Singular Varieties Research Articles

Singularities of secant varieties from a Hodge theoretic perspective

AbstractWe study the singularities of secant varieties of smooth projective varieties using methods from birational geometry when the embedding line bundle is sufficiently positive. More precisely, we study the Du Bois complex of secant varieties and its relationship with the sheaves of differential forms. Through this analysis, we give a necessary and sufficient condition for these varieties to have ‐Du Bois singularities (in a sense that was proposed in Shen, Venkatesh, and Vo [On k‐Du Bois and k‐rational singularities, arXiv e‐prints (June 2023), arXiv:2306.03977]). In addition, we show that the singularities of these varieties are never higher rational, by giving a classification of the cases when they are pre‐1‐rational. From these results, we deduce several consequences, including a Kodaira–Akizuki–Nakano type vanishing result for the reflexive differential forms of the secant varieties.

The Brasselet–Schürmann–Yokura Conjecture on L-Classes of Projective Rational Homology Manifolds

Abstract In 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson $L$-class $L_{*}(X)$ and the Hirzebruch homology class $T_{1,*}(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest.

The Morrison–Kawamata cone conjecture for singular symplectic varieties

The Morrison–Kawamata cone conjecture for singular symplectic varieties

On ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-Quasiordinary Surface Singularities and Their Resolution

Quasiordinary power series were introduced by Jung at the beginning of the 20th century, and were not paid much attention until the work of Lipman and, later on, Gao. They have been thoroughly studied since, as they form a very interesting family of singular varieties, whose properties (or at least many of them) can be encoded in a discrete set of integers, much as what happens with curves. Hironaka proposed a generalization of this concept, the so-called ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-quasiordinary power series, which has not been examined in the literature in such detailed way. This paper explores the behavior of these series under the resolution process in the surface case.

Generalized Hamiltonian systems on subvarieties: constant rank case

For the constraint variety in symplectic manifold, the solvable Hamiltonian vector fields on the constraint are investigated. According to P.A.M. Dirac [3], the space of solvable Hamiltonian systems is determined by the geometric restriction of the symplectic form to the constraint. Solvability condition of the generalized Hamiltonian systems is extended to singular varieties and applied under some assumption on singularities. The constraint being a smooth submanifold in a symplectic space was considered in [6]. In this paper, we investigate the solvability of generalized Hamiltonian systems and the constraint invariants on singular constraints in the constant rank case.

Milnor–Wood inequality for klt varieties of general type and uniformization

AbstractWe generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood‐type inequality and we characterize the corresponding maximal representations.

Singular contact varieties

In this note we propose the generalization of the notion of a holomorphic contact structure on a manifold (smooth variety) to varieties with rational singularities and prove basic properties of such objects. Natural examples of singular contact varieties come from the theory of nilpotent orbits: every projectivization of the closure of a nilpotent orbit in a semisimple Lie algebra satisfies our definition after normalization. We show the correspondence between symplectic varieties with the structure of a C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^*$$\\end{document}-bundle and the contact ones along with the existence of stratification à la Kaledin. In the projective case we demonstrate the equivalence between crepant and contact resolutions of singularities, show the uniruledness and give a full classification of projective contact varieties in dimension 3.

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampere operators. We then use the Perron method to solve Monge-Ampere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoe, Caffarelli-Viaclovsky and Hultgren-Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

Determinant morphism for singular varieties

Determinant morphism for singular varieties

Categorical absorptions of singularities and degenerations

We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. We construct (under appropriate assumptions) a categorical absorption for a projective variety $X$ with isolated ordinary double points. We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smooth curve $B$, the smooth part of the derived category of $X$ extends to a smooth and proper over $B$ family of triangulated subcategories in the fibers of $\mathcal{X}$.

On the embedded Nash problem

Abstract The embedded Nash problem for a hypersurface in a smooth algebraic variety is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal models of the pair contribute with such families. We solve the problem for unibranch plane curve germs, in terms of the resolution graph. These are embedded analogs of known results for the classical Nash problem on singular varieties.

Cohomological Descent for Faltings Ringed Topos

Abstract Faltings ringed topos, the keystone of Faltings’ approach to p-adic Hodge theory for a smooth variety over a local field, relies on the choice of an integral model, and its good properties depend on the (logarithmic) smoothness of this model. Inspired by Deligne’s approach to classical Hodge theory for singular varieties, we establish a cohomological descent result for the structural sheaf of Faltings topos, which makes it possible to extend Faltings’ approach to any integral model, that is, without any smoothness assumption. An essential ingredient of our proof is a variation of Bhatt–Scholze’s arc-descent of perfectoid rings.

