FJV2014 - SapporoThe 2nd Franco-Japanese-Vietnamese Symposium on
Singularities
Program
Schedule, title and abstract
Dates: Aug. 25– 29 (Aug. 30), 2014.Venue: Lecture room 7-310 in the building no. 7, Faculty of Science, Hokkaido Univ.URL: http://www.math.sci.hokudai.ac.jp/˜fj singularities/FJV2014/index.html
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Organizing CommitteeVincent Blanlœil (Strasbourg), Jean-Paul Brasselet (Marseille), Nguyen Viet Dung (Hanoi),Masaharu Ishikawa (Tohoku), Toru Ohmoto (Hokkaido), Masahiko Yoshinaga (Hokkaido).
Scientific CommitteeAlexandru Dimca (Nice), Toshizumi Fukui (Saitama), Toshitake Kohno (Tokyo), GooIshikawa (Hokkaido), Ta Le Loi (Dalat), Osamu Saeki (Kyushu), Jorg Schurmann (Mn-ster), Bernard Teissier (Paris), Michel Vaquie (Toulouse), Shoji Yokura (Kagoshima).
TopicsA wide range of interests in- Algebraic Geometry of Singular Varieties- Geometry and Topology of Singular Spaces and Maps.
On our activitiesIn the past 16 years, we had the Franco-Japanese singularity project (6 symposiums) andthe Japanese-Vietnamese singularity project (3 symposiums). In the last year 2013 inNice, we started a new project the Franco-Japanese-Vietnamese Symposium on Singular-ities, main activity of the International Research Group in Singularity Thoery (CNRS).
SponsorsThis conference is mainly supported by GDRI, GDR (CNRS), Kakenhi (JSPS) andHokkaido University.
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InformationRegistrationPlease come to our secretary desk in the front of the conference hall (7-310) and checkyour name on the list – in particular mark the check-box for BBQ party on 28th (seebelow). If you have any question, feel free to ask us (Local staff: Ohmoto, Yoshinaga andfour students – Kabata, Sano, Sasajima, Fujisawa).
Lecture HallAll talks will be given in the no. 310 lecture hall in the 7th building, Faculty of Science.Poster Session will be organized in the room no. 219 (2F) on 27th (Wed). Before thissession, we take a group photo in the lecture hall.For speakers: Projector, table camera and blackboards are available. If you do not bringyour PC but want to use projector, please pass to us the pdf file of your talk via USB orEmail in advance.
Coffee BreakTea room is the room no. 219 (2F): we prepare some hot coffee and tea, cold drinks,snacks, cakes, etc. Please use this room freely. On each floor of the building there aresmall ‘common rooms’, but those are for people working in laboratories, not for us.
Wi-FiYou can use Wi-Fi in the lecture hall and our tea room. Please come to the secretarydesk. We then give you an ID/password (one user), which is valid during the conference.
LunchNear the conference venue, there are the Faculty House “Enreisou” (ordinary restaurant)and Cafeteria “Chuo-Shokudo” (self-service cafeteria on 1F and 2F). Everyday we reserve10 seats in the Faculty House – if you wish to take one of those reserved seats in therestaurant, please write your name on an order form at the secretary desk by 10:00 onthat day (if the restaurant is not full, you can go anytime without reservation). Anotheroption is to go out and take some cafe or restaurants nearby the JR Sapporo station.
BBQOn 28th evening, we have BBQ party. The fee is 6,000 yen per each person (4,000 yen forstudents), so please pay for it at the secretary desk. The place is Sapporo Beer Garden.Chartered buses will come to the front of our building. The distance is less than 3km, soyou may walk to get there.
Seminar RoomWe reserve a small seminar room with a large brackboard in the 4th building during theconference. If you want to use it, please come to the desk.
