Perverse schobers and birational geometry

Bondal, Alexey; Kapranov, Mikhail; Schechtman, Vadim

doi:10.1007/s00029-018-0395-1

Cited by 28 publications

(25 citation statements)

References 45 publications

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“…Thus, Conjecture 3.3 predicts the existence of semiorthogonal decomposition (19) with components B such that HH • (B) = C. Such a decomposition is indeed easy to construct, it is enough to take B = O X , see §3.6.…”

Section: 2mentioning

confidence: 99%

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Semiorthogonal decompositions in families

Кузнецов¹

2021

Preprint

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We discuss recent developments in the study of semiorthogonal decompositions of algebraic varieties with an emphasis on their behaviour in families. First, we overview new results concerning homological projective duality. Then we introduce residual categories, discuss their relation to small quantum cohomology, and compute Serre dimensions of residual categories of complete intersections. After that we define simultaneous resolutions of singularities and describe a construction that works in particular for nodal degenerations of even-dimensional varieties. Finally, we introduce the concept of absorption of singularities which works under appropriate assumptions for nodal degenerations of odd-dimensional varieties.

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Section: 2mentioning

confidence: 99%

“…(i) There is a semiorthogonal decomposition (19), where the component B is generated by an Aut(X)-invariant exceptional collection of length N/m. (ii) The residual category R of (19) has a completely orthogonal Aut(X)-invariant decomposition…”

Section: Conjecture 35 ([79 Conjecture 13]mentioning

confidence: 99%

Semiorthogonal decompositions in families

Кузнецов¹

2021

Preprint

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“…schober [29]. The full theory is still under development [4,10,25,30,8,11]. For much of this section we restrict to the following special class of schobers.…”

Section: Spectral Networkmentioning

confidence: 99%

Spectral networks and stability conditions for Fukaya categories with coefficients

Haiden¹,

Katzarkov²,

Simpson³

2021

Preprint

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Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients.They are closely related to the spectral networks of Gaiotto-Moore-Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives.We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an A 2 quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

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“…A much more general instance where a combinatorial description of the category of perverse sheaves is known [KS16] is that of a complex affine space stratified by a complexified real hyperplane arrangement H and this leads to a corresponding definition of perverse schobers [BKS18]. See §3 for more details.…”

Section: Introductionmentioning

confidence: 99%

A class of perverse schobers in Geometric Invariant Theory

Špenko,

Bergh

2019

Preprint

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Perverse schobers are categorifications of perverse sheaves. We construct a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasisymmetric representation X of a reductive group G, extending the local system of triangulated categories exhibited by them. The triangulated categories appearing in our perverse schober are subcategories of the derived category of the quotient stack X/G.

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Perverse schobers and birational geometry

Cited by 28 publications

References 45 publications

Semiorthogonal decompositions in families

Semiorthogonal decompositions in families

Spectral networks and stability conditions for Fukaya categories with coefficients

A class of perverse schobers in Geometric Invariant Theory

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