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## Symplectic monodromy, Leray residues and quasi-homogeneous polynomials (2012)

Citations: | 1 - 1 self |

### Citations

636 | Singular points of complex hypersurfaces - Milnor - 1968 |

496 |
Introduction to Symplectic Topology
- McDuff, Salamon
- 1995
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Citation Context ...wise by iZΩ|TXz = θCn+1 |T (Xz ∩U) in a neighbourhood U ∩Xz of ∂Xz satisfies the required LZΩγ = Ωγ , for any path γ : [0, 1] → D∗δ , which implies that αz defines a contact form for any z ∈ D∗δ (see =-=[11]-=-). That the orthogonality requirement of 2.5 is satisfied follows from the fact that the gradient Zr := grad(r) of the radius function r on B2n+2r , 0 < r < 1 satisfies by direct calculation i 12ZrΩCn... |

459 |
Elements de geometrie algebrique
- Grothendieck
- 1966
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Citation Context ...−1(V ),F) (Rif∗ is identical to the right derived functor of the direct image functor f∗ and is calculated by injective or f∗-acyclic resolutions of F, for details see Hartshorne [30] or Grothendieck =-=[29]-=-). We have the following isomorphism, refer to Looijenga [27]. Note that the sections of the vectorbundle accociated to Rif∗CX (with fibres Hi(Xu,C) over S′) constitute the sheaf Rif∗CX ⊗CS OS. Lemma ... |

259 |
Stable mappings and their singularities
- Golubitsky, Guillemin
- 1973
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Citation Context ... ′ : Vτ → Vτ ′ , hτ = IdVτ so that for all τ ′ ∈ Iτ the diagram Vτ × J −−−−→ gτ′ Vτ ′ × Jypr1 ypr1 Vτ −−−−→ hτ′ Vτ ′ (67) commutes. Following [17] (Ch. VII, 8), (see also Guillemin and Golubitsky =-=[16]-=-, Ch. V, Theorem 4.2), this already follows if we can show that f̂τ,U is ’infinitesimally stable’ in the appropriate sense. We will sketch a proof of this below (Lemma 2.24) and assume for the moment ... |

183 | Fukaya categories and Picard-Lefschetz theory - Seidel |

103 |
Tangents to an analytic variety
- Whitney
- 1965
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Citation Context ... dim(Λ(x) ∩ Λs(x)) = k}, where codim(Mkx) = 1 2k(k + 1) (cf. Arnold [2]). Furthermore, for any x ∈ N , the sets {Mkx}k∈N+ furnish Mx with the structure of a stratified space (see Mather [34], Whitney =-=[48]-=-) with smooth top-stratum M1x of codimension 1, singular set M 2 x of at least codimension 3 in i ∗Lag(M,ω)x and strata Mkx. Set M k = ⋃ x∈N M k x, k ∈ N+. These remarks suggest the following Definiti... |

86 | Special Lagrangians, stable bundles and mean curvature flow
- Thomas, Yau
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Citation Context ...yθ R −−−−→ R/2πZ, (77) Proof. The proof follows from the fact that σQ = dθ on Q, that is H = Jd̃θ, where ·̃ here denotes metric duality T ∗Q→ TQ and H the mean curvature vector field and is proven in =-=[45]-=-, see also [5]. Lemma 3.4. There is a smooth extension Xke of X̃ k to the unpunctered disk fke : X k e → Dδ and an extension Ωke of Ω k to Xke so that (X k e , f k e ,Ω k e) defines an exact symplecti... |

66 | On the Maslov index - Cappell, Lee, et al. - 1994 |

50 | Lectures on Calabi-Yau and special Lagrangian geometry - Joyce |

48 |
Intégrales asymptotiques et monodromie
- Malgrange
- 1974
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Citation Context ...ism of the respective OD,0-modules. Finally recall that HnX/D |D∗ ≃ H′′|D∗, where HnX/D is the relative cohomology sheaf on D and the isomorphism is given by the Poincare-Leray-residue (cf. Malgrange =-=[21]-=-, Section 4). Then, the above facts gave rise to the following question: Is there a ’singularity-theory’ proof of the fact that the map π0(Symp(F, ∂F, ω))→ π0(Diff(F, ∂F )), for F being the Milnor fib... |