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Campus Map
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25th (Mon)
09:20 - 09:30 Openning address: Hiroaki Terao, the dean of Faculty of Scinence
09:30 - 10:20 Adam Parusiniski (Nice)
Local topological algebraicity of analytic function germs
10:30 - 11:20 Pham Tien Son (Dalat)
Effective Lojasiewicz inequalities for the largest eigenvalue
of real symmetric matrix polynomials
11:40 - 12:10 Fabien Priziac (Saitama, JSPS)
Equivariant blow-Nash equivalence and equivariant zeta functions
for invariant Nash germs
14:00 - 14:50 Le Dung Trang (Marseille-Fortaleza)
Lipschitz regularity
15:00 - 15:30 Aurelio Menegon Neto (Paraıba, Brazil)
Le’s polyhedron and the boundary of the Milnor fiber of non-isolated
singularities
16:00 - 16:50 Michel Vaquie (Toulouse)
Derived Algebraic Geometry
26th (Tue)
09:30 - 10:20 Hussein Mourtada (Paris)
Jet schemes and generating sequences of some divisorial valuations
10:30 - 11:20 Bernard Teissier (Paris)
Valuations and Toric Geometry
11:40 - 12:10 Christophe Eyral (Polish Acad. Sci., Warsaw)
Topological triviality of linear deformations with constant Le numbers
14:00 - 14:50 Eva Feichtner (Bremen)
Wonderful compactifications from a tropical viewpoint
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15:00 - 15:30 Nguyen Tat Thang (Tohoku, JSPS)
On stable linear deformations of Brieskorn singularities of two variables
16:00 - 16:50 Hiroaki Terao (Hokkaido)
Ideals of the roots posets and a new proof of the dual-partition
formula by Shapiro-Steinberg-Kostant-Macdonlald
27th (Wed)
09:00 - 09:50 Takuro Abe (Kyoto)
Recent developments on algebra of line arrangements
10:00 - 10:50 Dirk Siersma (Utrecht)
Projective hypersurfaces with 1-dimensional singularities
11:10 - 11:40 Kenta Hayano (Hokkaido)
Multisections of Lefschetz fibrations and mapping class groups of surfaces
11:45 - 12:50 Group Photo
12:00 - 12:50 Poster Session
13:00 – Free discussion
28th (Thu)
09:30 - 10:20 Krzysztof Kurdyka (Savoie)
Stratified-algebraic vector bundles
10:30 - 11:20 Goulwen Fichou (Rennes)
Regular functions after one blowing-up in the plane
11:40 - 12:10 Guillaume Valette (Krakow)
Arc-quasianalytic functions
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14:00 - 14:50 Kyoji Saito (IPMU)
Mirror Symmetry for Primitive Forms
15:00 - 15:30 Shinichi Tajima (Tsukuba) and Yoko Umeta (Tokyo Sci.)
Newton filtration and local cohomology
16:00 - 16:50 Yukio Matsumoto (Gakushu-in)
Teichmuller spaces as infinite polyhedra
18:30 - 20:30 BBQ Dinner
29th (Fri)
09:30 - 10:20 Kentaro Saji (Kobe)
Geometry of singularities of fronts
10:30 - 11:20 Benoit Guerville (Pau)
Topological invariant of line arrangements
11:40 - 12:10 Pauline Bailet (Nice)
Milnor fiber of hyperplane arrangements and mixed Hodge theory
14:00 - 14:50 Sampei Usui (Osaka)
A study of open mirror symmetry for quintic threefold through
log mixed Hodge theory
15:00 - 15:30 Ursula Ludwig (Paris Orsay)
The Witten deformation for singular spaces and radial Morse functions
16:00 - 16:50 Laurentiu Maxim (Wisconsin, Madison)
L2 Betti numbers of hypersurface complements
30th (Sat) Free form discussion
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Poster Session 27th (Wed)
Yutaro Kabata (Hokkaido):
Recognition of plane-to-plane map-germs and applications to generic differential geometry
Masayuki Kawashima (Tokyo Sci. Univ.):
On Zariski pair of different torus types
Shumi Kinjo (Shinshu):
Immersions of 3-sphere into 4-space associated with Dynkin diagrams of types A and D
Kazumasa Inaba (Tohoku):
On deformations of isolated singularities of polar weighted homogeneous mixed polyno-
mials
Nhan Nguyen (Marseille):
Lipschitz stratification in o-minimal structures
Takayuki Okuda (Kyushu):
Splitting of singular fibers and topological monodromies
Daiki Sumida (Kyushu):
Singularities of the maps associated with Milnor fibrations
Pho Duc Tai (Vietnam Nat. Univ.):
On the classification of smooth quartics and Zariski pairs from their dual curves
Michele Torielli (Hokkaido) :
Resonant bands, Aomoto complex and real 4-nets
Cristina Valle (Tokyo Met. Univ.):
On the blow-analytic equivalence of embedded singularities
Anna Valette (Krakow):
A generalized Sard theorem on real closed fields
Juan Viu Sos (Pau):
Algebraic Hilbert’s 16th problem and line arrangements
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25th (Mon)
Local topological algebraicity of analytic function germs
Adam Parusinski (Nice)
Abstract : T. Mostowski proved that every (real or complex) germ of an analytic set
is homeomorphic to the germ of an algebraic set. We show that every (real or complex)
analytic function germ, defined on a possibly singular analytic space, is topologically
equivalent to a polynomial function germ defined on an affine algebraic variety. The main
tools are: Artin approximation and Zariski equisingularity. (This is a joint work with
Marcin Bilski and Guillaume Rond).
