Singular Varieties Research Articles

Introduction: Self-Identity and Ambivalence

In this introduction to the first installment of the Common Knowledge symposium on xenophilia, the editor explains the rationale of the new project, citing increases in aggressive xenophobia internationally. He comments on the intergroup-relations theorist Todd Pittinsky's argument that, since tolerance is not logically the antithesis of negative feelings toward out-groups, even long-established traditions of toleration are inadequate to prevent intergroup aggression. Pittinsky proposes that tolerance be replaced, as a principle of peacekeeping, by the encouragement of positive feelings toward out-groups, and the author of this essay responds by showing how the Freudian theory of ambivalence and the history of literature on which Freud drew in constructing it support Pittinsky's viewpoint. Enemies regularly fall in love, above all in Shakespearean drama, which Freud explains by arguing that all human relations are love-hate relations. Thus, this essay suggests, it takes as much psychic energy to repress the love in what appear to be relationships of hatred as to repress the hatred in what appear to be relationships of love. There is a metaphysical component as well to this argument: the xenophobe and xenophile equally presuppose that the identity principle (a = a) is applicable to society as well as to mathematics; both assume that each discrete social group is self-identical and differs from all other groups more or less radically. The main difference between –phile and –phobe is the latter's relative incapacity to live with ambivalence. Given these arguments, one must expect to find negative as well as positive motives in the etiology and conduct of xenophilia—and the first installment of the Common Knowledge symposium is said to focus on cases of xenophilia in which varieties of enmity accompany that singular and underrated variety of love.

Two-dimensional gauge dynamics and the topology of singular determinantal varieties

We record an observation about the Witten indices in two families of gauged linear sigma models: the U(2) model for linear sections of Grassmannians, and the U(1) model for quadric complete intersections. We describe how the Witten indices are related to the Euler characteristics of the singular skew-symmetric or symmetric determinantal varieties featuring in the analysis of their opposite phases, and we discuss the extent to which these relationships can be reconciled with standard Born-Oppenheimer arguments.

Arithmetic of singular character varieties and their E‐polynomials

We calculate the E-polynomials of the S L 3 ( C ) - and G L 3 ( C ) -character varieties of compact oriented surfaces of any genus and the E-polynomials of the S L 2 ( C ) - and G L 2 ( C ) -character varieties of compact non-orientable surfaces of any Euler characteristic. Our methods also give a new and significantly simpler computation of the E-polynomials of the S L 2 ( C ) -character varieties of compact orientable surfaces, which were computed by Logares, Muñoz and Newstead for genus g = 1 , 2 and by Martinez and Muñoz for g ⩾ 3 . Our technique is based on the arithmetic of character varieties over finite fields. More specifically, we show how to extend the approach of Hausel and Rodriguez-Villegas used for non-singular (twisted) character varieties to the singular (untwisted) case.

Differential forms in positive characteristic avoiding resolution of singularities

This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than Kahler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdhtopology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with Kahler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number of examples highlight the difficulties that arise with torsion forms and with alternative candiates.

Stringy Chern classes of singular toric varieties and their applications

Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating d-dimensional reflexive polytopes to the number 12 in dimension d>3.

Extremal transition and quantum cohomology: Examples of toric degeneration

When a singular projective variety Xsing admits a projective crepant resolution Xres and a smoothing Xsm, we say that Xres and Xsm are related by extremal transition. In this article, we study a relationship between the quantum cohomology of Xres and Xsm in some examples. For 3-dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing Xsm is isomorphic to a certain subquotient of the quantum cohomology of a resolution Xres with the quantum variables of exceptional curves specialized to one. We observe that similar phenomena happen for toric degenerations of Fl(1,2,3), Gr(2,4), and Gr(2,5) by explicit computations.

The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces

We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex $K$-theory ring of a divisive weighted projective space (which is singular for non-trivial weights) is isomorphic to the ring of integral piecewise Laurent polynomials on the associated fan. Analogues of this description hold for other complex-oriented equivariant cohomology theories, as we confirm in the case of homotopical complex cobordism, which is the universal example. We also prove that the Borel versions of the equivariant $K$-theory and complex cobordism rings of more general singular toric varieties, namely those whose integral cohomology is concentrated in even dimensions, are isomorphic to rings of appropriate piecewise formal power series. Finally, we confirm the corresponding descriptions for any smooth, compact, projective toric variety, and rewrite them in a face ring context. In many cases our results agree with those of Vezzosi and Vistoli for algebraic $K$-theory, Anderson and Payne for operational $K$-theory, Krishna and Uma for algebraic cobordism, and Gonzalez and Karu for operational cobordism; as we proceed, we summarize the details of these coincidences.

Resolution of singularities of the cotangent sheaf of a singular variety

The main problem studied here is resolution of singularities of the cotangent sheaf of a complex- or real-analytic space X0 (or of an algebraic variety X0 over a field of characteristic zero). Given X0, we ask whether there is a global resolution of singularities σ:X→X0 such that the pulled-back cotangent sheaf of X0 is generated by differential monomials in suitable coordinates at every point of X (“Hsiang–Pati coordinates”). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of σ. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang–Pati 1985, Pardon–Stern 2001). Consequences include monomialization of the induced Fubini–Study metric on the smooth part of a complex projective variety X0; there have been important applications of the latter to L2-cohomology.

