AN OVERVIEW OF THE DECOMPOSITION THEOREM
DONGKWAN KIM
1. Statement and Proof of the Main Theorem
1.1. The Statement of the Main Theorem. We first state the main theorem.
Theorem 1 (Decomposition theorem). Lef f : X → Y be a proper morphism of algebraic
varieties over C. Then we have
X
p i
f∗ ICX =
H (f∗ ICX )[−i]
i∈Z
p
i
where each H (f∗ ICX ) is a semisimple perverse sheaf.
There are currently three known approaches to prove this theorem.
(1) The original proof by Beilinson, Bernstein, Deligne, and Gabber [BBD]
(2) Using mixed Hodge modules by Saito [S]
(3) Using classical Hodge theory by de Cataldo and Migliorini [dCM2]
Here we briefly sketch the original proof of [BBD]. The main reference of this talk would be
[dCM1], the expository article for this subject written by de Cataldo and Migliorini.
The main idea of [BBD] is that first we prove this theorem over a finite field and use ”spread
out” principle. One of the important property of this method is that we have an action of
the Galois group which acts on varieties and sheaves, which is topologically generated by the
Frobenius morphism. Here we introduce the notion of weights; it is basically the absolute
value of its eigenvalues. If the weights of some complex is in sufficiently good shape, we call
it mixed/pure, and we will see that it indeed gives a rigid structure on such complexes and
morphisms between them.
1.2. Some notations. First we work over a finite field. Let Fq be a finite field, F = Fq be
its algebraic closure, X0 be an algebraic variety over Fq , and X be X0 ×Fq F. Also we let F
be the geometric Frobenius with respect to Fq . We fix a prime ` prime to q, and consider the
bounded constructible derived category Dcb (X0 , Q` ). We usually put subscript 0 to denote
analogous objects over Fq corresponding to some objects over F.
Definition 2 ([BBD, 5.1.5]). For a Q` sheaf F0 on X0 , we call it pointwise/punctually pure
of weight w ∈ Z when for any n ≥ 1 and for any x ∈ X0 (Fqn ), all the eigenvalues and their
complex conjugates of (F n )∗ on Fx have absolute value q nw/2 . We call F0 mixed when it
admits a finite filtration of pointwise pure complexes. The weights of these nonzero successive
b
quotients are called pointwise weights of F0 . We denote by Dm
(X0 , Q` ) the subcategory of
b
Dc (X0 , Q` ) consisting of complexes whose cohomology sheaves are mixed, which we call
mixed complexes.
Date: November 9, 2015.
1
b
(X0 , Q` ) is stable under f∗ , f! , f ∗ , f ! , ⊗, Hom, and Verdier duality.
One can prove that Dm
Also it is stable under the usual truncation functor. Since the perverse t-structure can be
defined using ”glueing” procedure, we also see that it is stable under perverse truncations.
b
(X0 , Q` ) is also naturally a triangulated subcategory in both senses.
[BBD, 5.1.6] Thus Dm
Also, the category of mixed perverse sheaves, often denoted by Pm (X0 , Q` ), is abelian. One
can also show that Pm (X0 , Q` ) is closed under taking subquotients. [BBD, 5.1.7]
b
Definition 3 ([BBD, 5.1.8]). For K ∈ Dm
(X0 , Q` ), we say it is of weight ≤ w if the pointwise
i
weights of H K is ≤ w + i for all i ∈ Z. Also we say it is of weight ≥ w if its Verdier dual
DK is of weight −w. If K is of weight ≤ w and ≥ w, then we call it pure of weight w.
Example 4. if X0 is smooth and irreducible of dimension d and L[d] is a lisse sheaf on X0
concentrated on degree −d, then it is pure of weight w if H−d L[d] = L is of pointwise weight
≤ w − d and H−d D(L[d]) = D(H−d L[d]) = L∨ (d) is of pointwise weight ≤ −w − d, hence L∨
is of pointwise weight ≤ −w + d. (Note the Tate twist.) But it means that L is of pointwise
weight exactly w − d.
1.3. Weight and Six Functors.
Proposition 5 ([BBD, 5.1.14]). Let F0 be of weight ≤ w, F00 be of weight ≤ w0 , and G000 be
of weight ≥ w00 . Then
(1)
(2)
(3)
(4)
(5)
f! F0 , f ∗ F0 is of weight ≤ w.
f ! G0 , f∗ G0 is of weight ≥ w00 .