31 |
Lagrangian intersection theory: finite-dimensional approach, from: “Geometry of differential equations
- Eliashberg, Gromov
- 1998
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Citation Context ...(2.) is satisfied will also be proven in Appendix D. by representing a neighbourhood of each Qτ,x(t) in Qτ using generating families and subsequently using stability theory (see Eliashberg and Gromov =-=[13]-=- resp. Guillemin and Sternberg [17]) to show that certain connected components of the complement of the family of caustics NMτ,t ⊂ Qτ,x(t), t ∈ {0, 1}, τ ∈ [0, 1] do not vanish along the symplectic is... |

26 | Ramanujam The invariance of Milnor’s number implies the invariance of topological type - Tráng, P - 1976 |

25 |
Mixed Hodge structures and singularities, Cambridge Tracts
- Kulikov
- 1998
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Citation Context ...nvalues λi, i = 1 . . . , µ of the monodromy. Here, the normalization is determined by the asymptotic Hodge filtration on the ’canonical’ Milnor fibre, for that terminology, see for instance Kulikov (=-=[19]-=-) resp. Section 4. Since f is quasihomogeneous, in terms of a monomial basis (zα(i))µi=1, where α(i) ∈ Λ ⊂ Nn+1, |Λ| = µ, α(1) = 1, of the Milnor algebra M(f) := OCn+1,0/( ∂f ∂z0 , . . . , ∂f ∂zn )OCn... |

14 |
Closed forms on symplectic fibre bundles
- Gotay, Lashof, et al.
- 1983
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Citation Context ...ectic forms ωx and contact forms over the fibrewise boundary αx so that [ωx, αx] = 0 and so that there is a trivialization of a neighbourhood of the boundary as in (9). This implies (see Gotay et al. =-=[8]-=-) there 34 is at least locally on the base a smooth family of forms θz ∈ Λ1(T vXk2 )z , z ∈ U ⊂ D[δ,ǫ] (U open) so that dθz |(Xk2 )z = ωz, for any x ∈ D[s,ǫ].Then defining locally ΘU (x) = (f k 2 ) ∗(... |

11 | A note on mean curvature, Maslov class and symplectic area of Lagrangian immersions
- Cieliebak, Goldstein
- 2004
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Citation Context ...he family Qτ ev(sβ0 )(Qτ ) = iXf (π β)∗(dz0 ∧ · · · ∧ dzn)(Qτ ) = (eiθiXfvolQ)(Qτ ), (5) where d(f ◦πβ)∗(Xf ) = 1 and Xf is horizontal and eiθ : Q→ S1 is the ’phase’ of Q in Xβ. Now it is well-known (=-=[5]-=-) that dθ = σQ, where σQ is the mean curvature form of Q inX β. Then we can deform Q to a family of (Lagrangian) submanifolds Qt ⊂ Xβe , t ∈ [0, 1] fibred over S1δ(t) ⊂ D, δ(t)→ 0, t→ 1 into Lagrangia... |

7 |
Some topological invariants of isolated hypersurface singularities
- Nemethi
- 1996
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Citation Context ...∈ im i∗ : Hn(M,∂M,C) → Hn(M,C) implies by the exact sequence (19) above that [sx] lies in the image of b : H n(M,∂M,C) → Hn(M,∂M,C)∗. Now since the variation structure of f , compare for the notation =-=[24]-=- (see also [42]), is given by V(f) = ⊕ α∈Λ Wexp(2πil(α))((−1)[l(α)]+n) (20) and since the kernel of b is represented by (Lemma 4.4) monomials zα s.t. {α ∈ Λ ⊂ Nn+1 : l(α) ∈ Z} the claim follows, since... |

6 | Toward a general theory of linking invariants
- Chernov, Rudyak
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Citation Context ...Q1×{τ},Z)/Tor so that γ0,1 coincide with their previous definitions and further (γ∆τ = γτ − γ1) ∩ ∂N1,top = ∅ for any τ ∈ [0, 1]. Thus the family of pairs (∂N1,top, γ∆τ ) defines a link homotopy (see =-=[4]-=-) so that ∂N1,top ⊚ γ∆0 = ∂N 1,top ⊚ γ∆, and ∂N1,top ⊚ γ∆1 = 0, so we arrive at the assertion of Claim 1. To prove the second claim, recall that by (33) we have ΦXK (t) ∗sk = e2πiγtsk for t ∈ [0, 1], ... |