Effective Lojasiewicz inequalities for the largest eigenvalue ofreal symmetric matrix polynomials
Tien So.n Pha.m (Dalat)
Abstract : Let F (x) = (fij(x))i,j=1,...,p, be a real symmetric matrix polynomial of order
p and let f(x) be the largest eigenvalue function of the matrix F (x). By mf (x) we mean
the nonsmooth slope of f at x (note that mf (x) = ‖∇f(x)‖ if f is differentiable at x). In
this talk, we first give the following nonsmooth version of Lojasiewicz gradient inequality
for the function f with an explicit exponent: For any x ∈ Rn there exist c > 0 and ε > 0
such that
mf (x) ≥ c |f(x) − f(x)|1−1
R(2n+p(n+1),d+3) for all ‖x − x‖ ≤ ε,
where d := maxi,j=1,...,p deg fij and R(n, d) := d(3d − 3)n−1 if d ≥ 2 and R(n, d) := 0 if
d = 1. Then we discuss several aspects of Lojasiewicz inequalities, namely local and global
versions. Using the above inequality, we establish effective estimates for Lojasiewicz’s
exponents for the largest eigenvalue function f(x) of the matrix F (x).
The talk is based on recent joint work with Si Tie.p D- INH.
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25th (Mon)
Equivariant blow-Nash equivalence and equivariant zetafunctions for invariant Nash germs
Fabien Priziac (Saitama, JSPS)
Abstract : A crucial issue in the study of real analytic germs is the choice of a good
equivalence relation by which we can distinguish them. The topological equivalence does
not seem fine enough and the C1-equivalence has already moduli. T.-C. Kuo proposed
an equivalence relation for real analytic germs called blow-analytic equivalence, roughly
speaking analytic equivalence after composition with finite successions of blowings-up
with smooth centers, which seems to be a good equivalence relation in some sense.
In this talk, we are interested in the study of Nash germs, that is real analytic germs
with semialgebraic graph. G. Fichou defined a blow-Nash equivalence for Nash germs, that
is semialgebraic analytic equivalence after blowings-up, and invariants for this equivalence
relation, inspired by the motivic zeta functions of J. Denef and F. Loeser. We consider
Nash germs which are invariant under right composition with a linear action of a finite
group. For these invariant Nash germs, we define a generalization of the blow-Nash
equivalence involving equivariant data, which can also be seen as a refinement of the blow-
Nash equivalence. We then associate to each Nash germ its equivariant zeta functions,
which are defined using an invariant of equivariant real algebraic geometry as a motivic
measure. An important result is that the equivariant zeta functions are invariants for the
equivariant blow-Nash equivalence.
Lipschitz regularity
Le Dung Trang (Marseille-Fortaleza)
Abstract : The classical Theorem of Mumford states that a topologically regular complex
algebraic surface in C3 with an isolated singular point is smooth. Together with L.
Birbrair, A. Fernandes and J.E. Sampaio, I prove that any Lipschitz regular complex
algebraic set is smooth. No restriction on the dimension is needed. No restriction of
singularity to be isolated is needed.
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25th (Mon)
Le’s polyhedron and the boundary of the Milnor fiber ofnon-isolated singularities
Aurelio Menegon Neto (Paraıba, Brazil)
Abstract :We will apply Le’s construction of vanishing polyhedra to study the topology
of the boundary of the Milnor fiber of non-isolated singularities, as well as its degeneration
to the link. This is a joint work with J. Seade.
Derived Algebraic Geometry
Michel Vaquie (Toulouse)
Abstract : In this talk I will introduce derived algebraic geometry, which has been
developed by Bertrand Toen, Gabriele Vezzosi and Jacob Lurie in the last fifteen years.
