Canonical Morphism Research Articles

On Chow Rings of Quiver Moduli

Abstract We describe the point class and Todd class in the Chow ring of a moduli space of quiver representations, building on a result of Ellingsrud–Strømme. This, together with the presentation of the Chow ring by the second author, makes it possible to compute integrals on quiver moduli. To do so, we construct a canonical morphism of universal representations in great generality, and along the way point out its relation to the Kodaira–Spencer morphism. We illustrate the results by computing some invariants of some “small” Kronecker moduli spaces. We also prove that the first non-trivial (6-dimensional) Kronecker moduli space is isomorphic to the zero locus of a general section of $\mathcal{Q}^{\vee }(1)$ on $\textrm{Gr}(2,8)$.

On the sheafyness property of spectra of Banach rings

AbstractLet be a non‐Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that it is possible to associate to a homotopical Huber spectrum via the introduction of the notion of derived rational localization. The spectrum so obtained is endowed with a derived structural sheaf of simplicial Banach algebras for which the derived C̆ech–Tate complex is strictly exact. Under some hypothesis, we can prove that there is a canonical morphism of underlying topological spaces that is a homeomorphism in some well‐known examples of non‐sheafy Banach rings, where is the usual Huber spectrum of . This permits the use of the tools from derived geometry to understand the geometry of in cases when the classical structure sheaf is not a sheaf.

Bott-Chern hypercohomology and bimeromorphic invariants

AbstractThe aim of this article is to study the geometry of Bott-Chern hypercohomology from the bimeromorphic point of view. We construct some new bimeromorphic invariants involving the cohomology for the sheaf of germs of pluriharmonic functions, the truncated holomorphic de Rham cohomology, and the de Rham cohomology. To define these invariants, by using a sheaf-theoretic approach, we establish a blow-up formula together with a canonical morphism for the Bott-Chern hypercohomology. In particular, we compute the invariants of some compact complex threefolds, such as Iwasawa manifolds and quintic threefolds.

The Du Bois complex of a hypersurface and the minimal exponent

We study the Du Bois complex Ω_Z∙ of a hypersurface Z in a smooth complex algebraic variety in terms of its minimal exponent α˜(Z). The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein–Sato polynomial of Z, and refining the log-canonical threshold. We show that if α˜(Z)≥p+1, then the canonical morphism ΩZp→Ω_Zp is an isomorphism, where Ω_Zp is the pth associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if Z is singular and α˜(Z)>p≥2, we obtain non-vanishing results for some higher cohomologies of Ω_Zn−p.

Applications of Hochschild cohomology to the moduli of subalgebras of the full matrix ring

Let Moldn,d be the moduli of rank d subalgebras of Mn over Z. For x∈Moldn,d, let A(x)⊆Mn(k(x)) be the subalgebra of Mn corresponding to x, where k(x) is the residue field of x. In this article, we apply Hochschild cohomology to Moldn,d. The dimension of the tangent space TMoldn,d/Z,x of Moldn,d over Z at x can be calculated by the Hochschild cohomology H1(A(x),Mn(k(x))/A(x)). We show that H2(A(x),Mn(k(x))/A(x))=0 is a sufficient condition for the canonical morphism Moldn,d→Z being smooth at x. We also calculate Hi(A,Mn(k)/A) for several R-subalgebras A of Mn(R) over a commutative ring R. In particular, we summarize the results on Hi(A,Mn(k)/A) for all k-subalgebras A of Mn(k) over an algebraically closed field k in the case n=2,3.

Deformations and moduli of irregular canonical covers with K2=4pg-8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^2=4p_g-8$$\\end{document}

In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying K2=4pg-8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^2 = 4p_g-8$$\\end{document}, for any even integer pg≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_g\\ge 4$$\\end{document}. These surfaces also have unbounded irregularity q. We carry out our study by investigating the deformations of the canonical morphism φ:X→PN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi :X\\rightarrow {\\mathbb {P}}^N$$\\end{document}, where φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of X are unobstructed even though H2(TX)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^2(T_X)$$\\end{document} does not vanish. Consequently, X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality pg>2q-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_g > 2q-4$$\\end{document}, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with K2=2pg-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^2 = 2p_g- 4$$\\end{document}, studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus g≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\ge 3$$\\end{document}, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.

