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 foliation theory: a holomorphic foliation $\cal F$ in a ball $\B_r$ around the origin in $\C^3$, with an isolated singularity at 0, admits a $C^\infty$ normal section (away from 0) iff its multiplicity (or local index) is even, and this happens iff its normal bundle in $\B_r \setminus \{0\}$ is topologically trivial.
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…
There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and C.T.C.Wall. We derive this duality from the mirror symmetry, the Berglund-Hübsch transposition of invertible polynomials, and our previous results about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.
The famous $4n^2$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is higher than $4n^2μ$, where $μ$ is the multiplicity of the singular point.
We deform monomial space curves in order to construct examples of set-theoretical complete intersection space curve singularities. As a by-product we describe an inverse to Herzog's construction of minimal generators of non-complete intersection numerical semigroups with three generators.
C.T.C. Wall and the first author discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund-Hübsch transposition of invertible polynomials.
In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.
This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.
This paper will give some examples of diffeomorphic complex 5-dimensional complete intersections and remarks on these examples. Then a result on the existence of diffeomorphic complete intersections that belong to components of the moduli space of different dimensions will be given as a supplement to the results of P.Brückmann (J. reine angew. Math. 476 (1996), 209-215; 525 (2000), 213-217).
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "splice type singularities", which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q-homology sphere links, called "splice-quotient singularities". According to the Main Theorem, splice-quo…
We prove that the signature of the Milnor fiber of smoothings of a $2$-dimensional isolated complete intersection singularity does not exceed the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution, which implies Durfee's weak conjecture and a partial answer to Kerner--Némethi's conjecture.
We study the local symplectic algebra of the 0-dimensional isolated complete intersection singularities. We use the method of algebraic restrictions to classify these symplectic singularities. We show that there are non-trivial symplectic invariants in this classification.
In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in math.AG/0002083. The result can be applied to show the rigidity of all open swallowtails of dimension greater or equal than two. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show that lagrangian complete intersection in dimension greater than two cannot be regular in codimension one.
We prove that if X is a locally complete intersection variety, then X has all the jet schemes irreducible if and only if X has canonical singularities. After embedding X in a smooth variety Y, we use motivic integration to express the condition that X has irreducible jet schemes in terms of data coming from an embedded resolution of X in Y. We show that this condition is equivalent with having canonical singularities. In the appendix, this result is used to prove a generalization of Kostant's freeness theorem to the setting of jet schemes.
J. Wahl conjectured that every quasihomogeneous isolated normal singularity admits a positive grading for which there are no derivations of negative weighted degree. We confirm his conjecture for quasihomogeneous isolated complete intersection singularities of either order at least 3 or embedding dimension at most 5. For each embedding dimension larger than 5 (and each dimension larger than 3), we give a counter-example to Wahl's conjecture.
The Milnor number, μ(X,0), and the singularity genus, p_g(X,0), are fundamental invariants of isolated hypersurface singularities (more generally, of local complete intersections). The long standing Durfee conjecture (and its generalization) predicted the inequality μ(X,0) \geq (n+1)!p_g(X,0), here n=dim(X,0). Recently we have constructed counterexamples, proposed a corrected bound and verified it for the homogeneous complete intersections. In the current paper we treat the case of germs with Newton-non-degenerate principal part when the Newton diagrams are "large enough", i.e. they are large …
The method of intersection spaces associates rational Poincaré complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex …
We shall investigate a flipping contraction g : X -> Y from a 4-fold X with at most isolated complete intersection singularities. If Y has an anti-bi-canonical divisor (=bi-elephant) with only rational singularities, then g carries an inductive structure chained up by blow-ups (La Torre Pendente), and in particular the flip exists. This naturally contains Miles Reid's `Pagoda' as an anti-canonical divisor (=elephant) and its proper transforms.
