The general quadruple point formula (original) (raw)

2007

Sign up for access to the world's latest research

checkGet notified about relevant papers

checkSave papers to use in your research

checkJoin the discussion with peers

checkTrack your impact

Abstract

Maps between manifolds MmtoNm+ellM^m\to N^{m+\ell}MmtoNm+ell ($\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities.

Multiplicity as an Object of Study in Contemporary Differential Geometry

ANNALS OF THE ORADEA UNIVERSITY. Fascicle of Management and Technological Engineering., 2013

The study presents introductory concepts of multiplicity as one of the basic concepts of differential topology. The work gives the definition of Hausdorff space and some theorems concerning Hausdorff space.

Minicourse: Cohomology jump loci in geometry and topology. Part IV: Applications

The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.

The double point formula with isolated singularities and canonical embeddings

Journal of the London Mathematical Society, 2020

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. A concrete application is for surfaces with geometric genus pg = 5: the canonical model is embedded in P 4 if and only if we have a complete intersection of type (2, 4) or (3, 3). Contents 1. Introduction 2 2. The classical double point formula 6 3. Extension of Severi's Statement 9 4. Comparing normal sheaf and quotient bundle 12 5. Surfaces with rational double points 14 6. Double point formulae via symplectic approximations. 20 References 21

MULTIPLE-POINT FORMULAS—A NEW POINT OF VIEW

2002

On the basis of the Generalized Pontryagin-Thom construction (see Rimanyi & Szucs, 1998) and its application in computing Thom polynomials (see Rimanyi, 2001) here we introduce a new point of view in multiple-point theory. Using this approach we will first show how to reprove results of Kleiman and his followers (the corank 1 case) then we will prove some new multiple-point formulas which are not subject to the condition of corank≤ 1.

Minicourse: Cohomology jump loci in geometry and topology. Part II: Characteristic varieties

The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.

Quadratic Cohomology

Arnold Mathematical Journal, 2014

We study homological invariants of smooth families of real quadratic forms as a step towards a "Lagrange multipliers rule in the large" that intends to describe topology of smooth maps in terms of scalar Lagrange functions.

The link of {f(x, y) + z n = 0} and Zariski's conjecture

Compositio Mathematica, 2005

We consider suspension hypersurface singularities of type g = f (x, y) + z n , where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated hypersurface singularity is a rational homology 3-sphere then it determines the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.

Algebraic invariants in classification of 6-points in degenerations of surfaces

2010

In this paper, we find isomorphisms between certain invariant groups corresponding to different numerations on 6-points of surfaces. There is a combinatorial correspondence between four 6-point orderings obtained by exchanging two opposite labels. We derive isomorphisms between certain invariant quotient groups obtained from these 6-point nu-merations. This is a preliminary step towards an ultimate classification of 6-points invariants, and perhaps towards a proof that the invariant groups, or at least certain derived invariants, are independent of the arbitrary choice of the numeration.

Generalized Monodromy Conjecture in dimension two

Geometry & Topology, 2012

The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with an analytic germ f : (X, 0) → (C, 0) defined on a normal surface singularity (X, 0). The article targets the 'right' extension in the case when the link of (X, 0) is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function Z(f, ω; s) for any f and analytic differential form ω, which will play the key technical localization tool in the later definitions and proofs.

Thom polynomial computing strategies. A survey

Abstract. Thom polynomials compute the cohomology classes of degeneracy loci. In this paper we use a simple example to review the core ideas in different—mostly recently found—methods of computing Thom polynomials. Our goal is to show the underlying topology/geometry/algebra without involving combinatorics.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (33)