I will present the ideas which are involved in this theory, which is a generalization of the
classical algebraic geometry and I’ll show how this theory is suitable to singular or non
generic situations.
Then I’ll report on recent progress on generalization of symplectic geometry in the
world of derived algebraic geometry and I’ll discuss about the interaction with deformation
quantization.
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26th (Tue)
Jet schemes and generating sequences of some divisorialvaluations
Hussein Mourtada (Paris)
Abstract : I will talk on the one hand about the notion of a generating sequence of a
valuation, and on the other hand about the relation between jet schemes and divisorial
valuations. I will then describe how this relation allows one to construct generating se-
quences of some divisorial valuations; this provides a constructive approach to a conjecture
of Teissier on resolution of singularities.
Valuations and Toric Geometry
Bernard Teissier (Paris)
Abstract : I will explain the close relation between zero-dimensional valuations of excel-
lent local domains with an algebraically closed residue field and toric geometry.
In the case of Abhyankar valuations, this relation leads to a proof of local uniformization
and a toroidal description of the corresponding valuation rings.
A key ingredient is a valuative version of the Cohen structure theorem.
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26th (Tue)
Topological triviality of linear deformationswith constant Le numbers
Christophe Eyral (Polish Acad. Sci., Warsaw)
Abstract: Let f(t, z) = f0(z)+tg(z) be a holomophic function defined in a neighbourhood
of the origin in C × Cn. It is well known that if the one-parameter deformation family
{ft} defined by the function f is a µ-constant family of isolated singularities, then {ft} is
topologically trivial—a result of A. Parusinski. It is also known that Parusinski’s result
does not extend to families of non-isolated singularities in the sense that the constancy
of the Le numbers of ft at 0, as t varies, does not imply the topological triviality of
the family {ft} in general—a result of J. Fernandez de Bobadilla. In this talk, we show
that Parusinski’s result generalizes all the same to families of non-isolated singularities if
the Le numbers of the function f itself are defined and constant along the strata of an
analytic stratification of C× (f−10 (0)∩ g−1(0)). Actually, it suffices to consider the strata
that contain a critical point of f . This is a joint work with Maria Aparecida Soares Ruas.
Wonderful compactifications from a tropical viewpoint
Eva Feichtner (Bremen)
Abstract : Wonderful compactifications of arrangement complements as defined by De
Concini and Procesi in the early 90s have attracted considerable attention from an alge-
braic, geometric and combinatorial perspective. Notably, their combinatorial core data -
nested set complexes - is intimately linked to tropical geometry.
We will show how the tropical viewpoint provides for fresh insights and enriches our
understanding of these compactifications.
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26th (Tue)
On stable linear deformations of Brieskorn singularitiesof two variables
Nguyen Tat Thang (Tohoku, JSPS)
Abstract : Smooth maps h : X → Y between smooth manifolds X, Y is called stable
if for any map h′
in a small neighborhood of h in C∞(X, Y ) there exist a smooth map
Φ : X → X and Ψ : Y → Y such that h′ ◦ Φ = Ψ ◦ h. The map h is called generic if it
has only fold and cusp singularities. It is well-known that a stable map is a generic map.
A generic map may not be stable, but stable in general.
In this talk, we study the singularities of the mixed linear deformations of Brieskorn
singularities of two variables. The main result is the following:
Theorem 0.1. Let f(u, v) = up + vq, p, q ≥ 2 be a Brieskorn polynomial in two variables.
Then, for generic complex numbers a, b, the mixed polynomial map given by f(u, v)+au+
bv is generic.
This is joint work with K. Inaba, M. Ishikawa and M. Kawashima.
Ideals of the roots posets and a new proof of the dual-partitionformula by Shapiro-Steinberg-Kostant-Macdonlald
Hiroaki Terao (Hokkaido)
Abstract : A Weyl arrangement is the arrangement defined by the root system of a
finite Weyl group. When a set of positive roots is an ideal in the root poset, we call
the corresponding arrangement an ideal subarrangement. Our main theorem asserts that
any ideal subarrangement is a free arrangement and that its exponents are given by the
dual partition of the height distribution, which was conjectured by Sommers-Tymoczko.
In particular, when an ideal subarrangement is equal to the entire Weyl arrangement,
our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and
Macdonald. The proof of the main theorem is classification-free. It heavily depends on
the theory of free arrangements and thus greatly differs from the earlier proofs of the
formula. (This is a joint work with Takuro Abe, Mohamed Barakat, Michael Cuntz and
Torsten Hoge.)
