MacPherson Class Research Articles

THE ROLE OF ANTIBACTERIAL HYDROGEL COATING ON COMPLICATION RISKS AFTER COMPLEX PRIMARY AND CEMENTLESS HIP REVISION ARTHROPLASTY: A RETROSPECTIVE MATCHED COHORT STUDY

This study aimed to verify the hypothesis that an antibiotic loaded hydrogel, defensive antimicrobial coating (DAC), reduces overall complication and infection rates when used for high-risk primary and revision total hip arthroplasty (THA).This was a retrospective study matched cohort study of 238 patients, treated with cementless implants with and without DAC. A sub-group analysis of patients undergoing 2nd stage revision THA for prosthetic joint infection (PJI) was also conducted. Re-infection rates within 2 years, complications necessitating surgical intervention and radiographic analysis for aseptic loosening was assessed.The mean age was 68.3±11.5 years, with 39 (32.8%) Macpherson class A, 64 (53.8%) class B and 16 (13.4%) class C patients. 4 (3.4%) patients in the DAC group developed complications including 1 PJI and 1 delayed wound healing, while 13 (10.9%) patients in the control group developed complications including 5 PJIs and 3 delayed wound healing (p=0.032). PJI rates (p=0.136) and delayed wound healing rates (p=0.337) were not statistically significant. For 2nd stage revision THA for PJI there were 86 patients in the DAC group and 45 in the control group. 1 (1.2%) patient in the DAC group developed complications with no recurrences of infection or delayed wound healing, while 10 (22.2%) patients in the control group developed complications including 4 recurrent PJI and 1 delayed wound healing (p=0.003). Recurrent PJI rates were statistically significant (p=0.005) while delayed wound healing rates were not (p=0.165). Patients treated with DAC also had lower rates of aseptic loosening (0% vs 6.7%; p=0.015).Antibiotic impregnated hydrogel coatings on cementless implants showed decreased complication rates after complex primary or revision THA. In 2nd stage revision THA for PJI, it was associated with reduced risk of re-infection and aseptic loosening.

A Deterministic Method for Computing Bertini Type Invariants of Parametric Ideals

A new framework is proposed for computing general numerical invariants, called Bertini type invariants, associated to a family of polynomial ideals depending on deformation parameters. A deterministic algorithm is introduced for computing parameter dependency of Bertini type invariants. As an application, an algorithm for computing Chern–Schwartz–MacPherson classes of a family of singular projective hypersurfaces is obtained.

Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells

Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells

Conjectural Positivity of Chern–Schwartz–MacPherson Classes for Richardson Cells

Abstract Following some work of Aluffi–Mihalcea–Schürmann–Su for the CSM classes of Schubert cells and some elaborate computer calculations by Rimanyi and Mihalcea, I conjecture that the CSM classes of the Richardson cells expressed in the Schubert basis have nonnegative coefficients. This conjecture was principally motivated by a new product $\square $ coming from the Segre-SM classes in the cohomology of flag varieties (such that the associated Gr of this product is the standard cup product) and the conjecture that the structure constants of this new product $\square $ in the standard Schubert basis have alternating sign behavior. I prove that this conjecture on the sign of the structure constants of $\square $ would follow from my above positivity conjecture about the CSM classes of Richardson cells.

Stellahedral geometry of matroids

Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.

Motivic classes of degeneracy loci and pointed Brill‐Noether varieties

Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern–Schwartz–MacPherson classes, K-theory classes, and Cappell–Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch χ y $\chi _y$ -genus of classical and one-pointed Brill–Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.

Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

Abstract We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.

Structure constants for Chern classes of Schubert cells

A formula for the structure constants of the multiplication of Schubert classes is obtained in arXiv:1909.05283. In this note, we prove analogous formulae for the Chern--Schwartz--MacPherson (CSM) classes and Segre--Schwartz--MacPherson (SSM) classes of Schubert cells in the flag variety. By the equivalence between the CSM classes and the stable basis elements for the cotangent bundle of the flag variety, a formula for the structure constants for the latter is also deduced.

