We study certain mild degenerations of algebraic varieties which appear in the analysis of a large class of supersymmetric theories, including superstring theory. We analyze Witten's sigma-model and find that the non-transversality of the superpotential induces a singularization and stratification of the ground state variety. This stratified variety (the union of the singular ground state variety and its exo-curve strata) admit homology groups which, excepting the middle dimension, satisfy the "Kahler package" of requirements, extend the "flopped" pair of small resolutions to an "(exo)flopped" triple, and is compatible with mirror symmetry and string theory. Finally, we revisit the conifold transition as it applies to our formalism.
Intersection homology is obtained from ordinary homology by imposing conditions on how the embedded simplices meet the strata of a space $X$. In this way, for the middle perversity, properties such as strong Lefschetz are preserved. This paper defines local-global intersection homology groups, that record global information about the singularities of $X$. They differ from intersection homology in that stratified rather than ordinary simplices are used. An example of such is $σ_j\times Cσ_i$, where $σ_i$ and $σ_j$ are ordinary simplices, and $C$ is the coning operator. The paper concludes with …
We study the GKM theory for a equivariant stratified space having orbifold structures in tis successive quotients. Then, we introduce the notion of an \emph{almost simple polytope}, as well as a \emph{divisive toric variety} generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties.
We determine an upper bound for the cohomological dimension of the complement of a closed subset in a projective variety which possesses an appropriate stratification. We apply the result to several particular cases, including the Bialynicki-Birula stratification; in this latter case, the bound is optimal.
We prove that the Picard group of a regular simply connected variety over an algebraically closed field of arbitrary characteristic is finitely generated. The main difficulty to overcome is the unavailability of resolution of singularities. From this we deduce that in positive characteristic there exist no nontrivial stratified line bundles on such a variety, and we present a complex analog.
The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the homology groups of the transversal slices of the corresponding strata.
We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology, which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of log pairs in the direction of …
We show that for a complete complex algebraic variety the pure component of homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce "image homology" for noncomplete varieties.
By a theorem of A.Björner, for every interval $[u,v]$ in the Bruhat order of a Coxeter group $W$, there exists a stratified space whose strata are labeled by the elements of $[u,v]$, adjacency is described by the Bruhat order, and each closed stratum (resp., the boundary of each stratum) has the homology of a ball (resp., of a sphere). Answering a question posed by Björner, we suggest a natural geometric realization of these stratified spaces for a Weyl group $W$ of a semisimple Lie group $G$, and prove its validity in the case of the symmetric group. Our stratified spaces arise as links in th…
In this paper we introduce the Fourier-Mukai transform for Lawson homology of abelian varieties and prove an inversion theorem for the Lawson homology as well as the morphic cohomology of abelian varieties. As applications, we obtain the direct sum decomposition of the Lawson homology and the morphic cohomology groups with rational coefficients, inspired by Beauville's works on the Chow theory. An analogue of the Beauville conjecture for Chow groups is proposed and is shown to be equivalent to the (weak) Suslin conjecture for Lawson homology. A filtration on Lawson homology is proposed and con…
We generalize some classical results on Chow group of an abelian variety to semiabelian varieties and to motivic (co)homology, using a result of Ancona--Enright-Ward--Huber on a decomposition of the motive of a semiabelian variety in the Voevodsky's category.
We give an explicit formula for the zeroth $\mathbb{A}^1$-homology sheaf of a smooth proper variety. We also provide a simple proof of a theorem of Kahn-Sujatha which describes hom sets in the birational localization of the category of smooth varieties.
We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel-Moore homology.
We demonstrate that a conjecture of Teh which relates the niveau filtration on Borel-Moore homology of real varieties and the images of generalized cycle maps from reduced Lawson homology is false. We show that the niveau filtration on reduced Lawson homology is trivial and construct an explicit class of examples for which Teh's conjecture fails by generalizing a result of Schulting. We compare various cycle maps and in particular we show that the Borel-Haeflinger cycle map naturally factors through the reduced Lawson homology cycle map.
We construct rational projective 4-dimensional varieties with the property that certain Lawson homology groups tensored with Q are infinite dimensional Q-vector spaces. More generally, each pair of integers p and k, with k\geq 0, p>0, we find a projective variety Y, such that L_pH_{2p+k}(Y) is infinitely generated. We also construct two singular rational projective 3-dimensional varieties Y and Y' with the same homeomorphism type but different Lawson homology groups, specifically L_1H_3(Y) is not isomorphic to L_1H_3(Y')even up to torsion.
Let $X$ be a toric variety. Rationally Borel-Moore homology of $X$ is isomorphic to the homology of the Koszul complex $A^T_*(X)\otimes Λ^\x M$, where $A^T_*(X)$ is the equivariant Chow group and $M$ is the character group of $T$. Moreover, the same holds for coefficients which are the integers with certain primes inverted.