Deformations of Hyperelliptic and Generalized Hyperelliptic Polarized Varieties

The purpose of this article is twofold. Firstly, we address and completely solve the following question: Let (X, L) be a smooth, hyperelliptic polarized variety and let varphi : X longrightarrow Y subset textbf{P}^N be the morphism induced by |L|; when does varphi deform to a birational map? Secondly, we introduce the notion of “generalized hyperelliptic varieties” and carry out a study of their deformations. Regarding the first topic, we settle the non trivial, open cases of (X, L) being Fano-K3 and of (X, L) having dimension m ge 2, sectional genus g and L^m=2g. This was not addressed by Fujita in his study of hyperelliptic polarized varieties and requires the introduction of new methods and techniques to handle it. In the Fano-K3 case, all deformations of (X, L) are again hyperelliptic except if Y is a hyperquadric. By contrast, in the L^m=2g case, with one exception, a general deformation of varphi is a finite birational morphism. This is especially interesting and unexpected because, in the light of earlier results, varphi rarely deforms to a birational morphism when Y is a rational variety, as is our case. The Fano-K3 case contrasts with canonical morphisms of hyperelliptic curves and with hyperelliptic K3 surfaces of genus g ge 3. Regarding the second topic, we completely answer the question for generalized hyperelliptic polarized Fano and Calabi–Yau varieties. For generalized hyperelliptic varieties of general type we do this in even greater generality, since our result holds for Y toric. Standard methods in deformation theory do not work in the present setting. Thus, to settle these long standing open questions, we bring in new ideas and techniques building on those introduced by the authors concerning deformations of finite morphisms and the existence and smoothings of certain multiple structures. We also prove a new general result on unobstructedness of morphisms that factor through a double cover and apply it to the case of generalized hyperelliptic varieties.

De Rham Comparison and Poincaré Duality for Rigid Varieties

Over any smooth algebraic variety over a p-adic local field k, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham \({\mathbb {Z}}_p\)-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over k. In particular, we prove finiteness of étale cohomology with partial compact support of any \({\mathbb {Z}}_p\)-local systems, and establish the Poincaré duality for such cohomology after inverting p.

Symmetric states for C∗-Fermi systems

After introducing the infinite Fermi [Formula: see text]-tensor product of a single [Formula: see text]-graded [Formula: see text]-algebra as an inductive limit, we systematically study the structure of the so-called symmetric states, that is those which are invariant under the group consisting of all finite permutations of a countable set. Among the obtained results, we mention the extension of De Finetti theorem which asserts that a symmetric state is a “mixture” of product states, each of which is a product of a single even state. This result induces a canonical morphism of the simplexes made of the symmetric even states on the usual infinite [Formula: see text]-tensor product and the symmetric states on the infinite Fermi [Formula: see text]-tensor product. We then extend the so-called Klein transformation to the infinite Fermi [Formula: see text]-tensor product, available when the parity automorphism is inner. In such a situation, we investigate further properties of product states, the last being the extremal symmetric states on such an infinite Fermi [Formula: see text]-tensor product [Formula: see text]-algebra. This paper is complemented with a finite dimensional illustrative example for which the Klein transformation is not implementable, and then the Fermi tensor product might not generate a usual tensor product. Therefore, in general, the study of the symmetric states on the Fermi algebra cannot be easily reduced to that of the corresponding symmetric states on the usual infinite tensor product, even if both share many common properties.

Joyal's cylinder conjecture

For each pair of simplicial sets A and B, the category Cyl(A,B) of cylinders (also called correspondences) from A to B admits a model structure induced from Joyal's model structure for quasi-categories. In this paper, we prove Joyal's conjecture that a cylinder X∈Cyl(A,B) is fibrant if and only if the canonical morphism X⟶A⋆B is an inner fibration, and that a morphism between fibrant cylinders in Cyl(A,B) is a fibration if and only if it is an inner fibration. We use this result to give a new proof of a characterisation of covariant equivalences due to Lurie, which avoids the use of the straightening theorem. In an appendix, we introduce a new family of model structures on the slice categories sSet/B, whose fibrant objects are the inner fibrations with codomain B, which we use to prove some new results about inner anodyne extensions and inner fibrations.

Young's seminormal basis vectors and their denominators

We study Young's seminormal basis vectors of the dual Specht modules of the symmetric group, indexed by a certain class of standard tableaux, and their denominators. These vectors include those whose denominators control the splitting of the canonical morphism Δ(λ+μ)→Δ(λ)⊗Δ(μ) over Z(p), where Δ(ν) is the Weyl module of the classical Schur algebra labelled by ν.