F0 ⊗ F00 is of weight ≤ w + w0 .
Hom(F0 , G0 ) is of weight ≥ −w + w00 .
DF0 is of weight ≥ −w and DG0 is of weight ≤ −w00 .
Proof. f ∗ one is obvious. Since it is clear that exterior tensor product sums up weights with
the same direction, (3) is also clear. (5) is by definition. Then f ! one also follows. Since
Hom(F0 , G0 ) ' DF0 ⊗! G0 , (4) follows. Now f∗ , f! ones follow from adjunction. Or we can
use [D, 3.3.1, 6.2.3] by a dévissage argument to the cohomology with compact support and
use of Leray spectral sequence.
Remark. Note that this implies Riemann hypothesis part of the Weil conjecture. Suppose
X is a smooth projective variety of dimension d over Fq and consider the following zeta
function;
!
∞
X
#X(Fqn ) −ns
ζ(X, s) = exp
q
n
n=1
By Lefschets fixed point formula, we have
#X(Fqn ) =
2d
X
(−1)i tr((F n )∗ , Hci (X, Q` ))
i=0
2
If we denote λi,1 , · · · , λi,ik the eigenvalues of F ∗ on Hci (X, Q` ), then we have
!
∞
2n X
n
n
X
+
·
·
·
+
λ
λ
i,i
i,1
k −ns
(−1)i
q
ζ(X, s) = exp
n
i=0 n=1
=
2n
Y
exp (−1)i log(1 − λi,1 q −s ) · · · log(1 − λi,ik q −s )
i=0
=
P1 (q −s ) · · · P2n−1 (q −s )
P0 (q −s )P2 (q −s ) · · · P2n (q −s )
where
Pi (q −s ) = (1 − λi,1 q −s ) · · · log(1 − λi,ik q −s ).
Since the action of F is of finite order, λi,j are algebraic integers, thus Pi (q −s ) is an integral polynomial. Then by the property of f∗ and f! , λi,j (and its complex conjugates) has
eigenvalues whose absolute values are q i/2 since CX is pure of weight 0.
1.4. Semisimplicity.
b
(X0 , Q` ) and suppose K0 is of weight ≤ w
Proposition 6 ([BBD, 5.1.15]). Let K0 , L0 ∈ Dm
and L0 is of weight > w. Then for a : X0 → Spec Fq , we have
(1) a∗ Hom(K0 , L0 ) is of weight > 0.
(2) Homi (K0 , L0 ) = 0 for i > 0.
This time L0 is of weight ≥ w, then
(3) Homi (K, L)F = 0 for i > 0. In particular, the morphism Homi (K0 , L0 ) → Homi (K, L)
is trivial for i > 0.
Proof. The first assertion comes from the fact that Hom(K0 , L0 ) is of weight > 0 (resp. ≥ 0).
Also we note that there exists an exact sequence
0 → Homi−1 (K, L)F → Homi (K0 , L0 ) → Homi (K, L)F → 0
which comes from the spectral sequence
E2p,q = H p (Spec Fq , Hq M0 ) = H p (Gal(F/Fq ), Hq M ) ⇒ H p+q (Spec Fq , M0 ).
Now (2) follows from this exact sequence, since for i > 0 there is no invariance on Homi (K, L)
and no coinvariance on Homi−1 (K, L). (3) also directly follows from this sequence.
We cannot emphasize enough this proposition; if we put i = 1, then the vanishing of
Hom1 (K, L) means that there is no nontrivial extension. This combines with the concept of
purity to give a condition of semi-simplicity on the decomposition theorem as follows.
Theorem 7 ([BBD, 5.3.8]). If F0 is a pure perverse sheaf on X0 , then F on X is semisimple.
Proof. Let F 0 ⊂ F be the maximal semisimple subsheaf. Then it is F -stable, thus it corresponds to some F00 ⊂ F0 . We have an exact sequence
0 → F00 → F → F0 /F00 → 0.
But F00 and F0 /F00 have the same weight, from which we know that
Hom1 (F00 , F0 /F00 ) → Hom1 (F 0 , F/F 0 )
is a zero map. Thus F/F 0 = 0 by maximality and we obtain the result.
