ABSTRACTS, SHORT OUTLINES, INTRODUCTORY READING
ASI on New Developments in Singularity Theory
(preliminary version)
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Lecturer: V. Zakalyukin
Title: Flag contact singularities
We study the local contact of a distinguished flag (collection of nested submanifolds) with the image of a smooth mapping from certain manifold to the ambient space of the flag.
The flag contact singularities (orbits of right-left equivalencies which preserve the flag ) occupy the intermediate position between the singularities of hypersurfaces and that of complete intersections.
They naturally appear in various contexts of the singularity theory and its applications.
We describe the classification of simple singularities for certain distinguished flags, their bifurcation diagrams, spectral sequences of vanishing homologies.
We outline the relation of these classes to the boundary singularities, to the singularities of functions on a manifold with corners, to the diagrams of mappings, to relative minima function from control theory, to Lagrangian projections of reducible Lagrangian varieties.
Introductory reading :
1. Arnold V.I., Goryunov V.V., Lyashko O.V., Vassiliev V.A., Singularity Theory and applications II , Encyclopaedia of Math. Sciencies, vol. 39, Dynamical Systems 8 , Springer - Verlag, 1993.
2. Arnold V.I., Singularities of caustics and Wavefronts, Kluver Acad. Math. and Applic., 1990.
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Lecturer:A. Parusinski
Title: Preparation Theorem for Subanalytic Functions
We propose to look at the preparation theorem for subanalytic functions and its applications to the study of metric properties of subanalytic sets. The preparation theorem is a version of rectilinearization or Puiseux Theorem and allows to parametrize the sets in question piece by piece by fractional power series (maybe with negative coefficients). Thus it can be understood as a primitif version of desingularization.
The preparation theorem was introduced to prove the existence of Lipschitz stratification for subanalytic sets and later was applied by J.-M. Lion and J.-P. Rolin to study various properties of singular sets such as for instance: integration theory on subanalytic sets, o-minimality, order of contact between solutions of differential equations.
In the first part of the course we propose an elementary proof of the preparation theorem based on Puiseux theorem, that is which passes through the complex domain. In particular we obtain a complex analogue of the theorem. In the second part we propose as the first application to show the following result due to Lion and Rolin. Given a subanalyic family of k-dimensional compact subanalytic set $X_t$, where $t\in \R$ is a real parameter. Then the integral $\varphi (t) = \int_{X_t} f(t,x) \dvol_k$, for a subanalytic function $f(x,t)$, admits an asymptotic expansion at $t=0$ of the form $\varphi(t) = \sum _{i=0}^k g_i(t) (\ln t)^i$, where $g_i(t)$ are fractional power series. Then we discuss some other applications to the study of metric geometry of singular sets.
Introductory reading:
E. Bierstone, P. D. Milman, {\em Semianalytic and subanalytic sets}, Publ. I.H.E.S. 67 (1988), 5--42.
A. Parusi\'nski, \emph{Subanalytic functions}, Trans. Amer. Math. Soc 344 (2), 1994, 583--595
Further reading:
J.-M. Lion, J.-P. Rolin, {\em Th\'eor\`eme de pr\'eparation pour les fonctions logarithmico-exponentielles}, Ann. Inst. Fourier {\bf 47}, 3 (1997), 859--884.
J.-M. Lion, J.-P. Rolin, {\em Int\'egration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques}, Ann. Inst. Fourier {\bf 48}, 3 (1998), 755--767.
G. Comte, J.-M. Lion, J.-P. Rolin, {\em Nature log-analytique du volume des sous-analytiques}, Ill. Journ. Math., to appear
A.Parusi\'{n}ski, {\em Lipschitz stratifications of subanalytic sets}, Ann. Scient. Ec. Norm. Sup. 27(1994), 661--696.
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Lecturer: J. P. Brasselet
Title :Singularities and noncommutative Geometry.
Short outline : Classical results for manifolds and involving cohomological theory, such as Poincar\'e duality or de Rham theorem, fail to be true for singular varieties. Another example is the Connes result relating the de Rham cohomology of a compact manifold $M$ to the cyclic homology of the algebra of smooth functions on $M$. This result provides interesting applications in singularity theory and in noncommutative geometry, giving connection between geometry and algebraic methods. The aim of the course is to explain an equivalent of such a result in the case of singular spaces, using intersection cohomology instead of classical cohomology theory.