  1. V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasil'ev. Singularity theory. I. Translated from the 1988 Russian original. Encyclopaedia of Mathematical Sciences, Dynamical systems. VI. Springer, 1993, 1998.
  2. Eric Babson, Paul E. Gunnells, and Richard Scott. A smooth space of tetrahedra. Adv. Math., 165(2):285-312, 2002.
  3. G. Bérczi, L. Fehér, and R. Rimányi. Expressions for resultants coming from the global theory of singularities. In L. McEwan, J.-P. Brasselet, C. Melles, G. Kennedy, and K. Lautier, editors, Topics in Algebraic and Noncommutative Geometry, number 324 in Contemporary Mathematics. AMS, 2003.
  4. G. Bérczi and A. Szenes. Thom polynomials of Morin singularities. preprint, arXiv, math.AT/0608285, 2006.
  5. S. J. Colley. Lines having specified contact with projective varieties. In J. Carrell, A.V. Geramita, and P. Russell, editors, Proc. 1984 Vancouver conference on Algebraic Geometry, pages 47-70. AMS, Providence, RI, 1986.
  6. A. Collino and W. Fulton. Intersection rings of spaces of triangles. Mém. Soc. Math. France (N.S.), 38:75-117, 1989.
  7. Danielle Dias and Patrick Le Barz. Configuration spaces over Hilbert schemes and applications. Number 1647 in Lecture notes in mathematics. Springer, 1996.
  8. L. M. Fehér and B. Kőműves. On second order Thom-Boardman singularities. Fund. Math., 191:249-264, 2006.
  9. L. M. Fehér, A. Némethi, and R. Rimányi. Coincident root loci of binary forms. Michigan Math. J., 54:375- 392, 2006.
  10. L. M. Fehér and R. Rimányi. Thom series of contact singularities. In preparation.
  11. L. M. Fehér and R. Rimányi. Classes of degeneraci loci for quivers-the Thom polynomial point of view. Duke Math. J., 114(2):193-213, 2002.
  12. L. M. Fehér and R. Rimányi. Schur and Schubert polynomials as Thom polynomials-cohomology of moduli spaces. Cent. European J. Math, 4:418-434, 2003.
  13. L. M. Fehér and R. Rimányi. Calculation of Thom polynomials and other cohomological obstructions for group actions. In Real and Complex Singularities (Sao Carlos, 2002), number 354 in Contemp. Math., pages 69-93. Amer. Math. Soc., 2004.
  14. L. M. Fehér and R. Rimányi. On the structure of Thom polynomials of singularities. Bulletin of the London Mathematical Society, 39:541-549, 2007.
  15. William Fulton. Intersection Theory. Springer, 2nd Ed., 1998.
  16. William Fulton and Piotr Pragacz. Schubert varieties and degeneracy loci. Number 1689 in Lecture Notes in Mathematics. Springer-Verlag, 1998.
  17. M. Golubitsky and V. Guillemin. Stable mappings and their singularities, volume 14 of Graduate Texts in Mathematics. 1973.
  18. N. Katz. Pinceaux de Lefschetz; Theoreme d'existence. SGA, 7:212-253, 1973. Lect. Notes in Math. 340.
  19. M. Kazarian. Morin maps and their characteristic classes. 2006. preprint, www.mi.ras.ru/˜kazarian.
  20. M. É. Kazarian. Characteristic classes of singularity theory. In The Arnold-Gelfand mathematical seminars, pages 325-340. Birkhäuser Boston, 1997.
  21. M. É. Kazarian. Multisingularities, cobordisms, and enumerative geometry. (Russian). Uspekhi Mat. Nauk, (4(352)):29-88, 2003. translation in Russian Math. Surveys 58 (2003), no. 4, 665-724.
  22. S. Kleiman. Enumerative geometry of singularitites. In Real and complex singularities. Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, pages 297-396, 1976.
  23. S. Kleiman. Multiple-point formulas for maps. In Enumerative Geometry and Classical Algebraic Geometry, pages 237-252. Birkhäuser, 1982.
  24. Manfred Lehn. Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Inv. Math., 136(1):157-207, 1999.
  25. J. N. Mather. Stability of C ∞ mappings VI: the nice dimensions. In C. T. C. Wall, editor, Proceedings of Liverpool Singularities I, LNM 192. Springer, 1971.
  26. R. Rimányi. Thom polynomials, symmetries and incidences of singularities. Inv. Math., 143:499-521, 2001.
  27. R. Rimányi. Multiple-point formulas: A new point of view. Pacific Journal of Mathematics, 202(2):475-489, 2002.
  28. R. Rimányi. On right-left symmetries of stable singularities. Math. Z., 242:347-366, 2002.
  29. R. Rimányi and A. Szűcs. Generalized Pontrjagin-Thom construction for maps with singularities. Topology, 37(6):1177-1191, 1198.
  30. F. Ronga. La calcul des classes duales aux singularités de Boardman d'ordre 2. Comm. Math. Helv., 47:15-35, 1972.
  31. F. Ronga. On multiple points of smooth immersions. Comment. Math. Helv., 55:521-527, 1980.
  32. R. Thom. Les singularités des applications différentiables. Ann. Inst. Fourier, 6:43-87, 1956.
  33. A. S. Tikhomirov and T. L. Troshina. Top Segre class of a standard vector bundle E 4 D on the Hilbert scheme Hilb 4 S of a surface S. In Algebraic geometry and its applications (Yaroslavl, 1992), Aspects Math., pages 205-226, 1994.

On the triple points of singular maps

Commentarii Mathematici Helvetici, 2002

The number of triple points (mod 2) of a self-transverse immersion of a closed 2nmanifold M into 3n-space are known to equal one of the Stiefel-Whitney numbers of M. This result is generalized to the case of generic (i.e. stable) maps with singularities. Besides triple points and Stiefel-Whitney numbers, a certain linking number of the manifold of singular values with the rest of the image is involved in the generalized equation which corrects an erroneous formula in [9]. If n is even and the closed manifold is oriented then the equations mentioned above make sense over the integers. Together, the integer-and mod 2 generalized equations imply that a certain Stiefel-Whitney number of closed oriented 4k-manifolds vanishes. This Stiefel-Whitney number is in fact the first in a family which vanish on such manifolds.