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Note that this is false on X0 itself since F can have ”nontrivial Jordan blocks.” In a similar
spirit, one can prove the following theorem of which we will omit the proof.
b
(X0 , Q` ) is pure, then F is the direct sum of simple
Theorem 8 ([BBD, 5.4.6]). If F0 ∈ Dm
perverse sheaves of (possibly) shifted degree.
Now it remains to show that IC sheaves are pure.
1.5. Purity of IC Complexes: Gabber’s Theorem.
Proposition 9 ([BBD, 5.3.1]). For a mixed perverse sheaf of weight ≤ w (resp. ≥ w,) its
subquotients are also mixed of weight ≤ w (resp. ≥ w.)
Theorem 10 ([BBD, 5.3.2], Gabber purity). Let j : U0 ,→ X0 be an (affine) open embedding.
Then if F0 is a mixed perverse sheaf of weight ≤ w (resp. ≥ w) then j!∗ F0 is also mixed of
weight ≤ w (resp. ≥ w.) Therefore, especially, if F0 is pure, then j!∗ F0 is pure of the same
weight.
If j is affine then j!∗ F0 is a quotient of j! F0 which is also perverse and mixed, thus the
theorem follows from the proposition above. In general it is trickier.
Corollary 11 ([BBD, 5.3.4]). Mixed simple perverse sheaves are pure.
Every simple perverse sheaf is of the form j!∗ (L0 [d]). If it is mixed, then L0 is also mixed.
But it means it is pointwise pure since it is irreducible. Thus the result follows.
We see now that we proved the decomposition theorem over F; indeed, we have
(1) IC sheaves are pure by Gabber’s theorem.
(2) The pushforward of pure complexes by a proper map is also pure.
(3) A pure complex is, after base change to F, a direct sum of simple perverse sheaves of
shifted degree.
1.6. From F to C: Spread Out. To prove the result over C, we use ”spread out” principle,
i.e. if it is true for infinitely many primes, i.e. on a dense set of Spec Z, then it is also true
over Spec C. This principle is also used to prove the algebraic proof of the degeneration
of Hodge-to-de Rham spectral sequence by Deligne and Illusie. Here we simply state some
notations and the decomposition theorem over C without proof.
Definition 12 ([BBD, 6.2.4]). Let X be an algebraic variety over C. For a simple perverse
C-space over X(C), we say that it is of geometric origin if it belongs to the smallest set
which satisfies the following.
(1) It contains Cpt .
(2) It is stable under taking a simple constituent of p Hi of f∗ , f! , f ∗ , f ! , ⊗, Hom.
Definition 13 ([BBD, 6.2.4]). For F ∈ Dcb (X(C), C), we say it is semisimple of geometric
origin if it is a direct sum of simple perverse sheaves of geometric origin with possible degree
shift.
Theorem 14 ([BBD, 6.2.5]). If f : X → Y is proper, then f∗ sends semisimple complexes
of geometric origin to ones of the same kind.
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2. Semi-small and Small Maps
2.1. Definition of Semi-small Maps. We recall the notion of semi-small maps. Note that
we usually assume that the source of the semi-small map is smooth.
Definition 15. For a proper surjective morphism f : X → Y where X is irreducible and
smooth, we say it is semi-small if the following equivalent conditions hold. (n = dim X)
(1) f∗ CX [d] is perverse.
(2) dim X ×Y X ≤ n (or equivalently = n)
(3) dim{y ∈ Y | dim f −1 (y) ≥ i} ≤ dim X − 2i for all i ≥ 0
In particular, if it further satisfies
(3’) dim{y ∈ Y | dim f −1 (y) ≥ i} < dim X − 2i for all i > 0,
then we say it is small.
Note that it implies that f is generically finite, thus dim X = dim Y .
Definition 16. Suppose f : X → Y is semi-small and Y = tYα is an stratification where
f |f −1 Yα is a (topological) fiber bundle. (Such a stratification always exists.) Then we define
Lα := Hdim Y −dim Yα (f∗ CX |Yα ).
Note that it is nonzero if and only if dim Y − dim Yα = 2 dim f −1 (y) for some/any y ∈ Yα .
If so, we say that Yα is relevant.
Note that we have a decomposition
Lα =
M
⊕mφα
L φα
φα
where each of Lφα corresponds to the monodromy representation φα of π1 (Yα ). Since
(Lα )y = Hydim Y −dim Yα (f∗ CX ) is the top dimensional cohomology of the fiber of y, i.e.