On Analytic Todd Classes of Singular Varieties

AbstractLet $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ with $\pi :M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth }_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial }+\overline{\partial }^t$ on $M$. In the 2nd part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini–Study metric. First, assuming $\dim (V)\leq 2$, we compare the Baum–Fulton–MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial }$ complex. We show that there is no $L^2$-$\overline{\partial }$ complex on $(\operatorname{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum–Fulton–MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

THE CHERN–SCHWARTZ–MACPHERSON CLASS OF AN EMBEDDABLE SCHEME

The Chern–Schwartz–MacPherson class of a hypersurface in a nonsingular variety may be computed directly from the Segre class of the Jacobian subscheme of the hypersurface; this has been known for a number of years. We generalize this fact to arbitrary embeddable schemes: for every subscheme$X$of a nonsingular variety $V$, we define an associated subscheme$\mathscr{Y}$of a projective bundle$\mathscr{V}$over$V$and provide an explicit formula for the Chern–Schwartz–MacPherson class of$X$in terms of the Segre class of $\mathscr{Y}$in $\mathscr{V}$. If$X$is a local complete intersection, a version of the result yields a direct expression for the Milnor class of$X$.For$V=\mathbb{P}^{n}$, we also obtain expressions for the Chern–Schwartz–MacPherson class of $X$in terms of the ‘Segre zeta function’ of$\mathscr{Y}$.

The local Euler obstruction and topology of the stabilization of associated determinantal varieties

This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern–Schwartz–MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney’s $$m_{d}$$ multiplicity.

Chern–Schwartz–MacPherson classes of degeneracy loci

The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method---inspired by recent works of Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a resolution of singularities and often produces explicit (not sieve) formulas for CSM classes. Second, using the interpolation characterization we prove explicit formulas---including residue generating sequences---for the CSM and SSM classes of matrix Schubert varieties. Third, we suggest that a stable version of the SSM class of matrix Schubert varieties will serve as the building block of equivariant SSM theory, similarly to how the Schur functions are the building blocks of fundamental class theory. We illustrate these phenomena, and related stability and (2-step) positivity properties for some relevant representations.

Chern classes of logarithmic derivations for free divisors with Jacobian ideal of linear type

Let $X$ be a nonsingular variety defined over an algebraically closed field of characteristic $0$, and $D$ be a free divisor with Jacobian ideal of linear type. We compute the Chern class of the sheaf of logarithmic derivations along $D$ and compare it with the Chern–Schwartz–MacPherson class of the hypersurface complement. Our result establishes a conjecture by Aluffi raised in [Alu12b].

Homology theory formulas for generalized Riemann–Hurwitz and generalized monoidal transformations

In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann–Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincare duality is defined. We further consider possible applications and examples including branched covering maps, singular varieties involving virtual tangent bundles, the Chern–Schwartz–MacPherson class, the homology L-class, generalized signature, and the cohomology signature class.

Chern classes and characteristic cycles of determinantal varieties

Let K be an algebraically closed field of characteristic 0. For m≥n, we define τm,n,k to be the set of m×n matrices over K with kernel dimension ≥k. This is a projective subvariety of Pmn−1, and is called the (generic) determinantal variety. In most cases τm,n,k is singular with singular locus τm,n,k+1. In this paper we give explicit formulas computing the Chern–Mather class (cM) and the Chern–Schwartz–MacPherson class (cSM) of τm,n,k, as classes in the projective space. We also obtain formulas for the conormal cycles and the characteristic cycles of these varieties, and for their generic Euclidean Distance degree. Further, when K=C, we prove that the characteristic cycle of the intersection cohomology sheaf of a determinantal variety agrees with its conormal cycle (and hence is irreducible).Our formulas are based on calculations of degrees of certain Chern classes of the universal bundles over the Grassmannian. For some small values of m,n,k, we use Macaulay2 to exhibit examples of the Chern–Mather classes, the Chern–Schwartz–MacPherson classes and the classes of characteristic cycles of τm,n,k.On the basis of explicit computations in low dimensions, we formulate conjectures concerning the effectivity of the classes and the vanishing of specific terms in the Chern–Schwartz–MacPherson classes of the largest strata τm,n,k∖τm,n,k+1.The irreducibility of the characteristic cycle of the intersection cohomology sheaf follows from the Kashiwara–Dubson's microlocal index theorem, a study of the ‘Tjurina transform’ of τm,n,k, and the recent computation of the local Euler obstruction of τm,n,k.