For a large class of good moduli spaces $X$ of symmetric stacks $\mathcal{X}$, we define noncommutative motives $\mathbb{D}^{\text{nc}}(X)$ which can be regarded as categorifications of the intersection cohomology of $X$. These motives are summands of noncommutative resolutions of singularities $\mathbb{D}(X)\subset D^b(\mathcal{X})$ of $X$. The category $\mathbb{D}(X)$ is a global analogue of the noncommutative resolutions of singularities of $V//G$ for $V$ a symmetric representation of a reductive group $G$ constructed by Špenko-Van den Bergh.
We give a generalization of the duality of a zero-dimensional complete intersection to the case of one-dimensional almost complete intersections, which results in a {\em Gorenstein module} $M=I/J$. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity $f$ with a one-dimensional critical locus, we relate the signature on the jacobian module $I/J_f$ to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. A…
Suppose M is a complex manifold of dimension $n+1$ and K is a hypersurface in M. By Poincaré duality we define a residue morphism $res:H^{k+1}(M\setminus K)\longrightarrow H_{2n-k}(K)$ which generalizes the classical Leray residue morphism to cohomology for smooth K. We assume that K has isolated quasihomogeneous singularities. Suppose $ω$ is a holomorphic form of the type $(n+1,0)$ with the first order pole on K. The purpose of this note is to give a short, self contained proof of a criterion which tells us when the residue of $ω$ lifts to the intersection homology of K.
Let $X$ a complex projective variety of complex dimension $n$ with only isolated singularities of simply connected links. We show that we can endow the rational cohomology of the family of the $\overline{p}$-perverse intersection spaces $\{ I^{\overline{p}}X \}$ with compatible mixed Hodge structures.
We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $μ$ and the Tjurina number $τ$ of any isolated plane curve singularity satisfy the inequality $3μ<4τ$.
We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern-Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusinski and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern-Milnor classes of complete intersections with isolated singularities.
In this paper we study the intersection theory on surfaces with abelian quotient singularities and we derive properties of quotients of weighted projective planes. We also use this theory to study weighted blow-ups in order to construct embedded $\mathbf{Q}$-resolutions of plane curve singularities and abstract $\mathbf{Q}$-resolutions of surfaces.
We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized $4n^2$-inequality for complete intersection singularities and the technique of hypertangent divisors.
The present lecture notes give an introduction to works of García Barroso, González Pérez, Ruggiero and the author. The starting point of those works is a theorem of Płoski, stating that one defines an ultrametric on the set of branches drawn on a smooth surface singularity by associating to any pair of distinct branches the quotient of the product of their multiplicities by their intersection number. We show how to construct ultrametrics on certain sets of branches drawn on any normal surface singularity from their mutual intersection numbers and how to interpret the associated rooted trees i…
We prove that a Fano complete intersection of codimension $k$ and index 1 in the complex projective space ${\mathbb P}^{M+k}$ for $k\geqslant 20$ and $M\geqslant 8k\log k$ with at most multi-quadratic singularities is birationally superrigid. The codimension of the complement to the set of birationally superrigid complete intersections in the natural parameter space is shown to be at least $\frac12 (M-5k)(M-6k)$. The proof is based on the techniques of hypertangent divisors combined with the recently discovered $4n^2$-inequality for complete intersection singularities.
We prove that for two germs of analytic mappings $f,g\colon (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^p,0)$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $\{f_t\}$ of analytic maps with $f_0=f, f_1=g$ which has a so-called {\it uniform stable radius for the Milnor fibration}. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is…
The paper completes the study of symmetries of parabolic function singularities with relation to complex crystallographic groups that was started in \cite{GM,X9}. We classify smoothable automorphisms of $P_8$ singularities which split the kernel of the intersection form on the second homology. For such automorphisms, the monodromy groups acting on the duals to the eigenspaces with degenerate intersection form are then identified as some of complex affine reflection groups tabled in \cite{P}.
We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N=2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.