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27th (Wed)
Recent developments on algebra of line arrangements
Takuro Abe (Kyoto)
Abstract : Affine line arrangements in a two-dimensional vector space, or line arrange-
ments in the projective space have been one of the most simple, but interesting and
important objects in theory of hyperplane arrangements. They are simple compared with
higher dimensional cases, but still there are a lot of problems and open conjectures to be
studied.
In this talk, we foucs on thier algebraic aspects (in particular, the freeness), and in-
vestigate the relations with topology and combinatorics of line arrangements. Algebra
controls Betti numbers and the number of chambers of the complements, and vice versa.
Also, recent developments on juming lines of the logarithmic vector fileds, and inductive
(recursive) freeness will be reported.
Projective hypersurfaces with 1-dimensional singularities
Dirk Siersma (Utrecht)
Abstract : The homology of projective hypersurfaces is classically known for smooth
hypersurfaces. Due to results of Dimca the homology of a singular hypersurface with iso-
lated singularities is related to the homology of the smooth case as follows: the difference
is concentrated in one dimension and related to the direct sum of the Milnor lattices of
the singular points. In the talk we will treat 1-dimensional singularities. By using a one
parameter smoothing of an n-dimensional hypersurface we can compare with a smooth
hypersurface. We call this the vanishing homology of the smoothing. We will show that
this (relative) homology is concentrated in two dimensions only: n + 1 and n + 2. More-
over we will give precise information and bounds for the Betti numbers of the vanishing
homology in terms of properties of the singular set, the generic transversal singularities,
the ‘special ’non-isolated singularities and (if they occur) the isolated singularities. As
an example: the n + 2 Betti number is bounded by the sum of (generic) transversal Betti
numbers on each irreducible component of the 1-dimensional singular set. In several cases
this Betti number is zero. We discuss several examples. This is joint work (in progress)
with Mihai Tibar.
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27th (Wed)
Multisections of Lefschetz fibrations and mapping class groupsof surfaces
Kenta Hayano (Hokkaido)
Abstract : A Lefschetz fibration is a smooth map from a 4-manifold to a surface with
only Lefschetz critical points, and a multisection of it is a surface in the total space on
which the restriction of the fibration is a branched covering. In this talk we will explain
relation between multisections and the monodromy representation of a Lefschetz fibration.
As an application we will give new examples of Lefschetz fibrations with multisections.
This is joint work with Refik Inanc Baykur (University of Massachusetts).
17
28th (Thu)
Stratified-algebraic vector bundles
Krzysztof Kurdyka (Savoie)
Abstract : We investigate stratified-algebraic vector bundles on a real algebraic variety
X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed
subvarieties whose union is X. A topological vector bundle ξ on X is called a stratified-
algebraic vector bundle if, roughly speaking, there exists a stratification S of X such that
the restriction of ξ to each stratum S in S is an algebraic vector bundle on S. In particular,
every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-
algebraic vector bundles have many surprising properties, which distinguish them from
algebraic and topological vector bundles. Joint work with W. Kucharz.
Regular functions after one blowing-up in the plane
Goulwen Fichou (Rennes)
Abstract : We study real rational functions which admits a continuous extension at their
poles. These functions becomes regular after some blowings-up. We focus on the plane
case.
18
28th (Thu)
Arc-quasianalytic functions
Guillaume Valette (Krakow)
Abstract : I will present the results of a joint article with E. Bierstone and P. Milman.
We will focus on some quasi-analytic classes of functions. I will explain that if a function
f : U → R is C∞ along every definable arc and has quasianalytic graph then this function
becomes quasianalytic after finitely many local blowing-ups of smooth admissible centers.
This generalizes a theorem of the first two authors about arc-analytic functions.
Mirror Symmetry for Primitive Forms
Kyoji Saito (IPMU)
Abstract : Recently, joint with Changzheng Li and Si Li, we develop a perturbative
method to calculate primitive forms. Then, the method also perturbatively determine
the associated flat structure, including the flat coordinate system of the deformation
parameter space, and (genus 0) pre-potential function for the flat structure. On the other
side, recently, FJRW-theory was established as an intersection theory which counts the
number of solutions of Witten equation for a polynomial W. It was addressed by Ruan
and Chiodo to use it to the global mirror symmetry. In this talk, we confirm the mirror
symmetry between primitive form theory and FJRW-theory for 14 exceptional unimoduar
singularities and others.