6 |
Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in C3
- Saeki, Real
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Citation Context ...m of an isolated holomorphic singularity f : Cn+1 → C is a topological invariant for n ≤ 2. However, as Saeki shows, that result remains true for n = 3 if f is quasihomogeneous, moreover we have (cf. =-=[36]-=-, [46]): Theorem 4.8. Let f and g be quasihomogeneous polynomials with an isolated singularity at the origin in Cn+1 for n ≥ 1. Then the following four are equivalent: 1. f and g are connected by a µ-... |

5 | The equivariant signature of hypersurface singularities and eta–invariant - Némethi - 1995 |

5 |
Krylov: Kernel of the variation operator and periodicity of opebn books, math.GT 0308277v2
- Kauffman, N
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Citation Context ...resp. diffeomorphisms fixing the boundary ∂F pointwise (’relative’ here means the isotopy fixes the boundary pointwise). It is known as a consequence of work of Stevens [44] resp. Kauffman and Krylov =-=[31]-=- that if one represents the bundle Ỹ as a mapping cylinder Ỹ = F × [0, 1]/(x, 0) ∼ (ρ(x), 1) =: Fρ for some element ρ ∈ Diff(F, ∂F ), then under certain conditions, ρ is of finite order in π0(Diff(F... |

5 |
Periodicity of branched cyclic covers of manifolds with open book decomposition
- Stevens
- 1986
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Citation Context ...py group of symplectomorphisms resp. diffeomorphisms fixing the boundary ∂F pointwise (’relative’ here means the isotopy fixes the boundary pointwise). It is known as a consequence of work of Stevens =-=[44]-=- resp. Kauffman and Krylov [31] that if one represents the bundle Ỹ as a mapping cylinder Ỹ = F × [0, 1]/(x, 0) ∼ (ρ(x), 1) =: Fρ for some element ρ ∈ Diff(F, ∂F ), then under certain conditions, ρ ... |

4 | On transversality - Bruce - 1986 |

4 | The eta-invariant of variation structures - Nemethi - 1995 |

4 |
singular points on complete intersections
- Isolated
- 1984
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Citation Context ...44 we have chosen some basis (s1, . . . , sµ) spanning the sheaf of modules H n(f∗Ω·X̂/Dδ ) over ODδ so that the associated basis g1, . . . gµ ∈ OCn+1,0/M̂(f), where M̂(f) is a certain submodule (see =-=[27]-=-) of OCn+1,0 so that OCn+1,0/M̂(f) ≃ ωf,0/dΩn−1X/Dδ,0 (c.f. section 4), is chosen so that g1 = 1. As in the quasihomogeneous case, let ŝ ∈ Γ(Hn(f∗Ω·X̂/Dδ )) be associated to g1 and satisfying (17) an... |

4 | Gauß-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten - Der - 1975 |

4 |
Varchenko: The complex exponent of a singularity does not change along strata µ=const., Funct
- N
- 1982
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Citation Context ...h Hn(f∗Ω·X/S) outside of 0. Let now ω be a section ofH′′ over a neighbourhood S ⊂ C around s = 0 and consider this as a section of Hn(f∗Ω·X/S) on S ′ = S \ {0}, those sections are called by Varchenko =-=[46]-=- ’geometric sections’. Consider now over S′ the locally constant sheaf Hn = Rif∗CX′ and its dual Hn = Hom(H n,C) as 48 the sheaf of homomorphisms from Hn to C. For any s ∈ S′ we have Hn(s) ≃ Hn(Xs,C) ... |

3 |
On a generalization of the Hopf fibration
- Abe
- 1977
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Citation Context ...eans the isotopy fixes the boundary pointwise). It is known as a consequence of work of Stevens [44] resp. Kauffman and Krylov [31] that if one represents the bundle Ỹ as a mapping cylinder Ỹ = F × =-=[0, 1]-=-/(x, 0) ∼ (ρ(x), 1) =: Fρ for some element ρ ∈ Diff(F, ∂F ), then under certain conditions, ρ is of finite order in π0(Diff(F, ∂F )), namely if n is arbitrary and V (ρd) = 0, where V is the ’variation... |

