A note on resolution of rational and hypersurface singularities

Stepanov, D. A.

doi:10.1090/s0002-9939-08-09289-7

Cited by 17 publications

(27 citation statements)

References 19 publications

(23 reference statements)

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“…The papers of Stepanov [46] [47], concerning the analogous question for singularities, started a lot of renewed activity. Following these, a very instructive proof, which I first learned about from A. Ducros, was given by Thuillier [49].…”

Section: Dual Boundary Complexesmentioning

confidence: 99%

The dual boundary complex of theSL2character variety of a punctured sphere

Simpson¹

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Section: Dual Boundary Complexesmentioning

confidence: 99%

The dual boundary complex of theSL2character variety of a punctured sphere

Simpson¹

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…Thus the full transform of f Proof of Theorem 7.5. The proof is an extension of the method mentioned in [Stp2]. By theorem 7.7, we can connect D and D ′ by blow-ups with smooth centres which have SNC with intermediate divisors which are complements of U .…”

Section: Then There Exists a Weak Factorizationmentioning

confidence: 93%

Weights on cohomology, invariants of singularities, and dual complexes

2013

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Abstract. We study the weight filtration on the cohomology of a proper complex algebraic variety and obtain natural upper bounds on its size, when it is the exceptional divisor of a singularity. We also give bounds for the cohomology of links. The invariants of singularities introduced here gives rather strong information about the topology of rational and related singularities.Given a divisor on a variety, the combinatorics governing the way the components intersect is encoded by the associated dual complex. This is the simplicial complex with p-simplices corresponding to (p + 1)-fold intersections of components of the divisor. Kontsevich and Soibelman [KS, A.4] and Stepanov [Stp1] had independently observed that the homotopy type of the dual complex of a simple normal crossing exceptional divisor associated to a resolution of an isolated singularity is an invariant for the singularity. In fact, [KS], and later Thuillier [T] and Payne [P] have obtained homotopy invariance results for more general dual complexes, such as those arising from boundary divisors. In characteristic zero, all these results are consequences of the weak factorization theorem of W lodarczyk [Wlo], and Abramovich-Karu-Matsuki-W lodarczyk [AKMW]; Thuillier uses rather different methods based on Berkovich's non-Archimedean analytic geometry. As we show here, a slight refinement of factorization (theorems 7.6, 7.7) and of these techniques yields some generalizations this statement. This applies to divisors of resolutions of arbitrary not necessarily isolated singularities, and even in a more general context (discussed in the final section). We also allow dual complexes associated with nondivisorial varieties, such as fibres of resolutions of nonisolated singularites. Here is a slightly imprecise formulation of theorems 7.5 and 7.9. [KS], although we were unaware of this paper at the time these results were completed).To put the remaining results in context, we note that the present paper can be considered as the extended version of our earlier preprint [ABW], where we gave

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“…Our earlier paper contains a lemma on degeneration of a spectral sequence whose proof is incorrect. In this note we explain the mistake and provide a correction to it.In my paper "A note on resolution of rational and hypersurface singularities" [3], in the proof of Lemma 2.4, I claim that on a compact Kähler variety a restriction of a harmonic differential form onto a subvariety is harmonic. However, this is not correct.…”

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“…In my paper "A note on resolution of rational and hypersurface singularities" [3], in the proof of Lemma 2.4, I claim that on a compact Kähler variety a restriction of a harmonic differential form onto a subvariety is harmonic. However, this is not correct.…”

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Erratum to ‘‘A note on resolution of rational and hypersurface singularities”

Stepanov

2010

Proc. Amer. Math. Soc.

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Abstract. Our earlier paper contains a lemma on degeneration of a spectral sequence whose proof is incorrect. In this note we explain the mistake and provide a correction to it.In my paper "A note on resolution of rational and hypersurface singularities" [3], in the proof of Lemma 2.4, I claim that on a compact Kähler variety a restriction of a harmonic differential form onto a subvariety is harmonic. However, this is not correct. Here is a sketch of a counterexample due to D. Arapura.Example. Let X be a non-hyperelliptic curve of genus g > 2. X embeds into its Jacobian J. Let us give J a flat metric and X the induced metric. The harmonic forms on J are precisely the forms with constant coefficients, so they form an algebra. In particular, γ ij = α i ∧ᾱ j give a basis for the harmonic (1, 1)-forms on J, where α i = dz i and z i are the coordinates on C g . The restrictions of α i to X give a basis for the holomorphic 1-forms on X, and these separate points. Therefore we can find at least 2 linearly independent restrictions γ ij | X . However, the space of harmonic (1, 1)-forms on X is 1-dimensional. Thus one of these restrictions is not harmonic.On the other hand, our Lemma 2.4 is itself correct. In their recent preprint D. Arapura, P. Bakhtari, and J. W lodarczyk prove it as a consequence of their more general result ([1], 2.1, 2.2, 2.3). Our original argument can also be corrected, and here we present such a correction. The last paragraph of the proof of Lemma 2.4 from [3], starting with "Our aim is to show that d r = 0, r ≥ 2," should now be read as follows:Our aim is to show that d r = 0, r ≥ 2. The differential d 2 is trivial if the representative a ∈ K p,q can be chosen in such a way that δ(a) is exactly 0 but not only 0 modulo∂K p+1,q−1 . But this is true because there are harmonic differential forms in the classā ∈ H q ∂ (K p, * ) and we can take a to be a harmonic form of pure type (0, q). Notice that since we work on a compact Kähler variety, the form a is not only∂-closed but also ∂ +∂-closed, where ∂ +∂ is the complex de Rham differential. The form δa is defined by means of restrictions onto subvarieties and linear operations. We cannot claim that it is also harmonic, but it remains ∂ +∂-closed and of pure type (0, q). But it is 0 mod (∂K p+1,q−1 ), i.e.,∂-exact. It follows from the ∂∂-lemma [2, Ch. 1, sec. 2] that δa is also ∂-exact and thus is exactly 0.

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A note on resolution of rational and hypersurface singularities

Cited by 17 publications

References 19 publications

The dual boundary complex of theSL2character variety of a punctured sphere

The dual boundary complex of theSL2character variety of a punctured sphere

Weights on cohomology, invariants of singularities, and dual complexes

Erratum to ‘‘A note on resolution of rational and hypersurface singularities”

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