H dim Y −dim Yα (f −1 (y), Cf −1 (y) ), it has a basis which corresponds to the top dimensional irreducible components of f −1 (y). Then the monodromy on Lα comes from the permutation
action of π1 (Yα ) on these components. In other words, we have
M
H dim Y −dim Yα (f −1 (y), Cf −1 (y) ) '
Cmφα ⊗ φα
where mφα is the multiplicity of φα on H dim Y −dim Yα (f −1 (y), Cf −1 (y) ).
2.2. Decomposition Theorem for Semi-small Maps.
Theorem 17. We have
f∗ CX [d] =
M
IC(Yα , Lα ) =
Yα
M
IC(Yα , Lφα )⊕mφα
(Yα ,φα )
where the direct sum is over all relevant strata and mφα is as above. In particular, if f is
small, then
f∗ CX [d] = IC(Y, L).
where L is with respect to the open dense stratum of Y .
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2.3. Some Examples.
Definition 18. For a normal variety X, we say it has symplectic singularities if there exists a
closed nondegenerate 2-form ω on the smooth locus of X such that for some (any) resolution
of singularity X̃ → X the pull-back of ω extends to a symplectic form on the whole of X̃.
In this case we call it a symplectic resolution of X.
Theorem 19 ([K1, 1.2], [K2, 2.11]). A symplectic resolution is semismall.
Corollary 20. The Springer resolution Ñ → N is semismall.
This is because Ñ ' T ∗ B has a natural symplectic structure. Also one can use another
argument; indeed, it is not hard to prove that Ñ ×N Ñ is a union of conormal bundles over
each nilpotent orbit in N , and thus it is enough to show that the number of such orbits are
finite, e.g. [L]. Also this argument gives that every nilpotent orbit is relevant.
Likewise, we have the following property.
Proposition 21. The Grothendieck-Springer resolution g̃ → g is small.
3. Examples
3.1. Blow-ups and Group Actions.
Example 22. Suppose X ⊂ Pn is an irreducible projective variety of dimension d and let
Y ⊂ An+1 be its cone, and π : Ỹ → Y be the blowup at the origin. Then the decomposition
theorem says π∗ CỸ [d + 1] is semisimple, and also since π is an isomorphism away from the
origin, we have
π∗ CỸ [d + 1] = ICY ⊕ F0
where F0 is supported at the origin. If we take the stalk at the origin on both sides, we get
H ∗ (X, CX )[d + 1] = (ICY )0 ⊕ F0
Furthermore, note that π∗ CỸ [d] and ICY are self-dual, thus we have
π∗ CỸ [d + 1] = ICY ⊕ F0∨ .
In other words, F0 is also self-dual. Now we note that (ICY )0 sits on the degree in [−d−1, −1].
Thus we see that
F0 ' H 2d (X, CX )[−d + 1] ⊕ · · · ⊕ H d+1 (X, CX ) ⊕ · · · ⊕ H 2d (X, CX )[d − 1]
and rest of terms in H ∗ (X, CX )[d + 1] is contained in (ICY )0 .
If we know ICY on 0, then we can calculate the intersection cohomology of Y using
contraction principle. For example, if X = P1 × P1 as in Simon’s talk. Then we have
F0 = C[1] ⊕ C[−1] and ICY = CY ⊕ C0 [1].
Thus we have that
dim IH i (Y, C) = 1 if i = 0, 2,
0 otherwise.
If we use another blowup of Y along the plane P1 × A1 , then the fiber at 0 is just P1 and
this resolution is small. Thus we directly see that (ICY )0 = C[2] ⊕ C.
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Example 23. Suppose a finite group G acts freely on X. Then the categorical quotient
π : X → X//G is finite thus small. Thus we have
π∗ ICX = IC(X//G, L)
for some L on (X sm //G) ⊂ (X//G)sm since X//G has only one relevant stratum. Furthermore, if we restrict it to the smooth locus X sm , then the restriction of the LHS on X sm //G
is just a vector bundle with the G-action where each fiber is the regular representation of G.
Thus
M
π∗ IC(X, CX ) =
IC(X//G, Lχ )mχ
χ∈Ĝ
where Lχ corresponds to the irreducible character χ of G.
