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### Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem. Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$. Then for $\ell$ a linear form on $E$, we have a ...
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### Can I compute the cohomology of the complement of a log canonical divisor as if it were normal crossings?

Let $X$ be a smooth projective variety and $D$ a log-canonical divisor and let $U = X \setminus D$. I have heard the slogan "log-canonical is just as good as normal crossings for Hodge theory". This ...
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### Toric Degenerations and Nearby Cycles

Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) ...
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Let $U \subset \mathbb{R}^n$ be an open set and let $f:U \rightarrow \mathbb{R}$ be a smooth proper function. For $p \in \mathbb{R}^n$ let $$T_p(x_1,\ldots,x_n) = \sum_{i_1,\ldots,i_n=0}^{\infty} c_{... 0answers 464 views ### An elementary question in singularities The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ... 0answers 262 views ### Singularity structure of integrals of rational functions Suppose I have a convergent integral of the form \int_0^1dx_1\dots\int_0^1 dx_n \frac{P(x_i)}{Q(x_i)}, where P and Q are polynomial functions of n nonnegative real variables x_i. Let the ... 0answers 114 views ### Gysin exact sequence for a singular subvariety Let k be an algebraically closed field (I'm interested in a characteristic p>0 specific example) and let X be a (smooth if needed) algebraic variety. Let Y \subset X be a (possibly) ... 0answers 116 views ### Framed singular knots I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let D be ... 0answers 520 views ### Jacobian ideals reference Suppose that f : X \to V is a flat equidimensional (of dimension h) morphism of schemes of finite type and V is excellent (or a variety) For this one can formulate something called the Jacobian ... 0answers 551 views ### Isolated singularities and tangent cones Assume that I have an affine hypersurface X =V(f)\subset \mathbb{C}^4 of degree d with an isolated singularity of multiplicity m at the origin o=(0,0,0,0). Let$$f:=f_m + f_{m+1}+ \cdots +f_d$$... 0answers 319 views ### What is known about “singularity types” in the Murphy's Law sense? In his "Murphy's Law" paper, Vakil gives a definition equivalent to the following: The singularity type of a pointed scheme (X,p) its equivalence class, under the following equivalence relation: ... 0answers 257 views ### Fixed point sets that carry topology Let M be a closed smooth manifold. A generic diffeomorphism \phi: M\rightarrow M has non-degenerate fixed points, i.e. the intersections of its graph with the diagonal in M\times M are all ... 0answers 112 views ### Singular symplectic reduction in infinite dimension In 1991, Sjamaar and Lerman [1] introduced the notion of stratified symplectic spaces. Namely, if M is a symplectic manifold and G a Lie group acting properly (but not necessarily freely) on M ... 0answers 172 views ### Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical Let start with a definition Invariance of plurigenera: Choose m large enough so that mK_F has a non-zero global section for some fibre F. For any fibre F, we have K_F = K_{X/D}~_{|F}. So ... 0answers 123 views ### \mathbb{Q}-factoriality of singularities I would like to understand if a certain variety is \mathbb{Q}-factorial (i.e., if every Weil divisor D has a multiple mD that is Cartier). This property can be deduced by a local picture around ... 0answers 122 views ### Real structure in the mixed Hodge structure associated to an isolated singularity We know that a mixed Hodge structure on a complex vector space H with an integral lattice H_{\mathbb Z} consists of the weight filtration and the Hodge filtration. For an isolated hypersurface ... 0answers 190 views ### n-Fold Framed Functions Suppose that M is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on M, let's call it \mathrm{Fun}^\mathrm{fr}(M). This space is known to be ... 0answers 86 views ### Is there a correspondence between counting curves in P^2 blown up at a point and counting curves in P^2? Let X be \mathbb{P}^2 blownup at one point and \beta := d L -2E \in H_2(X, \mathbb{Z}), where L and E denote the class of a line and the exceptional divisor respectively. Let \mathcal{L}... 0answers 155 views ### Implicit function theorem for singularities I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild". For example, let 0 = z^2 - x y + z w + w^2 + \epsilon w define a ... 0answers 229 views ### Hypersurface with singularities I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?). It was said ... 