Decompositon Factors of Perverse Sheaves Iara Gonalves Department - - PowerPoint PPT Presentation

Decompositon Factors of Perverse Sheaves Iara Gonçalves

Department of Mathematics and Informatics, Universidade Eduardo Mondlane Department of Mathematics, Stockholm University

First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm 7–8 March 2017

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My Advisors

Rikard Bøgvad Andrei Shindyapin

Main advisor Assistant advisor Stockholm University Universidade Eduardo Mondlane

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Research Topic

The origin of the theory of perverse sheaves is Goresky and MaPherson’s theory of intersection homology. The aim was to find a topological invariant similar to cohomology that would carry over some of the nice properties of homology or cohomology of smooth manifolds also to singular spaces (especially Poincaré Duality). The intersection homology turns out to be the cohomology of a certain complex of sheaves, constructed by Deligne. This complex is the main example of a perverse sheaf.

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Introduction

Consider a union of lines S = ∪n

i=1Li, i = 1, ..., n, through the

• rigin in C2.

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Introduction

Consider a union of lines S = ∪n

i=1Li, i = 1, ..., n, through the

• rigin in C2. The fundamental group of C2 − S is

π1(C2 − S) = Γ1, . . . , Γn/R, where R is the group generated by the (cyclic) relations Γ1Γ2 . . . Γn = Γ2 . . . ΓnΓ1 = ΓnΓ1 . . . Γn−1.

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Introduction

Consider a union of lines S = ∪n

i=1Li, i = 1, ..., n, through the

• rigin in C2. The fundamental group of C2 − S is

π1(C2 − S) = Γ1, . . . , Γn/R, where R is the group generated by the (cyclic) relations Γ1Γ2 . . . Γn = Γ2 . . . ΓnΓ1 = ΓnΓ1 . . . Γn−1. Corollary Locally constant sheaves L of rank 1 on C2 − S are classified up to isomorphism by the element a = (a1, ..., an) ∈ Cn, such that for the monodromy representation Γie = aie. (1)

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Introduction

Consider a union of lines S = ∪n

i=1Li, i = 1, ..., n, through the

• rigin in C2. The fundamental group of C2 − S is

π1(C2 − S) = Γ1, . . . , Γn/R, where R is the group generated by the (cyclic) relations Γ1Γ2 . . . Γn = Γ2 . . . ΓnΓ1 = ΓnΓ1 . . . Γn−1. Corollary Locally constant sheaves L of rank 1 on C2 − S are classified up to isomorphism by the element a = (a1, ..., an) ∈ Cn, such that for the monodromy representation Γie = aie. (1) Denote the locally constant sheaf correspondent to a ∈ Cn by La.

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Let j be the inclusion j : C2 − S ֒ → C2, j∗ the direct image functor j∗ : Sh(C2 − S) → Sh(C2) and Rij∗ the i-th right derived functor of j∗ Rij∗ : D+(C2 − S) → Sh(C2)

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Let j be the inclusion j : C2 − S ֒ → C2, j∗ the direct image functor j∗ : Sh(C2 − S) → Sh(C2) and Rij∗ the i-th right derived functor of j∗ Rij∗ : D+(C2 − S) → Sh(C2) We want to study the irreducibility and the number of factors in the composition series of the perverse sheaf Rj∗La.

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Let j be the inclusion j : C2 − S ֒ → C2, j∗ the direct image functor j∗ : Sh(C2 − S) → Sh(C2) and Rij∗ the i-th right derived functor of j∗ Rij∗ : D+(C2 − S) → Sh(C2) We want to study the irreducibility and the number of factors in the composition series of the perverse sheaf Rj∗La. Definition A composition series of an object A in an abelian category is a sequence of subobjects A = X0 X1 · · · Xn = 0 such that each quotient object Xi \ Xi+1 is irreducible (for 0 ≤ i < n).

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Irreducibility

Theorem The perverse sheaf Rj∗La, where j : C2 − S → C2, is irreducible if, and only if, both of the following conditions are satisfied: ai = 1, for all i = 1, . . . , n; Πn

i=1ai = 1.

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Decomposition Factors

In the case the irreducibility conditions for Rj∗La are not satisfied, we can still ask for the number of decomposition factors of this

• bject. Let c(P•) represent the number of decomposition factors of

the perverse sheaf P•.