Introductionary readings:
Jean-Paul Brasselet, Aspects combinatoires des singularit\'es, Singularity Theory, World Scientific 1995, pp. 74 - 127.
Jean-Paul Brasselet and Andr\'e Legrand, Differential forms on singular varieties and cyclic homology, London Math. Soc. Lecture Notes Series 263, ed. B. Bruce and D. Mond, pp. 175 - 187.
Alain Connes, Noncommutative Geometry, Academic Press, 1994
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Lecturer G.M. Greuel
Title: The geometry of families of singular curves
It is a classical and interesting problem, which is still in the centre of theoretical research, to study the variety $V = V_d(S_1, \dots, S_r)$ of (irreducible) curves $C \subset \P^2=\P^2(\C)$ of degree $d$ having exactly $r$ singularities of prescribed (topological or analytical) types $S_1, \dots, S_r$. Among the most important questions are: \begin{itemize} \item Is $V \not= \emptyset$ (existence problem)? \item Is $V$ irreducible (irreducibility problem)? \item Is $V$ smooth of expected dimension ($T$--smoothness problem)? \end{itemize} A complete answer is only known in the special case of nodal curves, that is, for $V_d(r) = V_d(S_1, \dots, S_r)$ with $S_i$ ordinary nodes ($A_1$--singularities): $V_d(r) \not= \emptyset$ and $T$--smooth $\Longleftrightarrow r \le \tfrac{(d-1)(d-2)}{2}$ (Severi, 1921), $V_d(r)$ is irreducible (if $\not= \emptyset$) (Harris, 1985). Even for cuspidal curves there is no sufficient and necessary answer to any of the above questions and one can hardly expect such an answer.
In the lectures I should like to report on recent progress with respect to the above three questions. The emphasis on these results lies on sufficient conditions for existence, T-smoothness and irreducibility which are, asymptotically with respect to $d$, as close as possible to the known necessary conditions.
1st Lecture: Local versus global deformations and the use of zero-dimensional schemes.
When we consider singular projective curves in \mbox{$\P^2$}, we might be interested in the variety of all curves such that the singularities satisfy certain conditions, e.\ g., have a given number of singularities of fixed (topological or analytic) type, or just a fixed multiplicity at varying, or at fixed, points in $\P^2$.
In the first lecture I shall point out how questions about such varieties can be formulated in the framework of (global and local) deformation theory of curves \mbox{$C\subset \P^2$}. Many questions can already be answered by linearizing the problem, i.\ e.\ by considering {\em infinitesimal deformations}.
In many cases of interest, the imposed conditions for the singularities $(C,z_i)$ are encoded by an ideal sheaf \mbox{$\kj_X\subset \ko_{\P^2}$} defining a zero-dimensional scheme \mbox{$X(C) \subset \P^2$} concentrated on the singularities \mbox{$z_1,\dots , z_r$} of $C$. For instance, the ideals \begin{eqnarray*} \Iea(C,z_i)&:=&\langle f, f_x, f_y\rangle \subset \ko_{\P^2,z_i}\,,\;\;\;\text{respectively} \\ \Ies(C,z_i)&:=&\left\{g\in \ko_{\P^2,z_i} \,\Big|\, f+\varepsilon g \text{ is equisingular over } \Spec\C[\varepsilon]/(\varepsilon^2) \right\} \end{eqnarray*} (\mbox{$f(x,y)=0$} a local equation of $(C,z_i)$) define the tangent space to equianalytic, respectively to equisingular, deformations of $(C,z_i)$.
If $\kj_{X(C)}$, for \mbox{$X=\Xea$}, respectively \mbox{$X=\Xes$}, denote the corresponding ideal sheaves, then, as typical results, we have that \begin{itemize} \item \mbox{$H^0\bigl(\kj_{X(C)}(d)\bigr) \big/ H^0\bigl(\ko_{\P^2}\bigr)$} is isomorphic to the Zariski tangent space of $\Vd$ at $C$. \item \mbox{$H^1\bigl(\kj_{X(C)}(d)\bigr) =0$} if and only if $\Vd$ is $T$--smooth at $C$. \end{itemize} However, also the existence and irreducibility problem can be reduced to an $H^1$-vanishing problem for certain other ideal sheaves of 0-dimensional schemes.