Example 24. Suppose a finite group G acts on a smooth variety X of dimension d. Then for
π : X → Y = X//G we see that CY → R0 π∗ CX splits which corresponds to G-invariants on
R0 π∗ CX . Since R0 π∗ CX [d] ' π∗ CX [d] and it is self-dual, we see that CY [d] is also self-dual.
Thus we can easily check that CY [d] ' ICY .
Here is a detailed example. Suppose we have an action of Z/3 on A2 by
(x, y) 7→ (ωx, ω −1 y)
where ω is the third root of unity. Then we have ρ : A2 → X = A2 /Z3 and
M
ρ∗ CA2 [2] =
IC(X, Lα )
Lα
where each Lα corresponds to the representation of Z/3 since ρ is finite. (Note that on A2 \ 0
the action of Z/3 is free.) Also direct calculation shows that the blowup π : X̃ → X along
0 has fiber P1 ∨ P1 over 0. Thus we have
π∗ CX̃ [2] ' ICX ⊕ C2 and ICX = CX [2].
Now it is clear that IC(X, Lα )0 = 0 if α is not trivial.
3.2. Springer Correspondence. [CG] Originally the Springer correspondence tells that
we can find every representation of the Weyl group W of a reductive group G by looking
at the action of W on the cohomology of each Springer fiber Bx for some nilpotent element
x ∈ g. Here we will find a geometric analogue of such correspondence.
Let π : Ñ → N be the Springer resolution. Then since this is semismall, we have
M
π∗ CÑ [d] =
IC(O, LφO )mφO
O,φO
as before. Now we take granted that this is also true if we impose G-equivariant conditions.
That is, note that the stratification of N given by G-orbits are trivially G-equivariant and as
CÑ [d] and π are G-equivariant, each LφO should also be G-equivariant. Now for any x ∈ O,
we have a long exact sequence
· · · → π1 (G) → π1 (O) → π0 (Gx ) → π0 (G) → π0 (O) → 1
Since LφO are G-equivariant, if t ∈ π1 (O) is on the image of π1 (G), then the action of t on
each LφO is trivial. Thus we see that every φO factors through π0 (Gx ) = Gx /G0x . Note that
this action is basically the permutation action on the top dimensional cohomology of the
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fiber of O, i.e. the irreducible components of the Springer fiber of x. (Note that Springer
fibers are equidimensional.)
By Schur’s lemma, we take the endomorphism of both sides to get
M
End(π∗ CÑ [d]) =
End CmφO
O,φO
In other wods, End(π∗ CÑ [d]) is semisimple. Now we use the following lemma.
Lemma 25 ([CH, 2.21(2) and 2.23]). We have an isomorphism
End(π∗ CÑ [d]) ' HomBM
2d (Ñ ×N Ñ )
where the RHS is equipped with the usual convolution structure.
Meanwhile, note that we have the following diagram.
 i0
Ñ /
g̃
π0
π

i
N
/
g
Then we have π∗ CÑ [d] = i∗ π∗0 Cg̃ [d] = i∗ IC(g, L) where L is supported on grs equipped with
the natural action of the Weyl group W since g̃rs → grs is a principal W -bundle. Thus
π∗ CÑ [d] also has a natural W -action and this induces a homomorphism
C[W ] → End(π∗ CÑ [d]).
Borho and MacPherson [BM] showed that this is indeed an isomorphism Thus we have
M
C[W ] =
End CmφO
O,φO
which is the usual Springer correspondence.
3.3. Schubert Varieties and Kazhdan-Lusztig Polynomials. [H] We will prove the
following theorem of Kazhdan and Lusztig. Suppose G is a reductive group over Fq . For a
flag variety G/B, we have the Bruhat decomposition G/B = tw∈W X(w). Then we have
Theorem 26 ([KL, 4.2]). Hi ICX(w) = 0 if i is odd. If i is even and B 0 ∈ X(w)(Fqn ), then
i
ni/2
all the eigenvalues of F n on HB
.
0 ICX(w) are q
^ → X(w). For a reduced expression
Here we use the Bott-Samelson resolution π : X(w)
w = s1 · · · sl , we define
^ = {(g1 B, · · · , gn B) | g −1 gi ∈ Bsi B where g0 := id}
X(w)
i−1
and
^ → X(w) : (g1 B, · · · , gn B) 7→ gn B.