In this paper we develop homology and cohomology theories which play the same role for real projective varieties that Lawson homology and morphic cohomology play for projective varieties respectively. They have nice properties such as the existence of long exact sequences, the homotopy invariance, the Lawson suspension property, the homotopy property for bundle projection, the splitting principle, the cup product, the slant product and the natural transformations to singular theories. The Friedlander-Lawson moving lemma is used to prove a duality theorem between these two theories. This dualit…
A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so. Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was s…
We study the behavior of cohomological support loci of the canonical bundle under derived equivalence of smooth projective varieties. This is achieved by investigating the derived invariance of a generalized version of Hochschild homology. Furthermore, using techniques coming from birational geometry, we establish the derived invariance of the Albanese dimension for varieties having non-negative Kodaira dimension. We apply our machinery to study the derived invariance of the holomorphic Euler characteristic and of certain Hodge numbers for special classes of varieties. Further applications con…
We prove that the intersection homology Poincare' polynomial P(X) of an affine toric variety X is bounded below by the product P(Y)P(X/Y), where Y is the closure of any orbit in X and X/Y is a slice transverse to the orbit. This proves a combinatorial conjecture of Kalai for rational polytopes.
In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and algebraic optimization. Namely, we express the degree corresponding to the distance variable of the offset hypersurface in terms of the Euclidean Distance Degree of the starting variety, obtaining a new way to compute these degrees. Finally, we describe the non-properness locus of the offset construction and use this to describe the set of points that are topol…
Stratified-algebraic vector bundles on real algebraic varieties have many desirable features of algebraic vector bundles but are more flexible. We give a characterization of the compact real algebraic varieties having the following property: There exists a positive integer r such that for any topological vector bundle E on X, the direct sum of r copies of E is isomorphic to a stratified-algebraic vector bundle. In particular, each compact real algebraic variety of dimension at most 8 has this property. Our results are expressed in terms of K-theory.
Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric varieties.
Given a closed subvariety of an algebraic torus, the associated tropical variety is a polyhedral fan in the space of 1-parameter subgroups of the torus which describes the behaviour of the subvariety at infinity. We show that the link of the origin has only top rational homology if a genericity condition is satisfied. Our result is obtained using work of Tevelev and Deligne's theory of mixed Hodge structures.
We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.
This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n,n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-manifold and boundary curve system.
We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne's algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits, using the methods of Cartan and Borel. We also establish the degeneration of the Čech-de Rham spectral sequence for determinantal varieties, and compute explicitly the dimensions of de Rham cohomology groups of local cohomology …
We consider a possibility of the existence of intersection homology morphism, which would be associated to a map of analytic varieties. We assume that the map is an inclusion of codimension one. Then the existence of a morphism follows from Saito's decomposition theorem. For varieties with conical singularities we show, that the existence of intersection homology morphism is exactly equivalent to the validity of Hard Lefschetz Theorem for links. For varieties with arbitrary analytic singularities we extract a remarkable property, which we call Local Hard Lefschetz.
We compute the rational homology of the moduli stack $\mathcal{M}$ of objects in the derived category of certain smooth complex projective varieties $X$ including toric varieties, flag varieties, curves, surfaces, and some 3- and 4-folds. We identify Joyce's vertex algebra construction on $H_\ast(\mathcal{M},\mathbb{Q})$ with a generalized super-lattice vertex algebra associated to $K^0_{\rm top}(X^{\rm an}) \oplus K^1_{\rm top}(X^{\rm an})$.
We present a definition of intersection homology for real algebraic varieties that is analogous to Goresky and MacPherson's original definition of intersection homology for complex varieties.
The monodromy action in the homology (generally with twisted coefficients) of complements of stratified complex analytic varieties depending on parameters is studied. For a wide class of local degenerations of such families (stratified Morse singularities) local monodromy and variation operators are reduced to similar operators acting in the transversal slices of corresponding strata. In particular, these results imply a main part of F. Pham's generalized Picard--Lefschetz formulae.
A classical result of A. Connes asserts that the Frechet algebra of smooth functions on a smooth compact manifold X provides, by a purely algebraic procedure, the de Rham cohomology of X. Namely the procedure uses Hochschild and cyclic homology of this algebra. In the situation of a Thom-Mather stratified variety, we construct a Frechet algebra of functions on the regular part and a module of poles along the singular part. We associate to these objects a complex of differential forms and an Hochschild complex, on the regular part, both with poles along the singular part. The de Rham cohomology…
Let X be a smooth projective variety of dimension n on which rational and homological equivalence coincide for algebraic p-cycles in the range 0\leq p\leq s. We show that the homologically trivial sector of rational Lawson homology L_pH_k(X,Q)_{hom} vanishes for 0\leq n-p\leq s+2. This is an analogue of a theorem of C. Peters in "dual dimensions". Together with Peters' theorem we get that the natural transformation L_pH_k(X,Q)--> H_k(X,Q) is injective for all p and k when X is a smooth projective variety of dimension 4 and Ch_0(X) =Z.