Chern class of Schubert cells in the flag manifold and related algebras

We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra.

A direct algorithm to compute the topological Euler characteristic and Chern–Schwartz–MacPherson class of projective complete intersection varieties

Let V be a possibly singular scheme-theoretic complete intersection subscheme of Pn over an algebraically closed field of characteristic zero. Using a recent result of Fullwood (“On Milnor classes via invariants of singular subschemes”, Journal of Singularities) we develop an algorithm to compute the Chern–Schwartz–MacPherson class and Euler characteristic of V. This algorithm complements existing algorithms by providing performance improvements in the computation of the Chern–Schwartz–MacPherson class and Euler characteristic for certain types of complete intersection subschemes of Pn.

Computing characteristic classes of subschemes of smooth toric varieties

Let XΣ be a smooth complete toric variety defined by a fan Σ and let V=V(I) be a subscheme of XΣ defined by an ideal I homogeneous with respect to the grading on the total coordinate ring of XΣ. We show a new expression for the Segre class s(V,XΣ) in terms of the projective degrees of a rational map specified by the generators of I when each generator corresponds to a numerically effective (nef) divisor. Restricting to the case where XΣ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of XΣ via a dehomogenizing ideal we also give an expression for the projective degrees of this rational map in terms of the dimension of an explicit quotient ring. Under an additional technical assumption we construct what we call a general dehomogenizing ideal and apply this construction to give effective algorithms to compute the Segre class s(V,XΣ), the Chern–Schwartz–MacPherson class cSM(V) and the topological Euler characteristic χ(V) of V. These algorithms can, in particular, be used for subschemes of any product of projective spaces Pn1×⋯×Pnj or for subschemes of many other projective toric varieties. Running time bounds for several of the algorithms are given and the algorithms are tested on a variety of examples. In all applicable cases our algorithms to compute these characteristic classes are found to offer significantly increased performance over other known algorithms.

Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds

We obtain an algorithm computing the Chern–Schwartz–MacPherson (CSM) classes of Schubert cells in a generalized flag manifold$G/B$. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure–Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of$G/B$. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold$G/P$. We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting.

Dynamical Gelfand–Zetlin algebra and equivariant cohomology of grassmannians

We consider the rational dynamical quantum group [Formula: see text] and introduce an [Formula: see text]-module structure on [Formula: see text], where [Formula: see text] is the equivariant cohomology algebra [Formula: see text] of the cotangent bundle of the Grassmannian [Formula: see text] with coefficients extended by a suitable ring of rational functions in an additional variable [Formula: see text]. We consider the dynamical Gelfand–Zetlin algebra which is a commutative algebra of difference operators in [Formula: see text]. We show that the action of the Gelfand–Zetlin algebra on [Formula: see text] is the natural action of the algebra [Formula: see text] on [Formula: see text], where [Formula: see text] is the shift operator. The [Formula: see text]-module structure on [Formula: see text] is introduced with the help of dynamical stable envelope maps which are dynamical analogs of the stable envelope maps introduced by Maulik and Okounkov [Quantum Groups and quantum cohomology, preprint (2012) 1–276, arXiv:1211.1287]. The dynamical stable envelope maps are defined in terms of the rational dynamical weight functions introduced in [G. Felder, V. Tarasov and A. Varchenko, Solutions of the elliptic QKZB equations and Bethe ansatz I, in Topics in Singularity Theory V. I. Arnold’s 60th Anniversary Collection, Advances in the Mathematical Sciences, AMS Translations, Series 2, Vol. 180 (AMS, 1997), pp. 45–76.] to construct q-hypergeometric solutions of rational qKZB equations. The cohomology classes in [Formula: see text] induced by the weight functions are dynamical variants of Chern–Schwartz–MacPherson classes of Schubert cells.