In this note we prove a conjecture of Kashiwara, which states that the Euler
class of a coherent analytic sheaf F on a complex manifold X is the product of
the Chern character of F with the Todd class of X. As a corollary, we obtain a
functorial proof of the Grothendieck-Riemann-Roch theorem in Hodge cohomology
for complex manifolds.Comment: Final versio

Grivaux, Julien

English

arXiv

Final versionInternational audienceIn this note we prove a conjecture of Kashiwara, which states that the Euler class of a coherent analytic sheaf F on a complex manifold X is the product of the Chern character of F with the Todd class of X. As a corollary, we obtain a functorial proof of the Grothendieck-Riemann-Roch theorem in Hodge cohomology for complex manifolds

Hal-Diderot

On a conjecture of Kashiwara relating Chern and Euler classes of O-modules

Julien Grivaux

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On a conjecture of Kashiwara relating Chern and Euler
classes of O-modules
Julien Grivaux
To cite this version:
Julien Grivaux. On a conjecture of Kashiwara relating Chern and Euler classes of O-modules. Journal
of Differential Geometry, International Press, 2012, 90 (2), pp.267–275. ￿hal-01301388￿
ON A CONJECTURE OF KASHIWARA RELATING
CHERN AND EULER CLASSES OF O-MODULES
Julien Grivaux
Abstract
In this note we prove a conjecture of Kashiwara, which states
that the Euler class of a coherent analytic sheaf F on a complex
manifold X is the product of the Chern character of F with the
Todd class of X. As a corollary, we obtain a functorial proof of
the Grothendieck-Riemann-Roch theorem in Hodge cohomology
for complex manifolds.
1. Introduction
The notation used throughout this article is defined in §2.
Let X be a complex manifold, ωX be the holomorphic dualizing com-
plex of X, δX be the diagonal injection of X in X × X and Dbcoh(X)
be the full subcategory of the bounded derived category of analytic
sheaves on X consisting of objects with coherent cohomology. In the
letter [7] which is reproduced in Chapter 5 of [6], Kashiwara constructs
for every F in Dbcoh(X) two cohomology classes hhX(F) and thhX(F) in
H0supp(F)(X, δ
∗
XδX∗OX) and H0supp(F)(X, δ!XδX! ωX); they are the Hochs-
child and co-Hochschild classes of F .
Let us point out that characteristic classes in Hochschild homology
are well-known in homological algebra (see [8, §8]). They have been re-
cently intensively studied in various algebraico-geometric contexts. For
further details, we refer the reader to [3], [2], [13] and to the references
therein.
If f :X //Y is a holomorphic map, the classes hhX and thhX satisfy
the following dual functoriality properties:
– for every G in Dbcoh(Y ), hhX(f∗G) = f∗ hhY (G),
– for every F in Dbcoh(X) such that f is proper on supp(F),
thhY (Rf!F) = f! thhX(F).
The analytic Hochschild-Kostant-Rosenberg isomorphisms construc-
ted in [7] are specific isomorphisms
δ∗XδX∗OX '
⊕
i≥0
Ω iX [i] and δ
!
XδX! ωX '
⊕
i≥0
Ω iX [i]
PROOF COPY 1 NOT FOR DISTRIBUTION
2 JULIEN GRIVAUX
in Dbcoh(X). The Hochschild and co-Hochschild classes of an element
F in Dbcoh(X) are mapped via the above HKR isomorphisms to the so-
called Chern and Euler classes of F in ⊕
i≥0
Hisupp(F)(X,Ω
i
X), denoted by
ch(F ) and eu(F).
