A note on the dual complex of a resolution of singularities

Stepanov, D. A.

doi:10.1070/rm2006v061n01abeh004309

Cited by 17 publications

(29 citation statements)

References 8 publications

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“…In fact, we will prove somewhat sharper results in theorems 7.5, 7.8 and 7.9 below. Stepanov in [Stp1] showed this result in the particular case, where E is the exceptional divisor of a resolution of an isolated singularity. Actually, his proof works for any boundary divisors which are complements a fixed open subset U = X − E. Thuillier in [T] proves a similar result in any characteristic.…”

Section: Boundary Divisors and The Weak Factorization Theoremmentioning

confidence: 92%

“…We consider a generalization later in theorem 7.9, which allows for general fibres. The theorem above is a generalization of the version of theorem for isolated singularities given in [Stp2,Stp1]. The following theorems 7.5, 7.8 generalize theorem 7.4, and the results of [P] for log-resolutions.…”

Section: Composition Of Blow-ups Of All the Intersection Components Smentioning

confidence: 92%

“…This theorem and related ones in the final section were inspired by the work of Payne [P, §5], Stepanov [Stp1], and Thuillier [T] mentioned above (and in hindsight also by [KS], although we were unaware of this paper at the time these results were completed).…”

Section: Theorem 01 Suppose That X Is a Smooth Complete Variety Lementioning

confidence: 99%

“…As we will explain below, these results are established in a more refined form in the present paper. To explain the motivation, we recall the following from the end of [Stp1]:…”

Section: Theorem 01 Suppose That X Is a Smooth Complete Variety Lementioning

confidence: 99%

“…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.…”

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confidence: 99%

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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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Section: Boundary Divisors and The Weak Factorization Theoremmentioning

confidence: 92%

Section: Composition Of Blow-ups Of All the Intersection Components Smentioning

confidence: 92%

Section: Theorem 01 Suppose That X Is a Smooth Complete Variety Lementioning

confidence: 99%

“…As we will explain below, these results are established in a more refined form in the present paper. To explain the motivation, we recall the following from the end of [Stp1]:…”

Section: Theorem 01 Suppose That X Is a Smooth Complete Variety Lementioning

confidence: 99%

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confidence: 99%

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Weights on cohomology, invariants of singularities, and dual complexes

2013

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From Singularities to Graphs

Popescu-Pampu

2022

When Form Becomes Substance

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Harmonic Maps to Buildings and Singular Perturbation Theory

Katzarkov

Noll

Pandit

et al. 2015

Commun. Math. Phys.

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The notion of a universal building associated with a point in the Hitchin base is introduced. This is a building equipped with a harmonic map from a Riemann surface that is initial among harmonic maps which induce the given cameral cover of the Riemann surface. In the rank one case, the universal building is the leaf space of the quadratic differential defining the point in the Hitchin base.The main conjectures of this paper are: (1) the universal building always exists; (2) the harmonic map to the universal building controls the asymptotics of the Riemann-Hilbert correspondence and the non-abelian Hodge correspondence; (3) the singularities of the universal building give rise to Spectral Networks; and (4) the universal building encodes the data of a 3d Calabi-Yau category whose space of stability conditions has a connected component that contains the Hitchin base.The main theorem establishes the existence of the universal building, conjecture (3), as well as the Riemann-Hilbert part of conjecture (2), in the case of the rank two example introduced in the seminal work of Berk-Nevins-Roberts on higher order Stokes phenomena. It is also shown that the asymptotics of the Riemann-Hilbert correspondence is always controlled by a harmonic map to a certain building, which is constructed as the asymptotic cone of a symmetric space.

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A note on the dual complex of a resolution of singularities

Cited by 17 publications

References 8 publications

Weights on cohomology, invariants of singularities, and dual complexes

Weights on cohomology, invariants of singularities, and dual complexes

From Singularities to Graphs

Harmonic Maps to Buildings and Singular Perturbation Theory

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