Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterated residue identities are revealed.
In this paper we derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularities Σ^{i,j}. The formulas are given as linear combinations of Schur polynomials, and all coefficients are nonnegative.
Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algeb…
Combining the Kazarian approach to Thom polynomials via classifying spaces of singularities with the Fulton-Lazarsfeld theory of numerical positivity for ample vector bundles, we show that the coefficients of various Schur function expansions of the Thom polynomials of stable and unstable singularities are nonnegative.
Combining the "method of restriction equations" of Rimányi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin singularities $A_3: ({\bf C}^{\bullet},0)\to ({\bf C}^{\bullet + k},0)$ for any nonnegative integer $k$.
Thom (residual) polynomials in characteristic classes are used in the analysis of geometry of functional spaces. They serve as a tool in description of classes Poincaré dual to subvarieties of functions of prescribed types. We give explicit universal expressions for residual polynomials in spaces of functions on complex curves having isolated singularities and multisingularities, in terms of few characteristic classes. These expressions lead to a partial explicit description of a stratification of Hurwitz spaces.
We give the Thom polynomials for the singularities $I_{2,2}$ associated with maps $({\bf C}^{\bullet},0) \to ({\bf C}^{\bullet+k},0)$ with parameter $k\ge 0$. Our computations combine the characterization of Thom polynomials via the ``method of restriction equations'' of Rimanyi et al. with the techniques of Schur functions.
The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$. We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by an algebraic model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. We show that the polynomial GGL conjecture for a generic projective hypersurface of degree $\mathrm{…
We show that Thom polynomials of Lagrangian singularities have nonnegative coefficients in the basis consisting of Q-functions. The main tool in the proof is nonnegativity of cone classes for globally generated bundles.
We describe the positivity of Thom polynomials of singularities of maps, Lagrangian Thom polynomials and Legendrian Thom polynomials. We show that these positivities come from Schubert calculus.
We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the corresponding family of mixed hypersurfaces $f_t^{-1}(0)$ is Whitney equisingular (and hence topologically equisingular) and satisfies the Thom condition.
We combine recently developed intersection theory for non-reductive geometric invariant theoretic quotients with equivariant localisation to prove a formula for Thom polynomials of Morin singularities. These formulas use only toric combinatorics of certain partition polyhedra, and our new approach circumvents the poorly understood Borel geometry of existing models.
In this note we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and …
We develop algebro-combinatorial tools for computing the Thom polynomials for the Morin singularities $A_i(-)$ ($i\ge 0$). The main tool is the function $F^{(i)}_r$ defined as a combination of Schur functions with certain numerical specializations of Schur polynomials as their coefficients. We show that the Thom polynomial ${\cal T}^{A_i}$ for the singularity $A_i$ (any $i$) associated with maps $({\bf C}^{\bullet},0) \to ({\bf C}^{\bullet+k},0)$, with any parameter $k\ge 0$, under the assumption that $Σ^j=\emptyset$ for all $j\ge 2$, is given by $F^{(i)}_{k+1}$. Equivalently, this says that "…
We give the Thom polynomials for the singularities I_2,2 and A_3 associated with maps (C^n,0) -> (C^{n+k},0) with parameter k>=0. We give the Schur function expansions of these Thom polynomials. Moreover, for the singularities A_i (with any parameter k >= 0) we analyze the ``first approximation'' F^(i) to the Thom polynomial. Our computations combine the characterization of Thom polynomials via the ``method of restriction equations'' of Rimanyi et al. with the techniques of (super) Schur functions.
\'Akos K.\ Matszangosz, HUN-REN Alfr\'ed R\'enyi Institute of Mathematics, Re\'altanoda utca 13-15, 1053 Budapest, Hungary；L\'aszl\'o M. Feh\'er, Department of Analysis, Institute of Mathematics, E\"otv\"os Lor\'；University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary；HUN-REN Alfr\'ed R\'enyi Institute of Mathematics, Re\'altanoda utca 13-15, 1053 Budapest, Hungary
We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this gives an extension of our former result for stable singularities to nonnecessary stable ones. We also discuss some related aspects of Thom polynomials, which makes the article expository to some extent.
Grothendieck polynomials were introduced by Lascoux and Schützenberger, and they play an important role in K-theoretic Schubert calculus. In this paper, we give a new definition of double stable Grothendieck polynomials based on an iterated residue operation. We illustrate the power of our definition by calculating the Grothendieck expansion of K-theoretic Thom polynomials of $A_2$ singularities. We present the expansion in two versions: one displays its expected stabilization, while the other displays its expected finiteness property.