More precisely, already in early 90’s, Berglund and Hubsch introduced a mirror dual
polynomial W T for a certain class of weighted homogeneous polynomial W with an iso-
lated critical point, and then Krawitz constructed an identification of the Jacobi ring of
W T and the ring of state space HW of the FJRW-theory for W . Based on their identifica-
tion, we confirm that the 4th Tayler coefficients of the pre-potential of the flat structure
associated to the primitive form of W T coincides with the 4-point correlators of the genus
0 primitive FJRW-invariant for W . Due to the reconstruction of pre-potential based on
WDVV-euqation (Witten et al), this is sufficient to identify the pre-potential in both
sides. Then, further, the reconstruction theory of the higher genus ancestor potential
due to Givental, Teleman and Milanov, we get the full identification of the higher genus
potential functions in both sides, which was to be proven.
(joint work with Changzheng Li, Si Li and Yeffen Schen, http://arxiv.org/abs/1405.4530)
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28th (Thu)
Newton filtration and local cohomology
Shinichi Tajima (Tsukuba) and Yoko Umeta (Tokyo Sci.)
Abstract : We introduce a new framework for studying complex analytic properties of
hypersurface isolated singularities. We show, in this talk, an effective method to com-
pute algebraic local cohomology classes associated with Newton non-degenerate isolated
singularities and apply the method to study Tjurina numbers.
Teichmuller spaces as infinite polyhedra
Yukio Matsumoto (Gakushu-in)
Abstract : We consider Teichmuller space Tg,n of Riemann surfaces of genus g with
n punctures. We fix a sufficiently small number ε > 0 and hollow out an infinite real
analytic polyhedron P εg,n from Tg,n by setting
P εg,n = {[Σ] | Every simple closed geodesic in Σ has length = ε}.
Suppose 3g + n = 5. We will show by simple observations that the the group of automor-
phisms of P εg,n is the mapping class group Γg,n, which acts on P ε
g,n properly discontinously.
Each facet corresponds to a simplex σ of the Harvey’s curve complex Cg,n thus corresponds
to a boundery divisor Dσ of Deligne-Mumford’s compactified moduli space Mg,n. The sub-
group Nσ of Γg,n which preserves the facet Fσ is the normalizer of the free abelian group
Γσ generated by the Dehn twists about the simple closed curves in σ, and the ‘Weyl
group’ Wσ = Nσ/Γσ gives the natural orbifold structure to an open neighborhood of Dσ.
20
29th (Fri)
Geometry of singularities of fronts
Kentaro Saji (Kobe)
Abstract : In this talk, I deal with differential geometric properties of singularities of
fronts. Let M be an m-manifold and (N, g) a Riemannian (m + 1)-manifold. A C∞-map
f : M → N is a wave front (front) if for any p ∈ M , there exist a neighborhood U , and
a C∞-map ν : U → T1N along f such that for any p ∈ U ,
g(dfp(Xp), ν(p)) = 0
holds. Generally fronts have singularities. As an intrinsic formulation of fronts, coherent
tangent bundles will be introduced. Let M be an oriented m-manifold. A coherent tangent
bundle over M is a 5-tuple (M, E, 〈 , 〉 , D, ϕ), where
(1) E is a vector bundle of rank m over M with an inner product 〈 , 〉,(2) D is a metric connection on (E, 〈 , 〉),(3) ϕ : TM → E is a bundle homomorphism which satisfies DXϕ(Y ) − DY ϕ(X) −
ϕ([X, Y ]) = 0 for vector fields X and Y on M .
A point p ∈ M is a ϕ-singular point if ϕ : TpM → Ep is not a bijection, where Ep is the
fiber of E at p. One can characterize Ak-singular points of ϕ-singular points having the
similar properties of those of fronts.
If f is a front, then taking E = (ν)⊥, a coherent tangent bundle is obtained. The
singular curvature which is defined as a curvature of a cuspidal edge in a Riemannian 3-
manifold, can be defined in this setting and the Gauss-Bonnet type formulas hold. With
this setting, several Gauss-Bonnet type formulas and several results on the topology and
geometry of wave fronts will be presented.