How $H^1$-vanishing results lead to interesting results on the geometry of families of singular curves will be explained in the second lecture.
2nd Lecture: Asymptotically proper results for existence, T-smoothness and irreducibility
When looking for appropriate $H^1$-vanishing theorems for the problems under consideration, one has to be aware that the needed types of those are quite different. \begin{itemize} \item Existence problem: to show that \mbox{$\Vd\neq \emptyset$} it suffices to have an $H^1$-vanishing theorem for {\em generic} fat point, respectively cluster, schemes. \item T-smoothness problem: to prove the T-smoothness of $\Vd$ we need $H^1$-vanishing for the zero-dimensional schemes $X(C)$ (described above) associated with {\it any} curve \mbox{$C\in V$}. \item Irreducibility problem: to have irreducibility it suffices to show \mbox{$H^1\bigl(\kj_{X(C)}(d)\bigr) =0$} for {\it generic} curves \mbox{$C\in V$}. \end{itemize} Various $H^1$-vanishing criteria have been used in connection with the problems stated. The classical idea, already applied by Severi, Segre, Zariski is to restrict the ideal sheaf to the curve \mbox{$C\in V$} itself. For many cases one obtains already better results when replacing $C$ by a polar curve, or a special auxiliary curve.
Hirschowitz initiated a new approach by the so-called Horace method, applicable mainly to the existence problem. The Horace method, as well as its recent modifications, consists mainly in two ``procedures'': reduction (by a curve) and specialization (with respect to the curve).
Combining this approach with Xu's $H^1$-vanishing theorem for generic fat point schemes (based on Kodaira vanishing) and a version of the Viro method lead to the first asymptotically proper sufficient existence condition:
\smallskip \begin{theorem} Let $S_1,\dots, S_r$ be topological types. Then $V_d(S_1, \dots, S_r) \not=\emptyset$ if $$\sum^r_{i=1} \mu(S_i) \le \frac{1}{46}(d+2)^2$$ and two additional conditions for the five ``worst'' singularities hold true. \end{theorem}
\noindent Chiantini and Sernesi were the first to apply Bogomolov's theory of unstable rank two vector bundles on surfaces to the smoothness problem (in the case of nodal curves). Their approach could be extended in the sequel to curves with arbitrary singularities.
Barkats showed how to apply the Castelnuovo function theory for the computation of $h^1$ in relation to the irreducibility problem. His approach combined with new ideas lead to the following $H^1$-vanishing result:
\begin{theorem} Let \mbox{$C\subset \P^2$} be an irreducible curve of degree \mbox{$d$}, \mbox{$\Xea(C) \subset \P^2$} the zero-dimensional scheme defined by the Tjurina ideals $\Iea(C,z)$ and \mbox{$Y\subset \Xea(C)$} any subscheme. Then \mbox{$H^1\bigl(\kj_{Y/\P^2}(d)\bigr)$} vanishes if \begin{equation} \label{Smoothness condition 1} \sum\limits_{z\in \supp(Y)} \gamma \!\;(C;Y_z) \,\leq \, (d+3)^2\,, \end{equation} (with some new local invariant \mbox{$\gamma \!\;(C;Y_z)\leq (\deg(Y_z)+1)^2$}). \end{theorem}
\noindent As a corollary, one can deduce sufficient conditions for T-smoothness and irreducibility with the best known asymptotic behaviour. For T-smoothness these conditions are conjectured to be asymptotically proper; for curves with nodes and cusps the T-smoothness condition is indeed even exact up to linear terms in $d$: \begin{theorem} The variety of irreducible plane curves of degree $d$ with $n$ nodes and $k$ cusps is T-smooth (or empty) if $$ 4\,n+9\,k \:\leq\: (d+3)^2.$$ It is irreducible if $$ \frac{25}{2}\,n + 18\, k\:<\:d^2.$$ \end{theorem}
The three theorems were proved in joint papers with C.~Lossen and E.~Shustin.