The rationality of the singularities of the $A_n$-loci is the natural question that arises in the papers devoted to the study of the Thom polynomials and $K$-theoretic invariants of the said loci. In this paper we prove that, in general, the $A_2$-loci have singularities worse than rational, and therefore they can not be presented as a GIT quotient of a smooth variety with respect to a reductive group.
In this paper we propose a systematic study of Thom polynomials for group actions defined by M. Kazarian. On one hand we show that Thom polynomials are first obstructions for the existence of a section and are connected to several problems of topology, global geometry and enumerative algebraic geometry. On the other hand we describe a way to calculate Thom polynomials: the method of restriction equations. It turned out that though the idea is quite simple the method is very powerful. We reproduced and improved earlier result in several directions (singularities, Schubert calculus, quivers). Ho…
We discuss computations of the Thom polynomials of singularity classes of maps in the basis of Schur functions. We survey the known results about the bound on the length and a rectangle containment for partitions appearing in such Schur function expansions. We describe several recursions for the coefficients. For some singularities, we give old and new computations of their Thom polynomials.
We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [GB-P1].
We study universal polynomials of characteristic classes associated to the $\mathcal{A}$-classification (i.e. up to right-left equivalence) of holomorphic map-germs $(\mathbb{C}^2,0) \to (\mathbb{C}^n, 0)$ $(n=2,3)$. That enables us to systematically treat with classical enumerative problems of lines of prescribed contact with a given projective surface in $3$ and $4$-spaces.
Hilbert scheme topological invariants of plane curve singularities are identified to framed threefold stable pair invariants. As a result, the conjecture of Oblomkov and Shende on HOMFLY polynomials of links of plane curve singularities is given a Calabi-Yau threefold interpretation. The motivic Donaldson-Thomas theory developed by M. Kontsevich and the third author then yields natural motivic invariants for algebraic knots. This construction is motivated by previous work of V. Shende, C. Vafa and the first author on the large $N$ duality derivation of the above conjecture.
Let $(X,0)$ be an isolated complete intersection complex singularity ($X$ can also be smooth at 0). Let $K$ be its link, $\cal X$ its canonical contact structure and $\D_X$ the complex vector bundle associated to $\cal X$. We prove that the bundle $\D_X$ is trivial if and only if the Milnor number of $X$ satisfies $μ(X,0) \equiv (-1)^{n-1}$ modulo $(n-1)!$. This follows from a general theorem stating that the complex orthogonal complement of a vector field in $X$ with an isolated singularity at 0 is trivial iff the GSV-index of $v$ is a multiple of $(n-1)!$. We have also an application to foli…
A contact singularity is a normal singularity $(V,0)$ together with a holomorphic contact form $η$ on $V\backslash$ Sing $V$ in a neighbourhood of 0, i.e. $η\wedge (dη)^r$ has no zero, where dim $V=2r+1$. The main result of this paper is that there are no isolated contact singularities.
We give a criterion to test geometric properties such as Whitney equisingularity and Thom's $a_f$ condition for new families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.
It is known that the lens space $L(2n,1)$ supports a virtually overtwisted contact structure arising as the boundary of the Milnor fiber of a complex hypersurface singularity. In this article we study the problem of realizing other $(L(p,q),ξ)$ in such a way, obtaining a series of necessary conditions for this to happen.
Thom polynomial describes the cohomology class Poincaré dual to the locus of particular singularity of a generic holomorphic map. In this paper we derive a closed formula for the generating function of its coefficients. The method is based on a new construction of the embedding space of punctual Hibert scheme that we call the non-associative Hilbert scheme. The efficiency of the method is demonstrated on explicit computation of a number of Thom polynomials, including those associated with singularities of Thom-Boardman types~$Σ^{i,j}$, $Σ^{2,1,1}$, $Σ^{2,2,1}$, and~$Σ^{2,2,2}$.
We show that every convergent power series with monomial extended Jacobian ideal is right equivalent to a Thom-Sebastiani polynomial. This solves a problem posed by Hauser and Schicho. On the combinatorial side, we introduce a notion of Jacobian semigroup ideal involving a transversal matroid. For any such ideal we construct a defining Thom-Sebastiani polynomial. On the analytic side, we show that power series with a quasihomogeneous extended Jacobian ideal are strongly Euler homogeneous. Due to a Mather-Yau-type theorem, such power series are determined by their Jacobian ideal up to right equ…
We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when expanded in these bases. The method uses a suitable Lagrange Grassmann bundle on the product of projective spaces. This is an extension of a nonnegativity result for Lagrangian Thom polynomials obtained by the authors previously. For a fixed pecialization, other specializations of the parameter lead to upper bounds for the coefficients of the given basis. One g…
We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka's maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.