The natural morphism⊕
i≥0
Hisupp(F)(X,Ω
i
X)
//
⊕
i≥0
Hi(X,Ω iX)
maps ch(F) to the usual Chern character of F in Hodge cohomology,
which is obtained by taking the trace of the exponential of the Atiyah
class of the tangent bundle TX1.
The Chern and Euler classes satisfy the same functoriality properties
as the Hochschild and co-Hochschild classes, namely for every holomor-
phic map f from X to Y :
– for every G in D bcoh(Y ), ch(f∗G) = f∗ ch(G),
– for every F in D bcoh(X) such that f is proper on supp(F),
eu(Rf!F) = f! eu(F).
Furthermore, for every F in Dbcoh(X), eu(F) = ch(F) eu(OX). Put-
ting together the previous identity with the functoriality of the Euler
class with respect to direct images, Kashiwara obtained the following
Grothendieck-Riemann-Roch theorem:
Theorem 1.1. [7] Let f :X //Y be a holomorphic map and F be
an element of Dbcoh(X) such that f is proper on supp(F). Then the
following identity holds in
⊕
i≥0
Hif [supp(F)](Y,Ω
i
Y ):
ch(Rf!F) eu(OY ) = f!
[
ch(F) eu(OX)
]
.
Then Kashiwara stated the following conjecture (see [6, §5.3.4]):
Conjecture. [7] For any complex manifold X, the class eu(OX) is
the Todd class of the tangent bundle TX.
This conjecture was related to another conjecture of Schapira and
Schneiders comparing the Euler class of a DX -module m and the Chern
class of the associated OX -module Gr(m) (see [12], [1]).
The aim of this note is to give a simple proof of Kashiwara’s conjec-
ture:
Theorem 1.2. For any complex manifold X, eu(OX) is the Todd
class of TX.
1This property has been proved in [2] for algebraic varieties using different defini-
tions of the HKR isomorphism and of the Hochschild class. In Kashiwara’s setting,
this is straightforward.
PROOF COPY NOT FOR DISTRIBUTION
ON A CONJECTURE OF KASHIWARA 3
In the algebraic setting, an analogous result is established in [11] (see
also [9]).
As a corollary of Theorem 1.2, we obtain the Grothendieck-Riemann-
Roch theorem in Hodge cohomology for abstract complex manifolds,
which has been already proved by different methods in [10]:
Theorem 1.3. Let f :X //Y be a holomorphic map between com-
plex manifolds, and let F be an element of Dbcoh(X) such that f is proper
on supp(F). Then
ch(Rf!F) td(Y ) = f!
[
ch(F) td(X)].
in
⊕
i≥0
Hif [supp(F)](Y,Ω
i
Y ).
However, the proof given here is simpler and more conceptual. Be-
sides, we would like to emphasize that it is entirely self-contained and
relies only on the results appearing in Chapter 5 of [6].
Acknowledgement. I want to thank Masaki Kashiwara and Pierre
Schapira for communicating their preprint [6] to me. I also want to
thank Pierre Schapira for useful conversations, and Joseph Lipman for
interesting comments.
2. Notations and basic results
We follow the notation of [6, Ch.5].
If X is a complex manifold, we denote by Db(X) the bounded derived
category of sheaves of OX -modules and by Dbcoh(X) the full subcategory
of Db(X) consisting of complexes with coherent cohomology.
If f :X //Y is a holomorphic map between complex manifolds, the
four operations f ∗ :Db(Y ) //Db(X), Rf∗ , Rf! :D
b(X) //Db(Y ) and
f ! :Db(Y ) //Db(X) are part of the formalism of Grothendieck’s six
operations. Let us recall their definitions:
– f ∗ is the left derived functor of the pullback functor by f, that