Recent results about curvatures of singular surfaces and Gauss-Bonnet type formulas
will also be discussed.
Topological invariant of line arrangements
Benoit Guerville (Pau)
Abstract : The boundary manifold of a line arrangement can be defined as the boundary
of a regular neighbourhood of the arrangement in CP 2. The study of its inclusion in the
complement allows to construct a new topological invariant of the arrangement. It can be
view as an analogue of the linking number in links theory. It is simply computable and
allows to differentiate two arrangements with the same combinatorial structure but with
different embedding in CP 2 (i.e. Zariski pair).
21
29th (Fri)
Milnor fiber of hyperplane arrangementsand mixed Hodge theory
Pauline Bailet (Nice)
Abstract : We consider the mixed Hodge structure of the cohomology groups Hq(F, C)
of the Milnor fiber F of a central hyperplane arrangement A ⊂ Cn+1, and the mixed
Hodge numbers ha,b(Hq(F, C)).
We say that a complex variety Y is cohomologically Tate, if for any cohomology group
Hq(Y, C), we have the following vanishing of mixed Hodge numbers:
ha,b(Hq(Y, C)) = 0, for a 6= b.
Using the action of the monodromy on the cohomology groups of the Milnor fiber and
the spectrum of an arrangement, we study the following open question:
Is the Milnor fiber F of a central hyperplane arrangement cohomologically Tate? This
question has already been considered, and we have results for arrangements in C2 and C3.
Here we etablish an equivalence between triviality of the monodromy, Tate properties, and
the nullity of the non integer spectrum’s coefficients, for central and essential arrangements
in C4.
A study of open mirror symmetry for quintic threefold throughlog mixed Hodge theory
Sampei Usui (Osaka)
Abstract : We study open mirror symmetry for quintic threefold through log mixed
Hodge theory, especially by the recent result on Neron models for admissible normal
functions with non-torsion extensions in the joint work with K. Kato and C. Nakayama.
We positively use local systems with graded polarizations over the boundary points.
22
29th (Fri)
The Witten deformation for singular spacesand radial Morse functions
Ursula Ludwig (Paris Orsay)
Abstract : About 30 years ago motivated by ideas in quantum field theory, Witten
introduced a beautiful new approach to proving the famous Morse inequalities based on
the deformation of the de Rham complex (see “Supersymmetry and Morse Theory”, J. of
Diff. Geometry, 17). His ideas were fruitfully extended in different situations since, e.g.
to the holomorphic setting, for manifolds with boundaries ... The Witten deformation
combined with local index techniques allowed Bismut and Zhang (Asterisque 205, 1992)
to give generalisations of the famous theorem of Cheeger and Muller on the comparison
between analytic and topological torsion. The aim of this talk is to present a generalisation
of the Witten deformation to a singular space X with cone-like singularities and radial
Morse functions. As a result one gets Morse inequalities for the L2-cohomology, or dually
for the intersection homology of the singular space X. Moreover, as in the smooth theory,
one can relate the Witten complex, i.e. the complex generated by the eigenforms to small
eigenvalues of the Witten Laplacian, to an appropriate geometric complex (a singular
analogue of the smooth Morse-Thom-Smale complex). Radial Morse functions are inspired
from the notion of a radial vector field on a singular space. Radial vector fields have first
been used by Marie-Helene Schwartz to define characteristic classes on singular varieties.
L2 Betti numbers of hypersurface complements
Laurentiu Maxim (Wisconsin, Madison)
Abstract : I will present vanishing results for the L2 cohomology of complements to
complex affine hypersurfaces.
23
Poster Session
Recognition of plane-to-plane map-germs and applications togeneric differential geometry
Yutaro Kabata (Hokkaido)
Abstract : We show a complete set of criteria for plane-to-plane map-germs of corank one
with A-codimension≤ 6. As applications of our criteria to generic differential geometry,
we study surfaces through singularities arising in projection of them.
On Zariski pair of different torus types
Masayuki Kawashima (Tokyo Sci. Univ.)
Abstract : Let C be a plane curve in P2. We are interested in the topology of C and sev-
eral topological invariants of C: the fundamental group of the complement, the Alexander
polynomial and Characteristic varieties. In this poster, we construct Zariski pairs of dif-
ferent torus types of degree pq such that they have different Alexander polynomials. As
an application, we can construct Zariski triple of degree 8 such that they have different
Characteristic varieties.