Quadratic functions in geometry, topology,and M-theory

2002

We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension 4k+24k+24k+2 in terms of a holomorphic line bundle on the abelian variety H2k+1(M)otimesR/ZH^{2k+1}(M)\otimes R/ZH2k+1(M)otimesR/Z. Our results are inspired by work of Witten on the fivebrane partition function in MMM-theory (hep-th/9610234, hep-th/9609122). Our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics, and is based on our theory of "differential functions." These differential functions generalize the differential characters of Cheeger-Simons, and the bulk of this paper is devoted to their study.

The Euler-Poincaré Characteristic and Mixed Multiplicities

Kyushu Journal of Mathematics, 2015

This paper defines mixed multiplicity systems; the Euler-Poincare characteristic and the mixed multiplicity symbol of N d-graded modules with respect to a mixed multiplicity system, and proves that the Euler-Poincare characteristic and the mixed multiplicity symbol of any mixed multiplicity system of the type (k1,. .. , k d) and the (k1,. .. , k d)-difference of the Hilbert polynomial are the same. As an application, we get results for mixed multiplicities.

COHOMOLOGY JUMP LOCI OF 3-MANIFOLDS

Manuscripta Mathematica , 2022

The cohomology jump loci of a space X are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems , and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for X. We explore here the geometry of these varieties and the delicate interplay between them in the context of closed, orientable 3-dimensional manifolds and link complements. The classical multivariable Alexander polynomial plays an important role in this analysis. As an application, we derive some consequences regarding the formality and the existence of finite-dimensional models for such 3-manifolds.

homology

homology In this book our attention will be focused on compact, simply connected, dif-ferentiable 4-manifolds. The restriction to the simply connected case certainly rules out many interesting examples :indeed it is well-known that any finitely presented group can occur as the fundamental group of a 4-manifold.Furthermore the techniques we will develop in the body of the book are in reality rather insensitive to the fundamental group and much of our discussion can easily be generalized. The main issues however can be reached more quickly in the simply connected case. We shall see that for many purposes 4-manifolds with trivial fundamental group are of beguiling simplicitly but nevertheless the most basic questions about the differential topology of these manifolds lead us into new uncharted waters where the results described in this book serve at present,as isolated markers. After the fundamental group we have the homology and cohomology groups of a 4-manifold. For a closed oriented 4-manifold Poincare duality gives an isomorphism between homology and cohomology in complementary dimensions i and 4-i. So, when X is simply connected the first and third homology groups vanish and all the homological information is contained in H 2. The universal coefficient theorem for cohomology implies that when H 1 is zero H 2 is a free abelian group.In turn by Poincare duality the homology group is free. There are three concrete ways in which we can realize 2-dimensional ho-mology or cohomology classes on a 4-manifold and it is useful to be able to translate easily between them. The first is complex line bundles complex vector bundles of rank 1. On any space X a line bundle L is determined up to the bundle isomorphism by its Chern class and this sets up a bijection between the isomorphism classes of line bundles. The second realization is by smoothly embedded two-dimensional oriented surfaces in the manifold X. Such a surface carries a fundamental homology class ,given a line bundle L we can choose a general smooth section of the bundle whose zero set is a surface representing the homology class dual to c 1 (L) Third we have the de Rham representation of real cohomology classes by differential forms. Let X be a compact oriented simply connected four-manifold. The Poincare duality isomorphism between homology and cohomology is equivalent to a bi-linear form Q. This is the intersection form of the manifold. It is a unimodular symmetric form.Geometrically two oriented surfaces in X placed in general position will meet in a finite set of points. To each point we associate a sign ±1 according to the matching of the orientations in the isomorphism of the tangent bundles at that point. The intersection number is given by the total number of points counted with the signs. The pairing passes to homology to yield the form Q.Going over to cohomology the form translates into the cup product. Thus the form is an invariant of the oriented homotopy type of X. In terms of de Rham cohomology if w 1 and w 2 are closed 2-forms representing classes dual to the surfaces the intersection number is given by the integral X w 1 ∧ w 2 .

The monodromy of a series of hypersurface singularities

Commentarii Mathematici Helvetici, 1990

Let {f = 0} be a hypersurface in C "+ t with a l-dimensional singular set Z. We consider the series of hypersurfaces {f + exN= 0} where x is a generic linear form. We derive a formula, which relates the characteristic polynomials of the monodromies of f and f+ex u. Other ingredients in this formula are the horizontal and the vertical monodromies of the transversal (isolated) singularities on each branch of the singular set. We use polar curves and the carrousel method in the proof. The formula is a generalization of the Iomdin formula for the Milnor numbers:

Non-Zero Degree Maps Between 2n-Manifolds

Acta Mathematica Sinica, English Series, 2004

Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism φ : H n (L; Z) → H n (M; Z) can be realized by a map f : M → L of degree k for closed (n − 1)-connected 2n-manifolds M and L, n > 1. A corollary is that each (n − 1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 1. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms, in particular there is a map f : M → L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M.