0answers 209 views ### References for resolutions of ordinary singular points Let X be a n-dimensional complex projective algebraic variety, let us suppose that X has only isolated singularities. Edit: Let us say that an ordinary m-ple singular point is an isolated ... 0answers 191 views ### Hypersurfaces with Gorenstein singular loci Recall that a hypersurface D in a complex manifold X is called a free divisor if the Lie algebroid \mathcal{T}_X(-\log D) of vector fields tangent to D is locally free. This condition is ... 0answers 68 views ### on Neron defect of smoothness for groups schemes Let G a semisimple simply connected group over \mathbb{C}. Let \gamma\in G(\mathbb{C}[[t]]) such that \gamma is regular semisimple on G(\mathbb{C}((t))). We consider I_{\gamma} the group ... 0answers 321 views ### Blowing up a projective surface Let X be a smooth degree d (d>5) surface in \mathbb{P}^3. We now blow up X at a point, embed it in some projective space, and and consider a projection of it into \mathbb{P}^3. The ... 0answers 129 views ### Equivalence of Level Sets Consider the zero level set of f : \mathbb{R}^3 \to \mathbb{R}, where 0 is a regular value. Consider also the space of planes passing through the origin, i.e. \mathbb{RP}^2. For a fixed plane P ... 0answers 216 views ### Good covers on complex algebraic varieties with normal crossings singularities Let X be a topological space. A good cover on X is an open cover such that all finite non-empty intersections are contractible. It is a theorem of Hironaka that (complex) algebraic sets admit ... 0answers 146 views ### Tangent cones to Severi strata Let \mathbb{C}[[x,y]]/f(x,y) be a reduced plane curve singularity. The base of a versal family can be taken to be (an open subset in) \Lambda = \mathbb{C}[x,y]/(f,\partial_x f, \partial_y f); the ... 0answers 573 views ### Sebastiani-Thom isomorphism for D-modules Considering f:X\to \mathbb{C}, g:X\to \mathbb{C} and f\oplus g:(x,y)\mapsto f(x)+g(y). The Sebastiani-Thom isomorphism is an isomorphism \Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes \... 0answers 41 views ### Orbit spaces of Coxeter groups and singularities I have often seen in the literature the statement that the orbit spaces of irreducible finite Coxeter groups are equivalent to unfoldings of singularities. For instance, taken from Dubrovin, ... 0answers 161 views ### Ricci curvature in resolution of singularities Let X and X' are Kahler variety and f: (X',\omega')\to (X,\omega) be the resolution of singularities of X then from K_X=f^*K_X'+E how can we find the relation between Ric(\omega) and ... 0answers 97 views ### Singularities of algebraic curves, and torsion of the pull-back of the differential module by the normalisation The problem in the following : given an algebraic curve C, it's well-known that a smooth projective model of C can be construct as the set of discrete valuations v on it's function field \... 0answers 69 views ### quasi-ordinary singularities on a versal deformation? Let V be a variety over \mathbb{C} and suppose O is a singular point of V. Are there conditions on (V,O) such that a versal deformation W of (V,O) has only quasi-ordinary singularities. ... 0answers 400 views ### Weil Petersson metric on moduli space of Calabi Yau manifolds Let f:(X,D)\to Y be a holomorphic fibre space where D is divisor with conic singularities and let fibres (X_s,D_s) are log Calabi-Yau pair .i.e K_X+D is nummerically trivial, then we have ... 0answers 251 views ### generalization Abhyankar's lemma This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question. Let ... 0answers 263 views ### A strong form of implicit function theorem (what happens when the derivative is degenerate?) (this can be considered as some ad) Consider the system of equations F(x,y)=0. (Here x, y are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/C^\infty ... 0answers 392 views ### Blowdown and contraction I am sorry, my question is very naive. 2nd Edit: Let us suppose that V is a smooth complex projective variety, and Y\subset V is a smooth divisor and has an ample conormal line bundle. We would ... 0answers 157 views ### geometric irregularities in pde's The following question is intended for a person more acquainted with the works of Yves Laurent. see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French) http://link.springer.com/... 0answers 87 views ### Lagrangean equations for the generating function of quadrangulations Let M(z) be the generating function of edge-rooted connected quadrangulations, with z marking the number of edges. I derived the following Lagrangean equations for M(z):$$M(z) = \psi(L(z)),~\...
We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor. For a ...