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Lecturer: Barysnikov
Title: Stokes polyhedra
The central topic of this lecture will be the study of the structure of Stokes (anti-Stokes) sets for polynomial phase functions - sets of parameters where there is a leaf of the real (imaginary) part level foliation connecting two critical points. Their study has been initiated by M. Berry and coauthors in the context of asymptotic analysis.
It turns out that within the combinatorics of the Stokes sets some remarkable families of polyhedra are hidden, including the Stasheff polyhedra (associahedra).
Some related problems will be discussed (bifurcation diagrams of quadratic differentials, spaces of defects in crystals, bifurcation diagrams of Smale functions) as well as the sightings of Stokes polyhedra and their relatives in other parts of mathematics.
Suggested reading
V. Arnold, Computation of snakes, Russ. Math. Surveys (Uspekhi) 47 (1992).
M. Berry, Stokes' Phenomenon, Publ. Math. IHES 68 (1988).
J. Hubbard, H. Masur, Quadratic Differentials and Foliations, Acta Math. 142 (1979).
M. Kapranov, M. Saito, Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions, Contemp. Math., 227 (1998).
R. Thom, L'equivalence d'une foncition differentiable et d'un polynome, Topology 3 (1965).
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Lecturer: J. Damon
Title: Nonlinear Sections of Nonisolated Complete Intersections
Outline 1 Nonisolated Singularities as Nonlinear Sections
1) Singularities arising as Sections of Complete Intersections
2) K_V -equivalence
3) Role of Freeness for Divisors and Complete Intersections
2 Topology of Nonlinear Sections
4) Topology of Singular Milnor Fibers
5) Higher Multiplicities
6) Module Version of the Le-Greuel Formula
7) Relation with Buchsbaum-Rim Multiplicities
3 Topology of Singular Milnor Fibers via Differential Forms
8) Differential Forms on Free and Almost Free Divisors
4 Discriminants
9) Discriminants for Deformations of Sections
10) Morse-type Singularities for Sections and Mappings on Complete Intersections
11) Beyond Freeness - Free* Divisor Structures
12) Cohen-Macaulay Reductions for Groups of Equivalences
Papers for reference: There are a considerable number of papers dealing with Free Divisors which will be listed in the bibliography of the paper for the conference. Listed here are just a few papers beginning with Looijenga's excellent treatment of the results for isolated singularities which serves as a guide for results for nonisolated singularities, the book of Orlik- Terao which treats free arrangements giving results from many papers of Terao and others, the seminal paper of Saito on free divisors, a fundamental paper of Arnold which treats discriminants of A_k singularities implicitly using freeness, a paper joint with Mond which first considered the vanishing topology of discriminants by viewing them as sections, an AMS Memoir which covers basic results for the topology of nonlinear sections, and the paper of Mond to appear which treats differential forms on singular spaces to obtain the topology.
Looijenga, E.J.N. Isolated singular points on complete intersections, London Math. Soc. Lecture Notes (1984) Cambridge Univ. Press
Orlik, P. and Terao, H. Arrangements of Hyperplanes, Grundlehren der Math. Wiss. 300 (1992) Springer Verlag
Saito, K. "Theory of logarithmic differential forms and logarithmic vector fields", J. Fac. Sci. Univ. Tokyo Sect. Math. 27, 1980, 265-291
Arnold, V. I. "Wave front evolution and equivariant Morse lemma", Comm. Pure App. Math. 29, 1976, 557-582
Damon, J. Higher Multiplicities and Almost Free Divisors and Complete Intersections, Memoirs Amer. Math. Soc. vol 123 no 589, 1996
Damon, J. and Mond, D. "A-codimension and the vanishing topology of discriminants", Invent. Math. 106, 1991, 217-242
Mond, D. "Differential Forms on Free and Almost Free Divisors", to appear Jour. London Math. Soc.
