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 1 / 13
My Advisors Rikard Bøgvad Andrei Shindyapin Main advisor Assistant advisor Stockholm University Universidade Eduardo Mondlane 2 / 13
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. 3 / 13
Introduction Consider a union of lines S = ∪ n i = 1 L i , i = 1 , ..., n , through the origin in C 2 . 4 / 13
Introduction Consider a union of lines S = ∪ n i = 1 L i , i = 1 , ..., n , through the origin in C 2 . The fundamental group of C 2 − S is π 1 ( C 2 − S ) = � Γ 1 , . . . , Γ n � / R , where R is the group generated by the (cyclic) relations Γ 1 Γ 2 . . . Γ n = Γ 2 . . . Γ n Γ 1 = Γ n Γ 1 . . . Γ n − 1 . 4 / 13
Introduction Consider a union of lines S = ∪ n i = 1 L i , i = 1 , ..., n , through the origin in C 2 . The fundamental group of C 2 − S is π 1 ( C 2 − S ) = � Γ 1 , . . . , Γ n � / R , where R is the group generated by the (cyclic) relations Γ 1 Γ 2 . . . Γ n = Γ 2 . . . Γ n Γ 1 = Γ n Γ 1 . . . Γ n − 1 . Corollary Locally constant sheaves L of rank 1 on C 2 − S are classified up to isomorphism by the element a = ( a 1 , ..., a n ) ∈ C n , such that for the monodromy representation Γ i e = a i e . (1) 4 / 13
Introduction Consider a union of lines S = ∪ n i = 1 L i , i = 1 , ..., n , through the origin in C 2 . The fundamental group of C 2 − S is π 1 ( C 2 − S ) = � Γ 1 , . . . , Γ n � / R , where R is the group generated by the (cyclic) relations Γ 1 Γ 2 . . . Γ n = Γ 2 . . . Γ n Γ 1 = Γ n Γ 1 . . . Γ n − 1 . Corollary Locally constant sheaves L of rank 1 on C 2 − S are classified up to isomorphism by the element a = ( a 1 , ..., a n ) ∈ C n , such that for the monodromy representation Γ i e = a i e . (1) Denote the locally constant sheaf correspondent to a ∈ C n by L a . 4 / 13
Let j be the inclusion j : C 2 − S ֒ → C 2 , j ∗ the direct image functor j ∗ : Sh ( C 2 − S ) → Sh ( C 2 ) and R i j ∗ the i-th right derived functor of j ∗ R i j ∗ : D + ( C 2 − S ) → Sh ( C 2 ) 5 / 13
Let j be the inclusion j : C 2 − S ֒ → C 2 , j ∗ the direct image functor j ∗ : Sh ( C 2 − S ) → Sh ( C 2 ) and R i j ∗ the i-th right derived functor of j ∗ R i j ∗ : D + ( C 2 − S ) → Sh ( C 2 ) We want to study the irreducibility and the number of factors in the composition series of the perverse sheaf Rj ∗ L a . 5 / 13
Let j be the inclusion j : C 2 − S ֒ → C 2 , j ∗ the direct image functor j ∗ : Sh ( C 2 − S ) → Sh ( C 2 ) and R i j ∗ the i-th right derived functor of j ∗ R i j ∗ : D + ( C 2 − S ) → Sh ( C 2 ) We want to study the irreducibility and the number of factors in the composition series of the perverse sheaf Rj ∗ L a . Definition A composition series of an object A in an abelian category is a sequence of subobjects A = X 0 � X 1 � · · · � X n = 0 such that each quotient object Xi \ X i + 1 is irreducible (for 0 ≤ i < n ). 5 / 13
Irreducibility Theorem The perverse sheaf Rj ∗ L a , where j : C 2 − S → C 2 , is irreducible if, and only if, both of the following conditions are satisfied: a i � = 1 , for all i = 1 , . . . , n; Π n i = 1 a i � = 1 . 6 / 13
Decomposition Factors In the case the irreducibility conditions for Rj ∗ L a are not satisfied, we can still ask for the number of decomposition factors of this object. Let c ( P • ) represent the number of decomposition factors of the perverse sheaf P • . 7 / 13
Decomposition Factors In the case the irreducibility conditions for Rj ∗ L a are not satisfied, we can still ask for the number of decomposition factors of this object. Let c ( P • ) represent the number of decomposition factors of the perverse sheaf P • . Theorem Assume that a 1 , . . . , a k = 1 and a k + 1 , . . . , a n � = 1 . If Π n i = 1 a i = 1 , then c ( Rj ∗ L a ) = n + k − 1 . If Π n i = 1 a i � = 1 , then c ( Rj ∗ L a ) = k + 1 . 7 / 13
Length of Direct Images Let A be an arrangement of m + 1 hyperplanes H 0 , . . . , H m in C n and L = L ( A ) the set of nonempty intersections of the hyperplanes. We define a partial order on L by X ≤ Y ⇐ ⇒ Y ⊆ X . 8 / 13
Length of Direct Images Let A be an arrangement of m + 1 hyperplanes H 0 , . . . , H m in C n 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 � µ ( X )( − t ) codim X Π( A , t ) = X ∈ L where µ represents the Möbius function and t is an indeterminate. 8 / 13
Theorem Let H 0 , . . . , H m be the hyperplanes of the arrangement A in W = C n , j the inclusion j : C n − ∪ n i = 0 H i → C n and Rj ∗ the direct image Rj ∗ : M ( C n − ∪ n i = 0 H i ) → M ( C n ) . Then � c ( Rj ∗ C [ n + 1 ]) = Π( A , 1 ) = | µ ( X ) | X ∈ L ( A ) 9 / 13
Theorem Let H 0 , . . . , H m be the hyperplanes of the arrangement A in W = C n , j the inclusion j : C n − ∪ n i = 0 H i → C n and Rj ∗ the direct image Rj ∗ : M ( C n − ∪ n i = 0 H i ) → M ( C n ) . Then � c ( Rj ∗ C [ n + 1 ]) = Π( A , 1 ) = | µ ( X ) | X ∈ L ( A ) Theorem With the same conditions of the previous theorem, we have that: � dim ( H l ( Rj ∗ C [ n ])) c ( Rj ∗ C [ n + 1 ]) = Π( A , 1 ) = l ≥ 0 9 / 13
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 2 ( cardJ ( X ) − r ( X )) � � � b A ( a 1 , . . . , a p ) = a i + r ( X ) + j j = 0 X ∈ L ′ ( A ) i ∈ J ( X ) where L ′ ( A ) = { X ∈ L ( A ) : A X is irreducible } . 10 / 13
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 2 ( cardJ ( X ) − r ( X )) � � � b A ( a 1 , . . . , a p ) = a i + r ( X ) + j j = 0 X ∈ L ′ ( A ) i ∈ J ( X ) where L ′ ( A ) = { X ∈ L ( A ) : A X is irreducible } . Theorem ∈ Z , for all X ∈ L ′ ( A ) . Rj ∗ L a is irreducible if � i ∈ J ( X ) a i / 10 / 13
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 , H i ( F | S ) is a local system. 11 / 13
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. 12 / 13
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. 12 / 13
Tack så mycket! Obrigada! Thank you! 13 / 13
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