Let $(Z,o)$ be a three-dimensional terminal singularity of type $cA/r$. We prove that all exceptional divisors over $o$ with discrepancies $\le 1$ are rational.
By generalizing the Landau-Ginzburg/Calabi-Yau correspondence for hypersurfaces, we can relate a Calabi-Yau complete intersection to a hybrid Landau-Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader-Ross have provided a equivalence between these invariants and Gromov-Witten invariants of complete intersections. For Calabi-Yau complete intersections of two cubics, we show that this equivalence is directly related - via Chen character - to t…
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena such as for instance non-isolated singularities which are finitely determined, or smoothings with low connectivity; already the union of the coordinate axes in $(\mathbb{C}^3,0)$ is determinantal, but not a complete intersection. We start with the algebraic background and then continue by discussing the subtle interplay of unfoldings and deformations in this s…
Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological space. This result applies to a large class of examples, such as normal surface singularities, varieties with ordinary multiple points, hypersurfaces with isolated singularities and more generally, complete intersections with isolated singularities. We obtain analogous results for contractions of subvarieties.
For a projective variety $Z$ and for any integer $p$, define the $p$-th Néron-Severi group $NS_p(Z)$ of $Z$ as the image of the cycle map $A_{p}(Z)\to H_{2p}(Z; \mathbb{C})$. Now let $X\subset \Ps^{2m+1}$ ($m\geq 1$) be a projective variety of dimension $2m-1$, with isolated singularities, complete intersection of a smooth hypersurface of degree $k$, with a hypersurface of degree $n>max\{k, 2m+1\}$, and let $F$ be a general hypersurface of degree $n$ containing $X$. We prove that the natural map $NS_m(X)\to NS_m(F)$ is surjective, and that if $dim NS_m(F)=1$ then $dim NS_m(X)=1$. In particular…
Isolated Cohen-Macaulay codimension 2 singularities share many common features with isolated complete intersection singularities, but they also exhibit some striking new behaviour. One such instance was recently observed by Damon and Pike in their study of the vanishing topology and Euler characteristic, where they took this class of singularities as examples. In this article, we explore their findings further by determining the Betti numbers explicitly and explain the new phenomena. An important tool here is the Tjurina modification relating a Cohen-Macaulay codimension 2 singularity to a fin…
We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by X. Gómez-Mont and G.-M. Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare…
We discuss and prove a number of cohomological results for Milnor fibers, real links, and complex links of local complete intersections with singularities of arbitrary dimension.
Drinfeld's relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularities consisting of stable maps from nodal deformations of the curve into twisted flag varieties. As an application, we prove that the twisted intersection cohomology sheaf on Drinfeld's compactification is universally locally acyclic over the moduli stack of $G$-bundles at points sufficiently antidominant relative to their defect.
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…
We observe that an interesting method to produce non-complete intersection subvarieties, the generalized complete intersections from L. Anderson and coworkers, can be understood and made explicit by using standard Cech cohomology machinery. We include a worked example of a generalized complete intersection Calabi-Yau threefold.
This paper studies intersection theory on the compactified moduli space M(n,d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus g > 1 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of M(n,d). Based on our earlier work, we give a precise formula for the intersection cohomology pairing and provide a method to calculate pairings on the partial resolution of s…
Green and Griffiths have introduced several notions of singularities associated with normal functions, especially in connection with middle dimensional primitive Hodge classes. In this note, by using the more elementary aspects of the Decomposition Theorem, we define global and local singularities associated with primitive middle dimensional cohomology classes and by using the Relative Hard Lefschetz Theorem, we show that these singularities detect the global and local triviality of the primitive class. In a final section, we write-up a classical inductive argument relating the Hodge conjectur…
The aim of this note is to give a simple definition of genus zero virtual orientation classes (or fundamental classes) for projective complete intersections or, more generally, for complete intersections in convex varieties, and to prove a push forward formula for them.
