal. Sabbah showed that S is contained in a union of translated subtori of codimension one in a complex affine torus. Budur-Wang showed recently that S is a union of torsion-translated subtori. We show here that S is always a hypersurface, and that it admits a formula in terms of log resolutions. As an application, we give a criterion in terms of log resolutions for the (semi-)simplicity as perverse sheaves, or as regular holonomic D-modules, of the direct images of rank one local systems under an open embedding. For hyperplane arrangements, this criterion is combinatorial.

Linear differential equations on the Riemann sphere and representations of quivers

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and determine the existence of solutions of the generalized additive Deligne-Simpson problems. Moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere. Namely, open embedding of the moduli spaces into quiver varieties is given and the non-emptiness condition of the moduli spaces is determined. Furthermore the connectedness of the moduli spaces is shown.

On the Period Map for Polarized Hyperkähler Fourfolds

Abstract We study smooth projective hyperkähler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy’s Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW (Eisenbud–Popescu–Walter) sextics. In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperkähler fourfolds with Picard number 1 or 2.

The Influence of Heat Flow on the Nonuniformity of the Stresses in the Surface of Oxide Ceramic with Fully Developed Relief

The influence of a heat flow on the nonuniformity of stresses in the surface layer of oxide ceramic with fully developed relief is studied. A relationship between a symmetric heat flow and the characteristics of the nonuniformity of the stresses in the surface of elements of the structure of oxide ceramic for the case of open embedding of the spherical grain is discovered.