3 |
Brieskorn:Die Monodromie der isolierten Singularitaeten von Hyperflchen
- V
- 1969
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Citation Context ...Then the map φ sending zα(i) to the class given by φ(zα(i)) = zα(i)dz0 ∧ · · · ∧ dzn in H′′0 , defines a C-isomorphism of vectorspaces φ : M(f) ≃ H′′0/fH′′0 , being, since H′′ is coherent, even free (=-=[9]-=-), also an isomorphism of the respective OD,0-modules. Finally recall that HnX/D |D∗ ≃ H′′|D∗, where HnX/D is the relative cohomology sheaf on D and the isomorphism is given by the Poincare-Leray-resi... |

3 |
Differentialtopologie und Singularitten, vieweg Wiesbaden Braunschweig 2001
- Funktionentheorie
(Show Context)
Citation Context ...is used in Section 2, the proof relies on a two-fold application of Moser’s technique and classical results about vanishing cycles of Milnor fibres and will be sketched below, for details see Ebeling =-=[12]-=- and Seidel [41], note that the Lemma is valid for any isolated singularity f : (Cn+1, 0)→ (C, 0) where f is holomorphic. Lemma 5.1. Let µ be the Milnor number of Fu. Then (Fu;ω) contains a collection... |

3 |
Morse theory and global coexistence of singularities on wave fronts
- Ferrand, Pushkar
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Citation Context ...er τ . Note that these considerations are connected with a question posed by Arnol’d concerning the persistence of caustics of wavefronts in families of Lagrangian embeddings (see Ferrand and Pushkar =-=[15]-=-, Entov [14]). We finally define a loop c : [0, 1]/{0, 1} → Q by fixing point z0 ∈ Qx, x ∈ S1δ above, and defining a map c̃ : [0, 1]→ Q̂ c̃(τ) = { (z0, δ, τ), τ ∈ [0, 1− r]( (ρkψ(τ))(z0), δ, τ ) , τ ∈... |

3 |
R.Mazzeo:Hodge cohomology of gravitational instantons, mathDG 0207169
- Hausel, Hunsicker
- 2003
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Citation Context ...inition of κF , this implies (88). Here we used that by the horizontality of F and the ’flatness’ of Xk2 in a neighbourhood of its common boundary with X̃ k, we have ∇uR = ∇LR using Proposition 14 of =-=[22]-=- (the second fundamental form of the fibre in the direction of R and the curvature of T hXke |U vanishes, here we use the submersion metric on Xke constructed in the remark below the proof of Lemma 3.... |

3 |
Eta invariants, spectral flow and sympletic monodromy of quasihomogeneous hypersurface singularities, Dissertation Thesis
- Klein
- 2010
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Citation Context ...t ∈ [0, 1] denotes parallel transport in X̃k along t 7→ ǫe2πit using the horizontal 13 distribution defined by the Euler vector field π∗k(K) = π ∗ k(2πi ∑ iwizi ∂ ∂zi ) on X̃k then by Lemma 3.1.11 in =-=[33]-=- ΦXK (t) ∗sk = e2πiγtsk, (33) where γ = m( ∑ i βi − β) ∈ Z and k = mβ as above, so γ ∈ Z \ {0} by Assumption 2.9. Note that this equation continues to hold on cohomology classes when replacing ΦXK by ... |

3 | Varchenko: The Gauss-Manin connection of isolated singular point and the Bernstein polynomial - N - 1980 |

2 |
Une classe characteristique intervenant dans les conditions de quanti cation
- Arnold
- 1998
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Citation Context ... section of det2 over i∗s2 in i∗∆(J)∗. Then Λs := u ◦ (i∗s)2 : N → i∗Lag(M,ω) defines the Maslov cycle M = ⋃ x∈N Mx, Mx = {Λ(x) ∈ i∗Lag(M,ω)x : Λ(x) ∩ Λs(x) 6= {0}}. Now it is well-known (see Arnol’d =-=[2]-=-), that M = PD[g̃∗β], where [g̃∗β] ∈ H1(i∗Lag(M,ω),Z) is determined by g̃ : i∗Lag(M,ω) → C∗ and det2(Λs(x)) = g̃(Λ(x)) · det2(Λ(x)) for any Λ(x) ∈ i∗Lag(M,ω)x, x ∈ N . On the other hand if m = dim(i∗L... |

