In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group $G$, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case $G = \mathrm{SL}_{2}(k)$, the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and in the parabolic scenarios.
In characteristic $p=2$, we compute the singularities of Kummer varieties arising from products of elliptic curves. This result is generalized to Kummer varieties associated to ordinary abelian varieties.
The paper is based on a talk. Complete exposition is given in "Equivariant Hirzebruch class for singular varieties". Starting from the classical theory we describe Hirzebruch class and the related Todd genus of a complex singular algebraic varieties. When the variety is equipped with an action of an algebraic torus we localize the Hirzebruch class at the fixed points of the action. We give some examples of computations.
We calculate the $E$-polynomials of the $SL_3(\mathbb{C})$ and $GL_3(\mathbb{C})$-character varieties of compact oriented surfaces of any genus and the $E$-polynomials of the $SL_2(\mathbb{C})$ and $GL_2(\mathbb{C})$-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 $SL_2(\mathbb{C})$-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 $g \ge 3$. Our technique is based…
We first construct explicit bases for the cotangent spaces at singular points on Hibi toric varieties, i.e., toric varieties associated to distributive lattices. We then determine the singular loci of these toric varieties.
Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood-Richardson Rule for flag varieties. We discuss three natural generalizations of Richardson varieties which we call projection varieties, intersection varieties, and rank varieties. In many ways, these varieties are more fundamental than Richardson varieties and are more easily amenable to inductive geometric constructions. In this paper, we study the singularities of each type of generalization. Like Richardson varieties, projection varieties are normal with rational singula…
For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain Segre-Plücker embeddings in series of papers. In this work we give a classification of the irreducible components of the singular locus of the dual Grassmannian. Basically, it admits two components: cusp type and node type which are degeneracies of a certain Hessian matrix, and the closure of the set of tangent planes having more than one critical point, respectivel…
Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert varieties, they contain distinguished points called permutation flags. In this paper, we give a combinatorial characterization for a permutation flag of a regular nilpotent Hessenberg variety to be a singular point. We also apply this result to characterize regular nilpotent Hessenberg varieties which are normal algebraic varieties.
This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus.
We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A short survey on the structure of the Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's question on the rationality of the irreducible components of Chow varieties.
Let $N$ be a point of an orbit closure $\bar{O_M}$ in a module variety such that its orbit $O_N$ has codimension two in $\bar{O_M}$. We show that under some additional conditions the pointed variety $(\bar{O_M},N)$ is smoothly equivalent to a cone over a rational normal curve.
We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their computation.
Let X be a projective variety, possibly singular. A generalized Albanese variety of X was constructed by Esnault, Srinivas and Viehweg over algebraically closed base field as a universal regular quotient of the relative Chow group of 0-cycles by Levine-Weibel. In this paper, we obtain a functorial description of the Albanese of Esnault-Srinivas-Viehweg over a perfect base field, using duality theory of 1-motives with unipotent part.
Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and over the corresponding type $ADE$ singular surfaces. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles.
For the toric variety X associated to the Bruhat poset of Schubert varieties in a minuscule G/P, we describe the singular locus in terms of the faces of the associated polyhedral cone. We further show that the singular locus is pure of codimension 3 in X.
Although regular semisimple Hessenberg varieties are smooth and irreducible, semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of semisimple Hessenberg varieties corresponding to the standard Hessenberg space. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of thi…
This paper surveys the authors recent work on two variable elliptic genus of singular varieties. The last section calculates a generating function for the elliptic genera of symmetric products. This generalizes the classical results of Macdonald and Zagier.
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.
The Hirzebruch $td_y(X)$ class of a complex manifold X is a formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class. The related $χ_y$-genus admits a generalization for singular complex algebraic varieties. The equivariant version of the Hirzebruch class can be developed as well. The general theory applied in the situation when a torus acts on a singular variety allows to apply powerful tools as the Localization Theorem of Atiyah and Segal for equivariant K-theory and Berline-Vergne formula for equivariant cohomology. We obtain a meaningful inva…
In this note, we prove the generic Kobayashi volume measure hyperbolicity of singular directed varieties $(X, V)$, as soon as the canonical sheaf $\mathcal{K}\_V$ of $V$ is big in the sense of Demailly.
A theorem of Jannsen asserts that if a smooth projective variety has injective cycle class maps, it has surjective cycle class maps. The object of this note is to present a version of Jannsen's theorem for singular quasi--projective varieties.
We study the Chow group of zero-cycles on singular varieties using the cdh topology. We define the cdh versions of the zero-cycles and albanese maps. We prove results comparing these groups for a singular variety with the similar groups on the resolution of singularities. We use these to prove some results about the known Chow group of zero-cycles on surfaces and threefolds and some cases of arbitrary dimension.
This expository note surveys some results on equivariant K-theory of varieties with a torus action, focusing on recent work with Sam Payne and Richard Gonzales. It is based on my contribution to the Clifford Lectures at Tulane University in March 2015.
Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g \ge 3.$ In this paper we prove the connectedness of the singular sublocus of $\mathcal{A}_g$ consisting of those abelian varieties which possess an involution different from $-id$.
We correct the proof and slightly strengthen a Kodaira-type vanishing theorem for singular varieties originally due to Jaffe and the first author. Specifically, we show that if $L$ is a nef and big line bundle on a projective variety of characteristic zero, the $i^{\text{th}}$ cohomology of $L^{-1}$ vanishes for $i$ in a range determined by the depth and dimension of the singular locus.
Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities. This is shown as a corollary to the characterization of a singularity that is obtained as a $K3^{[2]}$-type contraction and has a unique symplectic resolution.
Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kaehler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kaehler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular S…
We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jarosław Włodarczyk.
What is generally known as the "Bloch--Srinivas method" consists of decomposing the diagonal of a smooth projective variety, and then considering the action of correspondences in cohomology. In this note, we observe that this same method can also be extended to singular and quasi--projective varieties. We give two applications of this observation: the first is a version of Mumford's theorem, the second is concerned with the Hodge conjecture for singular varieties.
We prove that any ample class on a primitive symplectic variety that is locally trivial deformation of O'Grady's singular 6 dimensional example is proportional to the first Chern class of a uniruled divisor. This result answers a question of Lehn, Mongardi and Pacienza, extending their result for primitive symplectic varieties of this deformation type.
For the toric variety X associated to the Bruhat poset of Schubert varieties in the Grassmannian, we describe the singular locus in terms of the faces of the associated polyhedral cone. We also determine the tangent cones at the maximal singularities of X. These turn out to be again toric varieties. In the case of X being associated to the Bruhat poset of Schubert varieties in the Grassmannian of 2-planes in a n-dimensional vector space (over the base field), we also prove a certain product formula relating the multiplicities at certain singular points.
We calculate using Macaulay 2 the multiplicities of the most singular point on Schubert varieties on Gl(n)/B for $n=5,6$. The method of computation is described and tables of the results are included.
We prove a generalization of Demailly-Ein-Lazarsfeld's subadditivity formula and Mustata's summation formula for multiplier ideals to the case of singular varieties, using characteristic $p$ methods. As an application of our formula, we improve Hochster-Huneke's result on the growth of symbolic powers of ideals in singular affine algebras.
We generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted projective spaces and singular toric varieties. We generalize the Riemann-Roch equations to weighted projective spaces. We compute counting matrices of smooth Fano threefolds with Picard group Z and anticanonical degrees 2, 8, and 16.
This is a second part of alg-geom/9611015. We construct a natural hyperkaehler desingularization for all singular hyperkaehler varieties. The desingularization theorem was proven in alg-geom/9611015 under additional assumption of local homogeneity. Here we show that local homogeneity is redundant: every singular hyperkaehler variety has locally homogeneous singularities.
We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in particular, implies that it is at least 2. We also prove that the commuting variety is rational.
We prove the Yau-Tian-Donaldson's conjecture for any $\mathbb{Q}$-Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the previous result for smooth Fano varieties to this class of singular $\mathbb{Q}$-Fano varieties, which include those admitting crepant log resolutions.
This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.
A homotopical treatment of intersection cohomology recently developed by Chataur-Saralegui-Tanré associates a "perverse algebraic model" to every topological pseudomanifold, extending Sullivan's presentation of rational homotopy to intersection cohomology. In this context, there is a notion of "intersection-formality", measuring the vanishing of Massey products in intersection cohomology. In the present paper, we study the perverse algebraic model of complex projective varieties with isolated singularities. We endow such invariant with natural mixed Hodge structures. This allows us to prove so…
We discuss the basic properties of various versions of two variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the elliptic genera of Witten's phases on $N=2$ theories.
In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight $-1$.
We introduce a certain index of a collection of germs of 1-forms on a germ of a singular variety which is a generalization of the local Euler obstruction corresponding to Chern numbers different from the top one.
This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.
Schmidt and Spieß described the abelian tame fundamental group of a smooth variety over a finite field by using Suslin homology. In this paper we show that their result generalizes to singular varieties if one uses Weil-Suslin homology instead.
The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.
It was recently established by the first two authors that multiplier ideals on a smooth variety satisfy some special syzygetic properties. The purpose of this note is to show how some of these can be extended to the singular setting.
We prove that the non-nef locus and the restricted base locus of a pseudoeffective divisor coincide on KLT pairs. We also extend to KLT pairs F. Russo's characterization of nef and abundant divisors by means of asymptotic multiplier ideals.
We give various examples of Q-factorial projective toric varieties such that the sum of the squared torus invariant prime divisors is positive. We also determine the generators for the cone of effective $2$-cycles on a toric variety of Picard number two. This result is convenient to explain our examples.