An approach to intersection theory on singular varieties using motivic complexes

We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed.

Semistability of the tangent sheaf of singular varieties

Semistability of the tangent sheaf of singular varieties

How many hypersurfaces does it take to cut out a Segre class?

We prove an identity of Segre classes for zero-schemes of compatible sections of two vector bundles. Applications include bounds on the number of equations needed to cut out a scheme with the same Segre class as a given subscheme of (for example) a projective variety, and a ‘Segre–Bertini’ theorem controlling the behavior of Segre classes of singularity subschemes of hypersurfaces under general hyperplane sections.These results interpolate between an observation of Samuel concerning multiplicities along components of a subscheme and facts concerning the integral closure of corresponding ideals. The Segre–Bertini theorem has applications to characteristic classes of singular varieties. The main results are motivated by the problem of computing Segre classes explicitly and applications of Segre classes to enumerative geometry.

Deformations of singular symplectic varieties and termination of the log minimal model program

Deformations of singular symplectic varieties and termination of the log minimal model program

An analogue of the Aluffi algebra for modules

P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism A↠B by taking a suitable inverse limit of graded algebras, one for each representation of A as a residue ring of a given “ambient” ring R. Since giving a ring surjection A↠B is tantamount to giving an ideal I⊂A, it would seem natural to ask for an analogous notion for A-modules. This is the central purpose of this work. Since a given module may not admit any embedding into a free module, a preparatory toil includes dealing with this technical point at the outset. On the bright side, the intrusion of modules raises a few algebraic questions interesting on their own. It is to expect that this extension to modules may be transcribed in terms of coherent sheaves, thus possibly providing an answer to a question by Aluffi in this regard. Two main bodies of examples are treated in detail to illustrate how the theory works and to show the relation to finer properties of other algebras.

Large N topologically twisted index: necklace quivers, dualities, and Sasaki-Einstein spaces

In this paper, we calculate the topological free energy for a number of ${\mathcal N} \geq 2$ Yang-Mills-Chern-Simons-matter theories at large $N$ and fixed Chern-Simons levels. The topological free energy is defined as the logarithm of the partition function of the theory on $S^2 \times S^1$ with a topological A-twist along $S^2$ and can be reduced to a matrix integral by exploiting the localization technique. The theories of our interest are dual to a variety of Calabi-Yau four-fold singularities, including a product of two asymptotically locally Euclidean singularities and the cone over various well-known homogeneous Sasaki-Einstein seven-manifolds, $N^{0,1,0}$, $V^{5,2}$, and $Q^{1,1,1}$. We check that the large $N$ topological free energy can be matched for theories which are related by dualities, including mirror symmetry and $\mathrm{SL}(2,\mathbb{Z})$ duality.

The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction

The Kutta condition is applied to aerofoils with sharp trailing edges to allow for viscous effects to be considered within a simplified system of equations that are inviscid. This paper discusses in detail the inclusion of an unsteady Kutta condition at a sharp trailing edge during gust–aerofoil interaction and illustrates how the analytic solution for the far-field noise generated by this interaction changes if the unsteady Kutta condition is neglected, or more precisely, if the unsteady pressure is permitted to be singular at the trailing edge. The analytic solution, both with and without the unsteady Kutta condition, is compared with numerical results that have no imposed unsteady Kutta condition. Importantly the results agree well only when the unsteady Kutta condition is neglected in the analytic solution. This paper highlights where the far-field acoustics are most affected by neglecting the unsteady Kutta condition for a variety of singularities that can occur in the unsteady pressure at the trailing edge and shows that results permitting different behaviour in the unsteady surface pressure at the trailing edge could give significantly different far-field noise predictions, even though the surface pressure elsewhere along the aerofoil surface agrees with benchmark solutions.

On singular Fano varieties with a divisor of Picard number one

In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.

On cscK resolutions of conically singular cscK varieties

In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.

Algebraic Versus Homological Equivalence for Singular Varieties

Let Y ⊆ ℙN be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimYsing ≤ min {d + h − 1, n − 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H2(d+h)(Y; ℚ) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.

A closer look at mirrors and quotients of Calabi-Yau threefolds

Let X be the toric variety (P1)4 associated with its four-dimensional polytope 1. Denote by X˜ the resolution of the singular Fano variety Xo associated with the dual polytope 1o. Generically, anticanonical sections Y of X and anticanonical sections Y˜ of X˜ are mirror partners in the sense of Batyrev. Our main result is the following: the Hodge-theoretic mirror of the quotient Z associated to a maximal admissible pair (Y, G) in X is not a quotient Z˜ associated to an admissible pair in X˜ . Nevertheless, it is possible to construct a mirror orbifold for Z by means of a quotient of a suitable Y˜. Its crepant resolution is a Calabi-Yau threefold with Hodge numbers (8, 4). Instead, if we start from a non-maximal admissible pair, in the same case, its mirror is the quotient associated to an admissible pair.

Weak solutions to degenerate complex Monge–Ampère flows II

Studying the (long-term) behavior of the Kähler–Ricci flow on mildly singular varieties, one is naturally led to study weak solutions of degenerate parabolic complex Monge–Ampère equations.The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Monge–Ampère flows on compact Kähler manifolds. Our general theory allows in particular to define and study the (normalized) Kähler–Ricci flow on varieties with canonical singularities, generalizing results of Song and Tian.