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Lecturer: J. Denef
Title: GEOMETRY ON ARC SPACES, TOPOLOGICAL ZETA FUNCTIONS, AND THE MILNOR FIBRE
Short outline: These talks are an introduction to the geometry of arc spaces. For an algebraic variety over the field C of complex numbers, one considers the arc space L(X), whose points are the C[[t]]-rational points on X, and the truncated arc spaces L_n (X), whose points are the C[[t]]/t^n-rational points on X. The geometry of these spaces yields several new geometric invariants of X and brings new light to some classical invariants. For example, the Hodge spectrum and the topological zeta functions of a critical point of a polynomial can be expressed in terms of geometry on arc spaces (work of Denef and Loeser). In a different direction, Batyrev used arc spaces to prove a conjecture of Reid on quotient singularities (the McKay correspondence), and to construct his stringy Hodge numbers used in mirror symmetry. All these developments are based on Kontsevich?s construction of a measure on the arc space L(X). This is called the motivic measure, which is an analogue of the p-adic measure on a p-adic variety. In this way, several arithmetical results on p-adic integration and Igusa?s local zeta functions, find their natural counterpart in complex geometry.
Introductory reading: Only the basic notions from algebraic geometry and singularity theory are needed to understand the talks.(The notions of Milnor fibre, monodromy, and mixed Hodge structures are assumed to be known).
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Authors: J.H.M. Steenbrink, M. Schulze
Title: Computing Hodge-theoretic invariants of singularities
Short outline: We discuss the Gauss-manin connection of isolated hypersurface singularities as constructed by Brieskorn in his 1970 paper. We report about the work of Scherk and Steenbrink, who used this approach to give a description of the mixed Hodge structure on the cohomology of the Milnor fibre. Finally, we describe an algorithm to compute the discrete invariants of this mixed Hodge structure (the spectrum) and some related invariants.
Background: the papers E. Brieskorn, Die Monodromie der isolierten Singularit\"aten von Hyperfl\"achen. Manuscr. Math. 2, 103-161 (1970) J. Scherk, J.H.M. Steenbrink, On the mixed Hodge structure on the cohomology of the Milnor fibre, Math. Ann. 271, 641-665 (1985) J. Brian\,con, Ph. Maisonobe, Id\'eaux de germes d'op\'erateurs diff\'erentiels \`a une variable, L'Enseign. Math. 30 (1984), 7-38
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Lecturer C.T.C. Wall
Title: The geometry of unfolding}
\author{A.A. du Plessis and C.T.C. Wall}
One may regard a stable map $F:N\to P$ between smooth manifolds as a versal unfolding of its restriction $F|F^{-1}(y)$ to any fibre. Partitioning the target manifold $P$ according to the types of singularity of $F$ on the corresponding fibre gives a foliated stratification, whose parts we call leaves. The tangent space to a leaf at a point $y$ is spanned by the values at $y$ of the liftable vector fields; its codimension equals the sum of the Tjurina numbers of $F$ at the singular points on the fibre over $y$.
In the particular case of a weighted homogeneous function $f_0$ one has very explicit rules to write down the versal unfolding $F$, a basis for the liftable vector fields, and hence the discriminant matrix. We can use this to write down a Koszul complex, whose degree 1 terms represent vector fields that annihilate $F$, and calculate its homology.
If we consider the non-versal unfolding of $f_0$ where the unfolding monomials of weight greater than some fixed value $d$ are omitted, certain minors of the discriminant matrix give a set of defining equations for the locus of points where the corresponding germ fails to be stable, so versality breaks down.
This picture has a number of applications. We will focus on the special case where the unfolding consists of the family of all polynomials of degree $d$, corresponding to the family of all hypersurfaces $V$ of degree $d$ in complex projective space, and investigate the surprisingly close relation between the Tjurina number of $V$ (coming from the singularities) and the least degree $r$ of any homogeneous vector field tangent to $V$.
The arguments involve estimates in commutative algebra, and the results in high dimensions would be much more complete if a conjecture due to Eisenbud, Green and Harris could be established in general.\\
\centerline{\large Background references} \vspace{2mm}
\noindent Eisenbud, D., {\it Commutative Algebra}, Springer Graduate texts {\bf 150}, Springer-Verlag, 1995.\\ Looijenga, E.J.N., {\em Isolated singular points on complete intersections}, London Math. Soc. Lecture Notes no. {\bf 77}, Cambridge University Press, 1984.