X(w)
^ is an iterated P1 -bundle, thus is smooth. Also, it is known that for any
The variety X(w)
−1
y ∈ X(w), π (y) admits an affine paving. Thus by the decomposition theorem, first we see
that π∗ Q` X(w)
^ is semisimple. Also, it should contain ICX(w) as the restrictions of both on
X(w) coincide. Therefore, the first assertion follows since a variety paved by affine spaces
8
i
has vanishing odd cohomology. Also we see that all the eigenvalues of (F n )∗ on HB
0 π∗ Q ` ^
X(w)
are q ni/2 .
i p 0
From that we can prove that HB
0 H π∗ Q` ^ also has the same property. But note that
X(w)
this is not a consequence of the decomposition theorem; the theorem only works over F, not
over Fqn . Instead, one can use the distinguished triangle
p
p ≥j
p ≥j+1
Hj π∗ Q` X(w)
π∗ Q` X(w)
π∗ Q` X(w)
^ → τ
^ → τ
^
and the fact that this becomes the direct sum after we base change from Fqn to F. It is
even more nontrivial to show ICX(w) has the same property, since again the decomposition
only works over F. One needs to prove that ICX(w) is a subquotient of p H0 π∗ Q` X(w)
^ , e.g.
[GH, 10.7].
References
[BBD] Alexander A. Beilinson, Joseph Bernstein, and Pierre Deligne, Faisceaux pervers, Astèrisque 100
(1982).
[BM] Walter Borho and Robert MacPherson, Représentations des groupes de Weyl et homologie
d?intersection pour les variétés nilpotentes, CR Acad. Sci. Paris 292 (1981), no. 15, 707–710.
[CG] Neil Chriss and Victor Ginzburg, Representation Theory and Complex Geometry, Reprint of the
1997 Edition, Birkhäuser, Boston, 2010.
[CH] Alessio Corti and Masaki Hanamura, Motivic decomposition and intersection Chow groups, I, Duke
Mathematics Journal 103 (2000), 459–522.
[dC] Mark Andrea A. de Cataldo, Perverse sheaves and the topology of algebraic varieties: Five lectures
at the 2015 PCMI (2015), available at https://pcmi.ias.edu/files/Mark%20De%20Cataldo%
20Lecture%20Series.pdf.
[D] Pierre. Deligne, La conjecture de Weil. II, Publications mathématiques de l’I.H.É.S. 52 (1980),
137–252.
[dCM1] Mark Andrea A. de Cataldo and Luca Migliorini, The decomposition theorem, perverse sheaves and
the topology of algebraic maps, Bulletin of the AMS 46.4 (2009), 535–633.
[dCM2]
, The Hodge Theory of Algebraic maps, Ann. Scient. École Norm. Sup. 38 (2005), 693–750.
[G] Iain G. Gordon, Symplectic Reflection Algebras, Trends in Representation Theory of Algebras and
Related Topics, EMS Series of Congress Reports, EMS, 2008, pp. 285–347.
[GH] Ulrich Görtz and Thomas J. Haines, The Jordan-Hölder series for nearby cycles on some Shimura
varieties and affine flag varieties, Journal für die reine und angewandte Mathematik 2007 (2007),
no. 609, 161–213.
[H] Thomas J. Haines, A proof of the Kazhdan-Lusztig purity theorem via the decomposition theorem of
BBD, available at http://www.math.umd.edu/~tjh/KL_purity1.pdf. Expository note.
[HTT] Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, D-Modules, Perverse Sheaves, and Representation Theory, translated by Kiyoshi Takeuchi, Progress in Mathematics, vol. 236, Birkhäuser,
Boston, 2008.
[K1] Dmitry Kaledin, Symplectic resolutions: deformations and birational maps, arXiv:math/0012008
[math.AG] (2000), available at http://arxiv.org/abs/math/0012008.
[K2]
, Symplectic singularities from the Poisson point of view, arXiv:math/0310186 [math.AG]
(2003), available at http://arxiv.org/abs/math/0310186.
[KL] David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Porceedings of Symposia in Pure Mathematics 36 (1980).
[L] George Lusztig, On the Finiteness of the Number of Unipotent Classes, Invent. Math. 34 (1976),
201–213.
[S] Morihiko Saito, Mixed Hodge modules, Publ. RIMS, Kyoto Univ. 26 (1990), 221–333.
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA
02139-4307, U.S.A.
E-mail address: sylvaner@math.mit.edu
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