On linear exactness properties

We study those exactness properties of a regular category C that can be expressed in the following form: for any diagram of a given ‘finite shape’ in C, a given canonical morphism between finite limits built from this diagram is a regular epimorphism. The main goal of the paper is to characterize essentially algebraic categories which satisfy this property via (essential versions of) linear Mal'tsev conditions, which are known to correspond to the so-called matrix properties. We then apply this characterization, along with our earlier work on preservation of exactness properties by pro-completions, to prove that these exactness properties can be reduced to matrix properties already in the general setting of regular categories.

Morphisms to noncommutative projective lines

Let $k$ be a field, let ${\sf C}$ be a $k$-linear abelian category, let $\underline{\mathcal{L}}:=\{\mathcal{L}_{i}\}_{i \in \mathbb{Z}}$ be a sequence of objects in ${\sf C}$, and let $B_{\underline{\mathcal{L}}}$ be the associated orbit algebra. We describe sufficient conditions on $\underline{\mathcal{L}}$ such that there is a canonical morphism from the noncommutative space ${\sf Proj }B_{\underline{\mathcal{L}}}$ to a noncommutative projective line in the sense of \cite{abstractp1}, generalizing the usual construction of a map from a scheme $X$ to $\mathbb{P}^{1}$ defined by an invertible sheaf $\mathcal{L}$ generated by two global sections. We then apply our results to construct, for every natural number $d>2$, a degree two cover of Piontkovski's $d$th noncommutative projective line by a noncommutative elliptic curve in the sense of Polishchuk.

A Motivic Segal Theorem for Pairs (Announcement)

In order to provide a new, more computation-friendly, construction of the stable motivic category SH(k), V. Voevodsyky laid the foundation of delooping motivic spaces. G. Garkusha and I. Panin based on joint works with A. Ananievsky, A. Neshitov, and A. Druzhinin made that project a reality. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field k and any k-smooth scheme X, the canonical morphism of motivic spaces $$ {C}_{\ast } Fr(X)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left({X}_{+}\right) $$ is a Nisnevich locally group-completion. In the present paper, a generalization of that theorem is established to the case of smooth open pairs (X,U), where X is a k-smooth scheme and U is its open subscheme intersecting each component of X in a nonempty subscheme. It is claimed that in this case the motivic space C*Fr((X,U)) is a Nisnevich locally connected, and the motivic space morphism $$ {C}_{\ast } Fr\left(\left(X,U\right)\right)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left(X/U\right) $$ is Nisnevich locally weak equivalence. Moreover, it is proved that if the codimension of S = X−U in each component of X is greater than r ≥ 0, then the simplicial sheaf C*Fr((X,U)) is locally r-connected.

Tensor topology

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.

Twisting structures and morphisms up to strong homotopy

We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the “strong homotopy” morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.

The global derived period map

We develop the global period map in the context of derived geometry, generalising Griffiths' classical period map as well as the infinitesimal derived period map. We begin by constructing the derived period domain which classifies Hodge filtrations and enhances the classical period domain. We analyze the monodromy action. Then we associate to any polarized smooth projective map of derived stacks a canonical morphism of derived analytic stacks from the base into the quotient of the derived period domain by monodromy. We conclude the paper by discussing a few examples and a derived Torelli problem. In the appendix we describe how to present derived analytic Artin stacks as hypergroupoids, which may be of independent interest.

On the stable Andreadakis problem

Let Fn be the free group on n generators. Consider the group IAn of automorphisms of Fn acting trivially on its abelianization. There are two canonical filtrations on IAn: the first one is its lower central series Γ⁎; the second one is the Andreadakis filtration A⁎, defined from the action on Fn. In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ⁎) and L(A⁎) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included.

Categories of Bott-Samelson Varieties

We consider all Bott-Samelson varieties BS(s) for a fixed connected semisimple complex algebraic group with maximal torus T as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms BS(s)↪BS(s′), where s is a subsequence of s′. Every morphism of the new category induces a map between the T-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded \(\phantom {\dot {i}\!}H^{\bullet }_{T}(\text {pt})\)-modules coinciding on the objects with the usual functor \(\phantom {\dot {i}\!}H_{T}^{\bullet }\) of taking T-equivariant cohomologies. We also discuss the problem how to define a functor to the category of T-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones).

Boardman's Whole-Plane Obstruction Group for Cartan-Eilenberg Systems

Each extended Cartan-Eilenberg system (H, \partial) gives rise to two exact couples and one spectral sequence. We show that the canonical colim-lim interchange morphism associated to H is a surjection, and that its kernel is isomorphic to Boardman's whole-plane obstruction group W , for each of the two exact couples.