Deformation of Singularities and the Homology of Intersection Spaces

Banagl, Markus; Maxim, Laurenţiu

doi:10.1142/s1793525312500185

Cited by 33 publications

(33 citation statements)

References 19 publications

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“…A natural question is: does a duality-satisfying homology-type theory exist that behaves well under smooth deformations? This question has been broached and, for hypersurfaces with isolated singularities, answered partially in the affirmative in [6] and [7], in which an alternate theory is utilized: the intersection space homology theory introduced in [2]. At this point, a universal generalization has not been discovered, because intersection space theory has not been defined for the vast majority of singular spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Special cases where the stratification of X is more elaborate have been studied, for example in [4], but no all-encompassing picture has been painted. Despite the limited collection of spaces for which it is defined, intersection space theory has had applications in multiple fields: fiber bundle theory [3], algebraic geometry and smooth deformations [6] and [7], perverse sheaves [8], and theoretical physics [2,Chapter 3].…”

Section: Introductionmentioning

confidence: 99%

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Algebraic intersection spaces

Geske

2019

J. Topol. Anal.

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We define a variant of intersection space theory that applies to many compact complex and real analytic spaces X, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze "local duality obstructions", which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in X of a tubular neighborhood of the singular set.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algebraic intersection spaces

Geske

2019

J. Topol. Anal.

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“…Now the calculations of [Ban10, Section 3.9], together with our de Rham theorem, show that rk HI 2 (V ) = 1, rk HI 3 (V ) = 204, rk HI 4 (V ) = 1, in perfect agreement with the Betti numbers of V s , s = 0. Indeed, jointly with L. Maxim, we have established the following Stability Theorem, see [BM11]: Let V be a complex n-dimensional projective hypersurface with one isolated singularity and let V s be a nearby smooth deformation of V . Then for all i < 2n, and i = n, HI i s (V ; Q) ≅H i (V s ; Q).…”

Section: Introductionmentioning

confidence: 99%

“…At least if H n−1 (L; Z) is torsionfree, where L is the link of the singularity, the isomorphism is induced by a continuous map IV → V s and is thus a ring isomorphism. We use this in [BM11] to endow HI • s (V ; Q) with a mixed Hodge structure so that the canonical map IV → V induces homomorphisms of mixed Hodge structures in cohomology. Even if the monodromy is not trivial, IV → V s induces a monomorphism on homology.…”

Section: Introductionmentioning

confidence: 99%

Foliated stratified spaces and a De Rham complex describing intersection space cohomology

Banagl

2016

J. Differential Geom.

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Abstract. The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincaré duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with stratification depth 1 and flat link bundles, the present paper provides a de Rham-theoretic description of the theory HI by a complex of global smooth differential forms on the top stratum. We prove that the wedge product of forms introduces a perversity-internal cup product on HI, for every perversity. Flat link bundles arise for example in foliated stratified spaces and in reductive Borel-Serre compactifications of locally symmetric spaces. A precise topological definition of the notion of a stratified foliation is given.

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“…This point of view was exploited in [3], where the first and third authors considered the case of a hypersurface X ⊂ CP n+1 with only isolated singularities. For simplicity, let us assume that X has only one isolated singular point x, with Milnor fiber F x and local monodromy operator T x : H n (F x ) → H n (F x ).…”

Section: Introductionmentioning

confidence: 99%

Intersection spaces, perverse sheaves and type IIB string theory

Banagl

Budur

Maxim

2014

Advances in Theoretical and Mathematical Physics

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The method of intersection spaces associates rational Poincar\'e complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincar\'e duality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities

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Deformation of Singularities and the Homology of Intersection Spaces

Cited by 33 publications

References 19 publications

Algebraic intersection spaces

Algebraic intersection spaces

Foliated stratified spaces and a De Rham complex describing intersection space cohomology

Intersection spaces, perverse sheaves and type IIB string theory

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