is G // G ⊗f−1OY OX ,
– Rf∗ is the right derived functor of the direct image functor f∗, it
is the left adjoint to the functor f∗,
– Rf! is the right derived functor of the proper direct image functor
f!,
– f ! is the exceptional inverse image, it is the right adjoint to the
functor Rf!.
If W is a closed complex submanifold of Y, the pullback morphism
from f ∗Ω iY [i] to Ω
i
X [i] induces in cohomology a map
f ∗ :
⊕
i≥0
HiW (Y,Ω
i
Y )
//
⊕
i≥0
Hif−1(W )(X,Ω
i
X).
PROOF COPY NOT FOR DISTRIBUTION
4 JULIEN GRIVAUX
If Z is a closed complex submanifold of X and if f is proper on Z,
the integration morphism from Ω i+dXX [i+ dX ] to Ω
i+dY
Y [i+ dY ] induces
a Gysin morphism
f! :
⊕
i≥−dX
Hi+dXZ (X,Ω
i+dX
X )
//
⊕
i≥−dY
Hi+dYf(Z) (Y,Ω
i+dY
Y ).
Let X be a complex manifold, ωX be the holomorphic dualizing com-
plex of X and δX be the diagonal injection. If F belongs to D bcoh(X), we
define the ordinary dual (resp. Verdier dual) of F by the usual formula
D′F = RHomOX (F ,OX) (resp. DF = RHomOX (F , ωX)).
The Hochschild and co-Hochschild classes of F , denoted by hhX(F)
and thhX(F), lie in H0supp(F)(X, δ∗XδX∗OX) and H0supp(F)(X, δ!XδX! ωX)
respectively. They are constructed by the chains of maps
hhX(F) : id // RHom(F ,F) ∼ // δ∗X(D
′F O F) // δ∗XδX∗OX
thhX(F) : id // RHom(F ,F) ∼ // δ!X(DF O F) // δ!XδX! ωX
where in both cases the last arrows are obtained from the derived trace
maps
D′F L⊗OXF //OX and DF
L⊗OXF //ωX
by adjunction.
If f : X //Y is a holomorphic map between complex manifolds,
there are pullback and push-forward morphisms
f∗ :f∗δ∗Y δY ∗OY //δ∗XδX∗OX and f! : Rf! δ!XδX!ωX //δ!Y δY !ωY .
Besides, there is a natural pairing
(1) δ∗XδX∗OX
L⊗OX δ
!
XδX! ωX
// δ!XδX! ωX
given by the chain
δ∗XδX∗OX
L⊗OX δ
!
XδX! ωX ' δ!X(δX∗OX
L⊗OX×X δX! ωX) // δ
!
XδX! ωX .
Theorem 2.1. [7]
(i) For all elements F and G in D bcoh(X) and D bcoh(Y ) respectively,
hhX(f
∗G) = f∗ hhY (G) and f! thhX(F) = thhY (Rf!F).
(ii) Via the pairing (1), for every F in D bcoh(X),
hhX(F) thh(OX) = thhX(F).
The Hochschild and co-Hochschild classes are translated into Hodge
cohomology classes by Kashiwara’s analytic Hochschild-Kostant-Rosen-
berg isomorphisms2
(2) δ∗XδX∗OX '
⊕
i≥0
Ω iX [i] and δ
!
XδX! ωX '
⊕
i≥0
Ω iX [i],
2For a detailed account of the HKR isomorphisms, we refer to the introduction of
[5] and to the references therein.
PROOF COPY NOT FOR DISTRIBUTION
ON A CONJECTURE OF KASHIWARA 5
the resulting classes are called Chern and Euler classes. If F is an ele-
ment of Dbcoh(X), then ch(F) and eu(F) lie in
⊕
i≥0
Hisupp(F)(X,Ω
i
X).
The first HKR isomorphism commutes with pullback and the second
one with push forward. Besides, the pairing (1) between δ∗XδX∗OX and
δ!XδX! ωX is exactly the cup-product on holomorphic differential forms
after applying the HKR isomorphisms (2).
Theorem 2.2. [7]
(i) For every F in D bcoh(X), ch(F) is the usual Chern character ob-
tained by the Atiyah exact sequence.
(ii) For all elements F and G in D bcoh(X) and D bcoh(X) respectively,
ch(f∗G) = f∗ ch(G) and f! eu(F) = eu(Rf!F).
(iii) For every F in D bcoh(X), eu(F) = ch(F) eu(OX).
For the proofs of Theorems 2.1 and 2.2, we refer to [6, Ch. 5].
For any complex manifold X, we denote by td(X) the Todd class of
the tangent bundle TX in
⊕
i≥0
