Decomposition theorem of Beilinson, Bernstein and Deligne

In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber or BBDG decomposition theorem is a set of results concerning the cohomology of algebraic varieties. 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

→

Y

{\displaystyle f:X\to Y}

of relative dimension d between two projective varieties

−

∪

η

i

:

R

d

−

i

f

∗

(

Q

)

→

≅

R

d

+

i

f

∗

(

Q

)

.

{\displaystyle -\cup \eta ^{i}:R^{d-i}f_{*}(\mathbb {Q} ){\stackrel {\cong }{\to }}R^{d+i}f_{*}(\mathbb {Q} ).}

Here

η

{\displaystyle \eta }

is the fundamental class of a hyperplane section,

f

∗

{\displaystyle f_{*}}

is the direct image (pushforward) and

R

n

f

∗

{\displaystyle R^{n}f_{*}}

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

f

−

1

(

U

)

{\displaystyle f^{-1}(U)}

, for

U

⊂

Y

{\displaystyle U\subset Y}

.

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

−

∪

η

i

:

H

d

−

i

(

X

,

Q

)

→

≅

H

d

+

i

(

X

,

Q

)

.

{\displaystyle -\cup \eta ^{i}:H^{d-i}(X,\mathbb {Q} ){\stackrel {\cong }{\to }}H^{d+i}(X,\mathbb {Q} ).}

This hard Lefschetz isomorphism induces canonical isomorphisms

R

f

∗

(

Q

)

→

≅

⨁

i

=

−

d

d

R

d

+

i

f

∗

(

Q

)

[

−

d

−

i

]

.

{\displaystyle Rf_{*}(\mathbb {Q} ){\stackrel {\cong }{\to }}\bigoplus _{i=-d}^{d}R^{d+i}f_{*}(\mathbb {Q} )[-d-i].}

Moreover, the sheaves

R

d

+

i

f

∗

Q

{\displaystyle R^{d+i}f_{*}\mathbb {Q} }

appearing in this decomposition are local systems, 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

→

Y

{\displaystyle 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.