2 | Fibred knots and algebraic singularities,Topology 13 - Durfee - 1974 |

2 |
Hartshorne,Algebraic Geometry,Springer
- unknown authors
- 1977
(Show Context)
Citation Context ...d to V ⊂ S, V 7→ Hp(f−1(V ),F) (Rif∗ is identical to the right derived functor of the direct image functor f∗ and is calculated by injective or f∗-acyclic resolutions of F, for details see Hartshorne =-=[30]-=- or Grothendieck [29]). We have the following isomorphism, refer to Looijenga [27]. Note that the sections of the vectorbundle accociated to Rif∗CX (with fibres Hi(Xu,C) over S′) constitute the sheaf ... |

2 | Symplectic monodromy, quasihomogeneous polynomials and spectral flow, preprint math.DG/1104.2095v2 - Klein |

2 | Sakamoto:The Seifert-matrices of Milnor fiberings defined by holomorphic functions - unknown authors - 1974 |

2 |
Graded Lagrangian submanifolds, math.SG 9903049
- Seidel
(Show Context)
Citation Context ...= id and Kf being a rational homology sphere are equivalent to V (ρβ) = 0, if ρ represents h in π0(Diff(F, ∂F )), implying that in such cases ρ has finite order in π0(Diff(F, ∂F )). Now Seidel shows (=-=[39]-=-) that if n ≥ 2 and under the condition m(f) = µ∑ i=1 βi − β 6= 0, (2) the symplectic monodromy ρ ∈ π0(Symp(F, ∂F, ω)) is of infinite order. Note that Y carries the structure of a symplectic fibration... |

2 |
Steenbrink:Intersection form for quasi homogeneous polynomials
- unknown authors
- 1977
(Show Context)
Citation Context ...∂M,C) → Hn(M,C) implies by the exact sequence (19) above that [sx] lies in the image of b : H n(M,∂M,C) → Hn(M,∂M,C)∗. Now since the variation structure of f , compare for the notation [24] (see also =-=[42]-=-), is given by V(f) = ⊕ α∈Λ Wexp(2πil(α))((−1)[l(α)]+n) (20) and since the kernel of b is represented by (Lemma 4.4) monomials zα s.t. {α ∈ Λ ⊂ Nn+1 : l(α) ∈ Z} the claim follows, since [sx] is by def... |

2 | and weight filtration for smoothings of isolated singularities - Monodromy - 1995 |

1 | on Lagrangian and Legendrian Singularities - Surgery - 1999 |

1 |
Mather: Stratifications and mappings, Dynamical Systems
- N
- 1973
(Show Context)
Citation Context ... i∗Lag(M,ω)x : dim(Λ(x) ∩ Λs(x)) = k}, where codim(Mkx) = 1 2k(k + 1) (cf. Arnold [2]). Furthermore, for any x ∈ N , the sets {Mkx}k∈N+ furnish Mx with the structure of a stratified space (see Mather =-=[34]-=-, Whitney [48]) with smooth top-stratum M1x of codimension 1, singular set M 2 x of at least codimension 3 in i ∗Lag(M,ω)x and strata Mkx. Set M k = ⋃ x∈N M k x, k ∈ N+. These remarks suggest the foll... |

1 |
Fibred symplectic homology and the Leray spectral sequence
- Oancea
(Show Context)
Citation Context ...or any δ chosen as above and restricted to D∗δ := Dδ \{0} (here, Ω is the restriction of ΩCn+2 to X), is a symplectic fibration in the following sense (the form of the definition is mainly drawn from =-=[35]-=-). Definition 2.4. Let π : F →֒ E → S be a locally trivial fibration over a symplectic manifold (S, β) with boundary, so that the fibre F is a compact manifold with boundary. Let ∂hE denote the union ... |

1 | Khovanov:Quivers and Floer cohomology, preprint 2002 - Seidel, M |

1 |
A Long exact sequence for Floer Homology, math.SG 0105186,(2002
- Seidel
(Show Context)
Citation Context ...nnected). Note that this fact is a priori formulated for a closed symplectic manifold, but since in a neighbourhood of ∂M , J is compatible with j in the sense of Ass. 2.5, one can argue as in Seidel =-=[41]-=- to deduce that the space of in that sense compatible complex structures J(M,ω, j) is contractible. So we can choose a path of loops τ 7→ (J1)τ,t, t ∈ S1, τ ∈ [r, s] of j-compatible complex structures... |