The uniqueness theorem for Gysin coherent characteristic classes of singular spaces

AbstractWe establish a general computational scheme designed for a systematic computation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly geometric and of a recursive nature terminating on genera of explicit characteristic subvarieties that we construct. It enables us, for example, to apply intersection theory of Schubert varieties to obtain a uniqueness result for such characteristic classes in the homology of an ambient Grassmannian. Our framework applies in particular to the Goresky–MacPherson ‐class by virtue of the Gysin restriction formula obtained by the first author in previous work. We illustrate our approach for a systematic computation of the ‐class in terms of normally nonsingular expansions in examples of singular Schubert varieties that do not satisfy Poincaré duality over the rationals.

Deformations of half-canonical Gorenstein curves in codimension four

Recent work of Ablett (2021) and Kapustka, Kapustka, Ranestad, Schenck, Stillman and Yuan (2021) outlines a number of constructions for singular Gorenstein codimension four varieties. Earlier work of Coughlan, Gołȩbiowski, Kapustka and Kapustka (2016) details a series of nonsingular Gorenstein codimension four constructions with different Betti tables. In this paper we exhibit a number of flat deformations between Gorenstein codimension four varieties in the same Hilbert scheme, realising many of the singular varieties as specialisations of the earlier nonsingular varieties.

Bloch’s cycle complex and coherent dualizing complexes in positive characteristic

Let X X be a separated scheme of dimension d d of finite type over a perfect field k k of positive characteristic p p . In this work, we show that Bloch’s cycle complex Z X c \mathbb {Z}^c_X of zero cycles mod p n p^n is quasi-isomorphic to the Cartier operator fixed part of a certain dualizing complex from coherent duality theory. From this we obtain new vanishing results for the higher Chow groups of zero cycles with mod p n p^n coefficients for singular varieties.

About relative polar varieties and Brasselet numbers

In this work, we study the consequences of an empty polar variety on the topology of a function-germ with (possibly) nonisolated singularities defined on a singular variety.

Reconstruction of Projective Curves from the Derived Category

We prove that integral projective curves with Cohen–Macaulay singularities over an algebraically closed field of characteristic zero are determined by their triangulated category of perfect complexes. This partially extends a theorem of Bondal and Orlov to the case of singular projective varieties and in particular shows that all integral complex projective curves are determined by their category of perfect complexes.

Phonological Inclusion, and Exclusion, Regarding South African English in the Online OED

This article approaches the subject of inclusivity and diversity from a phonological perspective, as applied to the online Oxford English Dictionary (OED). Specifically, the pronunciation guide within the OED entry for South African English (SAE) is investigated. On the one hand, that the OED provides a list of World Englishes is a step forward in terms of diversity, avoiding the linguistic hegemony of British and American English, for example. On the other hand, the pronunciation guide for SAE relies on a rather singular variety — that reflective of white South Africans. Moreover, the OED pronunciation guides for all varieties of English reflect "educated urban speakers of standard English". Therefore, the pronunciation guide is exclusive regarding race and class, and thus additional varieties of SAE are not represented. As such, while the OED, as a trusted source, has made efforts to address linguistic diversity, more work needs to be done in this area. Keywords: pronunciation, accent, White South African English, great trichotomy, exclusivity, language attitudes

A singular variety associated to the smallest degree Pinchuk map

A singular variety associated to the smallest degree Pinchuk map

Small irreducible components of arc spaces in positive characteristic

We study the Irreducibility Problem for the arc space X ∞ of an irreducible singular algebraic variety X defined over a perfect field k of characteristic p > 0 . The existence of reducible such X ∞ is related to the fact that Kolchin's Irreducibility Theorem fails to hold in positive characteristic. We obtain a complete description of the irreducible components of X ∞ when X is a surface. Section 5 introduces two main new problems in arbitrary dimension: (1) in the line of O. Zariski and H. Hironaka, blowing up any X to get Y → X with Y ∞ irreducible; (2) in the line of J. Nash's work on arcs, characterizing irreducibility in terms of Resolution of Singularities.