The Thom-Boardman symbol was first introduced by Thom in 1956 to classify singularities of differentiable maps. It was later generalized by Boardman to a more general setting. Although the Thom-Boardman symbol is realized by a sequence of non-increasing, nonnegative integers, to compute those numbers is, in general, extremely difficult. In the case of polynomial multiplication maps, Robert Varley conjectured that computing the Thom-Boardman symbol for polynomial multiplication reduces to computing the successive quotients and remainders for the Euclidean algorithm applied to the degrees of the…
The concept of the maximal contact is the key in Hironaka's resolution theory. It treats local theory, and it is not effective in positive characteristics. This is the essential reason why Hironaka's theory treats only the case of characteristic zero. In this article we propose the substitute for the maximal contact, which is effective in any characteristics of the ground field. We replace the maximal contact by a theorem in the theory of torus embeddings. Using essential ideas here, we would like to establish the theory of resolution of singularities in arbitrary characteristics in a global s…
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. support norm) of the canonical contact structure, for each member of this family.
We construct coarse moduli spaces of semiquasihomogeneous hypersurface singularities with respect to right equivalence and contact equivalence. We have to fix the principal part of the semiquasihomogeneous singularities. For the moduli spaces with respect to contact equivalence we also fix the Hilbert function of the Tjurina algebra induced by the weights.
We show that the generating series of Euler characteristics of Hilbert schemes of points on any algebraic surface with at worst $A_n$-type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the Dedekind eta function. In particular it is a Fourier development of a meromorphic modular form with possibly half integer weight. The key ingredient is to apply the flop transformation formula of Donaldson-Thomas type invariants counting two dimensional torsion sheaves on 3-folds proved in the author's previous paper.
Let X be an affine normal variety with a C^*-action having only positive weights. Assume that X_{reg} has a symplectic 2-form w of weight l. We prove that, when l is not zero, the w is a unique symplectic 2-form of weight l up to C^*-equivariant automorphism When $l = 0$, we have a counter-example to this statement. In the latter half of the article, we associate to X a projective variety P(X) and prove that P(X) has a contact orbifold structure. Moreover, when X has canonical singularities, the contact orbifold structure is rigid under a small deformation. By using the contact structure on P(…
We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution has strong implications for the symplectic cohomology and conversely. We also use our computations to give a contact invariant of the link of the singularities and thereby distinguish many contact structures on connected sums of $S^2 \times S^3$.
We study loci of arcs on a smooth variety defined by order of contact with a fixed subscheme. Specifically, we establish a Nash-type correspondence showing that the irreducible components of these loci arise from (intersections of) exceptional divisors in a resolution of singularities. We show also that these loci account for all the valuations determined by irreducible cylinders in the arc space. Along the way, we recover in an elementary fashion-without using motivic integration-results of the third author relating singularities to arc spaces. Moreover, we extend these results to give a jet-…
The main purpose of this article is to lay the foundations for a classification of isolated hypersurface singularities in positive characteristic. Although our article is in the spirit of Arnol'd who classified real an complex hypersurfaces in the 1970's with respect to right equivalence, several new phenomena occur in positive characteristic. Already the notion of isolated singularity is different for right resp. contact equivalence over fields of characteristic other than zero. The heart of this paper consists of the study of different notions of non-degeneracy and the associated piecewise f…
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Department of Mathematics, Stony Brook University（石溪大学数学系）
C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way, we extend Bass-Quillen's fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-the…
Mather proved that the smooth stability of smooth maps between manifolds is a generic condition if and only if the pair of dimensions of the manifolds are 'nice dimensions' while topologically stability is a generic condition in any pair of dimensions. And, by a result of du Plessis and Wall $C^1$-stability is also a generic condition precisely in the nice dimensions. We address the question of bi-Lipschitz stability in this article. We prove that the Thom-Mather stratification is bi-Lipschitz contact invariant in the boundary of the nice dimensions. This is done in two steps: first we explici…
Currently there are several approaches to resolution of singularities in positive characteristic all of which have hit some obstruction. One natural idea is to try to construct new meaningful examples at this point to gain a wider range of experience. To produce such examples we mimic the characteristic zero approach and focus on cases where it fails. In particular, this short note deals with an example-driven study of failure of maximal contact and the search for an appropriate replacement.
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of ordinary or partial differential equations. Furthermore, we show how a combination of this geometric theory with (differential) algebraic tools allows us to make parts of the theory algorithmic. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for th…
