scholarly journals Perverse Schobers and Wall Crossing

2017 ◽  
Vol 2019 (18) ◽  
pp. 5777-5810 ◽  
Author(s):  
W Donovan
Keyword(s):  
Invariant Theory ◽  
Local System ◽  
Vector Spaces ◽  
Geometric Invariant ◽  
Perverse Sheaf ◽  
Derived Equivalences ◽  
Grothendieck Groups ◽  
Wall Crossing ◽  
Git Quotient ◽  
Cohomology Complex

Abstract For a balanced wall crossing in geometric invariant theory (GIT), there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterize when this is an intersection cohomology complex of a local system on the punctured disk.

scholarly journals Gromov–Witten Theory of Toric Birational Transformations

2019 ◽  
Vol 2020 (20) ◽  
pp. 7037-7072
Author(s):  
Pedro Acosta ◽  
Mark Shoemaker
Keyword(s):  
Asymptotic Expansion ◽  
Linear Transformation ◽  
Invariant Theory ◽  
Geometric Invariant Theory ◽  
Complete Intersections ◽  
Geometric Invariant ◽  
Genus Zero ◽  
Birational Transformations ◽  
Witten Theory ◽  
Wall Crossing

Abstract We investigate the effect of a general toric wall crossing on genus zero Gromov–Witten theory. Given two complete toric orbifolds $X_{+}$ and $X_{-}$ related by wall crossing under variation of geometric invariant theory quotients, we prove that their respective $I$-functions are related by linear transformation and asymptotic expansion. We use this comparison to deduce a similar result for birational complete intersections in $X_{+}$ and $X_{-}$. This extends the work of the previous authors in [2] to the case of complete intersections in toric varieties and generalizes some of the results of Coates–Iritani–Jiang [15] on the crepant transformation conjecture to the setting of non-zero discrepancy.

scholarly journals Graded rings of rank 2 Sarkisov links

2010 ◽  
Vol 197 ◽  
pp. 1-44 ◽  
Author(s):  
Gavin Brown ◽  
Francesco Zucconi
Keyword(s):  
Invariant Theory ◽  
Geometric Invariant Theory ◽  
Explicit Methods ◽  
Graded Rings ◽  
Geometric Invariant ◽  
Git Quotient ◽  
Rank 2

We compute a class of Sarkisov links from Fano 3-folds embedded in weighted Grassmannians using explicit methods for describing graded rings associated to a variation of geometric invariant theory (GIT) quotient.

scholarly journals Graded rings of rank 2 Sarkisov links

2010 ◽  
Vol 197 ◽  
pp. 1-44 ◽  
Author(s):  
Gavin Brown ◽  
Francesco Zucconi
Keyword(s):  
Invariant Theory ◽  
Geometric Invariant Theory ◽  
Explicit Methods ◽  
Graded Rings ◽  
Geometric Invariant ◽  
Git Quotient ◽  
Rank 2

We compute a class of Sarkisov links from Fano 3-folds embedded in weighted Grassmannians using explicit methods for describing graded rings associated to a variation of geometric invariant theory (GIT) quotient.

scholarly journals A short survey on observability

2019 ◽  
Vol 53 (supl) ◽  
pp. 143-183
Author(s):  
Walter Ferrer Santos
Keyword(s):  
Invariant Theory ◽  
Intermediate Step ◽  
Geometric Invariant ◽  
Related Concept ◽  
Algebraic Subgroup ◽  
Tensor Categories ◽  
Current Survey ◽  
The Subject ◽  
The Rich ◽  
New Applications

The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in geometric invariant theory. The concept of observable subgroup was introduced in the early 1960s with the purpose of studying extensions of representations from an afine algebraic subgroup to the whole group. The extent of its importance in representation and invariant theory in particular for Hilbert's 14th problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of strong observability was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the concept of observable action of an afine algebraic group on an afine variety, launching a series of new applications and opening a surge of very interesting activity. In another direction around 2006, the related concept of observable adjunction was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned.

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