24
Poster Session
Immersions of 3-sphere into 4-space associated with Dynkindiagrams of types A and D
Shumi Kinjo (Shinshu)
Abstract : The Smale-Hirsch h-principle implies that the group of regular homotopy
classes of immersions of m-sphere into N -space is isomorphic to the m th homotopy
group of the Stiefel manifold of m frames in N -space. The isomorphism is given as taking
the differential at each point of m-sphere and is called the Smale invariant. In particular,
Ekholm and Takase have given a formula for the Smale invariant of an immersion of
3-sphere into 4-space by using a singular Seifert surface for the immersion.
In this session we will explain the construction of two infinite sequences of immersions
of the 3-sphere into 4-space, parametrized by the Dynkin diagrams of types A and D. The
construction is based on immersions of 4-manifolds obtained as the plumbed immersions
along the weighted Dynkin diagrams. We will compute their Smale invariants by using
Ekholm-Takase’s formula in terms of singular Seifert surfaces.
On deformations of isolated singularities of polar weightedhomogeneous mixed polynomials
Kazumasa Inaba (Tohoku)
Abstract : In this poster, we deform isolated singularities of fg, where f and g are
2-variable weighted homogeneous complex polynomials, and show that there exists a
deformation of fg which has only indefinite fold singularities and isolated singularities
topologically equivalent to a complex Morse singularity.
25
Poster Session
Lipschitz stratification in o-minimal structures
Nhan Nguyen (Marseille)
Abstract : In this paper we prove that there exists a Lipschitz stratifications in the sense
of Mostowski for sets which are definable in a polynomially bounded o-minimal structure.
Splitting of singular fibers and topological monodromies
Takayuki Okuda (Kyushu)
Abstract : A degeneration of Riemann surfaces is a complex one parameter family of
compact complex curves allowed to have one singular fiber. It is known that there is a
good relationship between the topological types of degenerations of Riemann surfaces and
the surface mapping classes, via topological monodromy. In this poster, we are interested
in splitting families for degenerations, that is, deformation families in which the original
singular fiber splits into several simpler singular fibers. We introduce the topological
monodromies of splitting families, and show that they play an important role on studying
the topology of splitting families.
26
Poster Session
Singularities of the maps associated with Milnor fibrations
Daiki Sumida (Kyushu)
Abstract : I consider the product map of the projections of Milnor fibrations. It can
be fold map, furthermore the type of singularity of its depends on the choice of mixed
polynomials which define Milnor fibrations. I give how to determine fold singularity of
the product maps and its examples.
On the classification of smooth quartics and Zariski pairs fromtheir dual curves
Pho Duc Tai (Vietnam Nat. Univ.)
Abstract : We study the geometry of smooth quartics and together their dual curves.
Our main result is giving a new list of Zariski pairs that appear in the above dual curves.
27
Poster Session
On the blow-analytic equivalence of embedded singularities
Cristina Valle (Tokyo Met. Univ.)
Abstract : We present the classification of embedded real curve singularities up to blow-
analytic equivalence. In the case of unibranched and bibranched plane curve germs, this
was done by Kobayashi and Kuo. Here we show that the number of equivalence classes
is finite for any number of branches and fixed the value of a discrete invariant. In the
tribranched case, we give some explicit classification results. Furthermore, motivated by
a question of Fukui, we outline the study of curve germs embedded in a real singular
surface up to blow-analytic homeomorphism.
A generalized Sard theorem on real closed fields
Anna Valette (Krakow)
Abstract : We work with semi-algebraic functions on arbitrary real closed fields.We
generalize the notion of critical values and prove a Sard type theorem in our framework.
(joint work with Guillaume Valette)
28
Poster Session
Algebraic Hilbert’s 16th problem and line arrangements
Juan Viu Sos (Pau)
Abstract : Hilbert’s sixteenth problem is concerned with counting the maximal number
of limit cycles for a planar polynomial vector field. Considering algebraic limit cycles,
we focus on the relation between real line arrangements invariants by polynomial vector
fields. Thus, we investigate the influence of the combinatorial structure of an arrangement
on the minimal degree of the logarithmic derivatives of the arrangement (fixing only a
finite number of lines). From here we investigate the bounds of the maximal number of
lines invariant by a polynomial vector field of fixed degree. (Joint work with J. Cresson
and B. Guerville-Ball)