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Decomposition Factors

In the case the irreducibility conditions for Rj∗La are not satisfied, we can still ask for the number of decomposition factors of this

• bject. Let c(P•) represent the number of decomposition factors of

the perverse sheaf P•. Theorem Assume that a1, . . . , ak = 1 and ak+1, . . . , an = 1. If Πn

i=1ai = 1, then c(Rj∗La) = n + k − 1.

If Πn

i=1ai = 1, then c(Rj∗La) = k + 1.

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Length of Direct Images

Let A be an arrangement of m + 1 hyperplanes H0, . . . , Hm in Cn and L = L(A) the set of nonempty intersections of the hyperplanes. We define a partial order on L by X ≤ Y ⇐ ⇒ Y ⊆ X.

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Length of Direct Images

Let A be an arrangement of m + 1 hyperplanes H0, . . . , Hm in Cn and L = L(A) the set of nonempty intersections of the hyperplanes. We define a partial order on L by X ≤ Y ⇐ ⇒ Y ⊆ X. Definition The Poincaré polynomial of A is defined by Π(A, t) =

• X∈L

µ(X)(−t)codim X where µ represents the Möbius function and t is an indeterminate.

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Theorem Let H0, . . . , Hm be the hyperplanes of the arrangement A in W = Cn, j the inclusion j : Cn − ∪n

i=0Hi → Cn and Rj∗ the direct

image Rj∗ : M(Cn − ∪n

i=0Hi) → M(Cn). Then

c(Rj∗C[n + 1]) = Π(A, 1) =

• X∈L(A)

|µ(X)|

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Theorem Let H0, . . . , Hm be the hyperplanes of the arrangement A in W = Cn, j the inclusion j : Cn − ∪n

i=0Hi → Cn and Rj∗ the direct

image Rj∗ : M(Cn − ∪n

i=0Hi) → M(Cn). Then

c(Rj∗C[n + 1]) = Π(A, 1) =

• X∈L(A)

|µ(X)| Theorem With the same conditions of the previous theorem, we have that: c(Rj∗C[n + 1]) = Π(A, 1) =

• l≥0

dim(Hl(Rj∗C[n]))

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Bernstein Polynomial

Theorem Let V be a vector space and A a linear and free hyperplane

• arrangement. Then, the Bernstein ideal of A is principal and

generated by the polynomial bA(a1, . . . , ap) =

• X∈L′(A)

2(cardJ(X)−r(X))

• j=0

 

i∈J(X)

ai + r(X) + j   where L′(A) = {X ∈ L(A) : AX is irreducible}.

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Bernstein Polynomial

Theorem Let V be a vector space and A a linear and free hyperplane

• arrangement. Then, the Bernstein ideal of A is principal and

generated by the polynomial bA(a1, . . . , ap) =

• X∈L′(A)

2(cardJ(X)−r(X))

• j=0

 

i∈J(X)

ai + r(X) + j   where L′(A) = {X ∈ L(A) : AX is irreducible}. Theorem Rj∗La is irreducible if

i∈J(X) ai /

∈ Z, for all X ∈ L′(A).

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Impact and Applications of My Research

Which are the good properties of perverse sheaves that make them so attractive? The perverse sheaves on a complex manifold X form an abelian category, i.e., the notions of injections, surjections, kernels, cokernels, exact sequences all make sense and have the usual properties. The category of perverse sheaves is Artinian, i. e., every perverse sheaf has a finite composition series whose successive quotients are irreducible perverse sheaves. Perverse sheaves are constructible (or have constructible cohomology sheaves) meaning that there exists a stratification such that for each stratum S, Hi(F|S) is a local system.

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Impact and Applications of My Research

Through my research we reach a better understanding of the tools that are fundamental for the topological study of singular spaces and we develop the theory of perverse sheaves in itself. We show how to handle this highly abstract objects in specific cases and how to obtain concrete results.

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Impact and Applications of My Research

Through my research we reach a better understanding of the tools that are fundamental for the topological study of singular spaces and we develop the theory of perverse sheaves in itself. We show how to handle this highly abstract objects in specific cases and how to obtain concrete results. We extend it to others areas, like combinatorics and algebraic geometry, providing new ways of computing results but also revealing not immediately obvious properties of objects in the different areas involved.

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Tack så mycket! Obrigada! Thank you!

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