In mathematics, especially

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

. It was originally conjectured by Gelfand and MacPherson.

Statement

Decomposition for smooth proper maps

The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map

f: X \to Y

of relative dimension ''d'' between two projective varieties :

- \cup \eta^i : R^f_* (\mathbb Q) \stackrel \cong \to R^ f_*(\mathbb Q).

Here

\eta

is the fundamental class of a

hyperplane section In mathematics, a hyperplane section of a subset ''X'' of projective space P''n'' is the intersection of ''X'' with some hyperplane ''H''. In other words, we look at the subset ''X'H'' of those elements ''x'' of ''X'' that satisfy the single line ...

f_*

is the direct image (pushforward) and

R^n f_*

is the ''n''-th

derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...

of the direct image. This derived functor measures the ''n''-th cohomologies of

f^(U)

, for

U \subset Y

. In fact, the particular case when ''Y'' is a point, amounts to the isomorphism :

- \cup \eta^i : H^ (X, \mathbb Q) \stackrel \cong \to H^ (X, \mathbb Q).

This hard Lefschetz isomorphism induces canonical isomorphisms :

Rf_* (\mathbb Q) \stackrel \cong \to \bigoplus_^ R^ f_*(\mathbb Q) d-i

Moreover, the sheaves

R^ f_* \mathbb Q

appearing in this decomposition are

local system In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between homology theory, cohomology with coefficients in a fixed abelian group ''A'', and general ...

s, i.e., locally free sheaves of Q-vector spaces, which are moreover semisimple, i.e., a direct sum of local systems without nontrivial local subsystems.

Decomposition for proper maps

The decomposition theorem generalizes this fact to the case of a proper, but not necessarily smooth map

f: X \to Y

between varieties. In a nutshell, the results above remain true when the notion of local systems is replaced by

perverse sheaves The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular. The concept w ...

. The hard Lefschetz theorem above takes the following form:. NB: To be precise, the reference is for the decomposition. there is an isomorphism in the

derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...

of sheaves on ''Y'': :

^p H^ (Rf_* \mathbb Q) \cong  ^p H^ (Rf_* \mathbb Q),

where

Rf_*

is the total derived functor of

f_*

and

^p H^i

is the ''i''-th truncation with respect to the ''perverse''

t-structure In the branch of mathematics called homological algebra, a ''t''-structure is a way to axiomatize the properties of an abelian subcategory of a derived category. A ''t''-structure on \mathcal consists of two subcategories (\mathcal^, \mathcal^) o ...

. Moreover, there is an isomorphism :

Rf_* IC_X^\bullet \cong \bigoplus_i ^p H^i (Rf_* IC_X^\bullet) i

where the summands are semi-simple perverse-sheaves, meaning they are direct sums of push-forwards of intersection cohomology sheaves. If ''X'' is not smooth, then the above results remain true when

\mathbb Q dim X /math> is replaced by the

intersection cohomology In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them ov ...

complex

IC

Proofs

The decomposition theorem was first proved by Beilinson, Bernstein, and Deligne. Their proof is based on the usage of weights on l-adic sheaves in positive characteristic. A different proof using mixed Hodge modules was given by Saito. A more geometric proof, based on the notion of semismall maps was given by de Cataldo and Migliorini. For semismall maps, the decomposition theorem also applies to Chow motives.

Applications of the theorem

Cohomology of a Rational Lefschetz Pencil

Consider a rational morphism

f:X \rightarrow \mathbb^1

from a smooth quasi-projective variety given by

_1(x):f_2(x) /math>. If we set the vanishing locus of f_1,f_2 as Y then there is an induced morphism \tilde = Bl_Y(X) \to \mathbb^1 . We can compute the cohomology of X from the intersection cohomology of Bl_Y(X) and subtracting off the cohomology from the blowup along Y . This can be done using the perverse spectral sequence
: E_2^ = H^l(\mathbb^1; ^\mathfrak\mathcal^m(IC_^\bullet(\mathbb)) \Rightarrow IH^(\tilde;\mathbb) \cong H^(X;\mathbb)

Local invariant cycle theorem

Let

f : X \to Y

be a

proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k ...

between complex algebraic varieties such that

X

is smooth. Also, let

y_0

be a regular value of

f

that is in an open ball ''B'' centered at

y

. Then the restriction map :

\operatorname^*(f^(y), \mathbb) = \operatorname^*(f^(B), \mathbb) \to \operatorname^*(f^(y_0), \mathbb)^

is surjective, where

\pi_

is the fundamental group of the intersection of

B

with the set of regular values of ''f''.

References

Survey Articles

* * *

Pedagogical References

* {{Citation , first1=Ryoshi , last1=Hotta , first2=Kiyoshi , last2=Takeuchi , first3=Toshiyuki , last3=Tanisaki , title= D-Modules, Perverse Sheaves, and Representation Theory