We prove a conjecture of Etingof and the second author for hypertoric varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson-de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson-de Rham-Poincare polynomial, and prove …
A stratified bundle is a vector bundle which is a D-module. We show that regular singularity of stratified bundles on smooth varieties in positive characteristic is preserved by pullback and that regular singularity can be checked on curves, if the ground field is large enough.
We study special values of zeta functions of singular varieties over finite fields. We give a new formula of special values by constructing a morphism of homology theories, which we call regulator, from higher Chow group to weight homology. Our regulator is defined by using the notion of weight complex for varieties over a perfect field, which was introduced by Gillet and Soule. The main idea of the proof of our formula of special values is to use weight spectral sequence of homology theories, whose E1 terms are homology groups for smooth projective schemes. Also, to calculate special values, …
This paper describes a paving by affines for regular nilpotent Hessenberg varieties in all Lie types, namely a kind of cell decomposition that can be used to compute homology despite its weak closure conditions. Precup recently proved a stronger result; we include ours because we use different methods. We then use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explic…
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.
In this paper, we introduce the notion of relation type of analytic and formal algebras and prove that it is well-defined and invariant by describing this notion in terms of the André-Quillen homology and using the Jacobi-Zariski long exact sequence of homology. In particular, the relation type is an invariant of schemes of finite type over a field, analytic varieties, and algebroid varieties.
We study the topological group structure (coming from loop multiplication) on an affine Grassmannian. In particular, we study finite-dimensional subvarieties that generate the homology ring. We show that there is a canonical family of generating Schubert varieties, namely those defined by the negative of the coroot associated to the highest root. These not only generate the homology, but generate the affine Grassmannian itself in a precise sense.
A stratified variety has a Kazhdan-Lusztig atlas if it can be locally modelled with Kazhdan-Lusztig varieties stratified by Schubert varieties in some Kac-Moody flag manifold via stratified isomorphisms. In this paper, we show that the partial flag manifold $G/P$ with the projected Richardson stratification has a Kazhdan-Lusztig atlas, with each chart stratified-isomorphic to a Kazhdan-Lusztig variety in the affine flag manifold of the formal loop group $\widehat{G}$ of $G$. This result generalizes that of Snider's on $Gr(k,n)$ with the positroid stratification, and is a geometric counterpart …
We construct an extended oriented $(2+ε)$-dimensional topological field theory, the character field theory $X_G$ attached to a affine algebraic group in characteristic zero, which calculates the homology of character varieties of surfaces. It is a model for a dimensional reduction of Kapustin-Witten theory ($N=4$ $d=4$ super-Yang-Mills in the GL twist), and a universal version of the unipotent character field theory introduced in arXiv:0904.1247. Boundary conditions in $X_G$ are given by quantum Hamiltonian $G$-spaces, as captured by de Rham (or strong) $G$-categories, i.e., module categories …
We generalize the Künneth formula for Chow groups to an arbitrary OBM-homology theory satisfying descent (e.g. algebraic cobordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a description for the homology groups of smooth toric varieties, that allows the calculation of algebraic cobordism groups of singular toric varieties. Some computations are carried out.
We consider derived invariants of varieties in positive characteristic arising from topological Hochschild homology. Using theory developed by Ekedahl and Illusie-Raynaud in their study of the slope spectral sequence, we examine the behavior under derived equivalences of various $p$-adic quantities related to Hodge-Witt and crystalline cohomology groups, including slope numbers, domino numbers, and Hodge-Witt numbers. As a consequence, we obtain restrictions on the Hodge numbers of derived equivalent varieties, partially extending results of Popa-Schell to positive characteristic.
We generalize the Harnack-Thom theorem to relate the ranks of the Lawson homology groups with $\Z_2$-coefficients of a real quasiprojective variety with the ranks of its reduced real Lawson homology groups. In the case of zero-cycle group, we recover the classical Harnack-Thom theorem and generalize the classical version to include real quasiprojective varieties. We use Weil's construction of Picard varieties to construct reduced real Picard groups, and Milnor's construction of universal bundles to construct some weak models of classifying spaces of some cycle groups. These weak models are use…
We study algebraic varieties parametrized by topological spaces and enlarge the domains of Lawson homology and morphic cohomology to this category. We prove a Lawson suspension theorem and splitting theorem. A version of Friedlander-Lawson moving is obtained to prove a duality theorem between Lawson homology and morphic for smooth semi-topological projective varieties. K-groups for semi-topological projective varieties and Chern classes are also constructed.