Hi(X,Ω iX).
3. Proof of Theorem 1.2
We proceed in several steps.
Proposition 3.1. Let Y and Z be complex manifolds such that Z
is a closed complex submanifold of Y , and let iZ be the corresponding
inclusion. Then, for every coherent sheaf F on Z, we have
iZ! [ch(F) td(Z)] = ch(iZ∗F) td(Y )
in
⊕
i≥0
HiZ(Y,Ω
i
Y ).
Proof. This is proved in the classical way using the deformation to
the normal cone as in [4, §15.2], except that we use local cohomology.
For the sake of completeness, we provide a detailed proof.
We start by a particular case:
– N is a holomorphic vector bundle on Z, and Y = P (N ⊕OZ),
– Z embeds in Y by identifying Z with the zero section of N.
Let d be the rank of N , pi be the projection of the projective bundle
P (N ⊕ OZ) and Q be the universal quotient bundle on Y ; Q is the
quotient of pi∗(N ⊕ OZ) by the tautological line bundle ON⊕OZ (−1).
Then Q has a canonical holomorphic section s which is obtained by the
composition
s : OY ' pi∗OZ // pi∗(N ⊕OZ) // Q.
This section vanishes transversally along its zero locus, which is exactly
Z. Therefore, we have a natural locally free resolution of iZ!OZ given
PROOF COPY NOT FOR DISTRIBUTION
6 JULIEN GRIVAUX
by the Koszul complex associated with the pair (Q∗, s∗):
0 // ∧dQ∗ // ∧d−1Q∗ // · · · // OY // iZ!OZ // 0.
This gives the equality
ch(iZ!OZ) =
d∑
k=0
(−1)kch(∧kQ∗) = cd(Q) td(Q)−1
in
⊕
i≥0
Hi(Y,ΩiY ), where cd(Q) denotes the d-th Chern class of Q (for the
last equality, see [4, § 3.2.5]). Since cd(Q) is the image of the constant
class 1 by iZ! and since i
∗
ZQ = N, we get
ch(iZ!OZ) = iZ!(i∗Ztd(Q)−1) = iZ!(td(N )−1).
For any coherent sheaf F on Z, we have iZ!F = iZ!OZ
L⊗OY pi
∗F so
that we obtain by the projection formula
(3) ch(iZ!F) = iZ!(ch(F)td(N )−1)
in
⊕
i≥0
Hi(Y,ΩiY ). Remark now that by Theorem 2.2 (ii) and (iii), we
have
ch(iZ!F) = iZ!(ch(F)eu(OZ) i∗Z eu(OY )−1)
in
⊕
i≥0
HiZ(Y,Ω
i
Y ). This proves that ch(iZ!F) lies in the image of
iZ! :
⊕
i≥0
Hi(Z,Ω iZ)
//
⊕
i≥0
Hi+dZ (Y,Ω
i+d
Y ).
Let us denote this image by W. The map
ι:W //
⊕
i≥0
Hi+d(Y,Ω i+dY )
obtained by forgetting the support is injective. Indeed, for every class
iZ!α in W, pi![ι(iZ!α)] = α. This implies that (3) holds in
⊕
i≥0
HiZ(Y,Ω
i
Y ).
We now turn to the general case, using deformation to the normal
cone. Let us introduce some notation:
– M is the blowup of Z × {0} in Y × P1, σ is the blowup map and
q = pr1◦σ,
– for any divisor D on M , [D] denotes its cohomological cycle class
in H1(M,Ω1M ),
– E = P(NZ/Y ⊕OZ) is the exceptional divisor of the blowup and
Y˜ is the strict transform of Y × {0} in M,
– µ is the embedding of Z in E, where Z is identified with the zero
section of NZ/Y ,
– F is the embedding of (the strict transform of) Z ×P1 in M and
for any t in P1, jt is the embedding of Mt in M,
PROOF COPY NOT FOR DISTRIBUTION
ON A CONJECTURE OF KASHIWARA 7
– k is the embedding of E in M.
Then M is flat over P1, M0 is a Cartier divisor with two smooth
components E and Y˜ intersecting transversally along P(NZ/Y ), and Mt
is isomorphic to Y if t is nonzero.
Let G = F!(pr∗1F). Since M is flat over P1, for any t in P1 \ {0},
j ∗t G = iZ!F and k ∗G = µ!F.
If ch(G) is the Chern character of G in ⊕
i≥0
HiZ×P1(M,Ω
i
M ), using the
identity (3) in
⊕
i≥0
HiZ(E,Ω
i
E), we get
jt! ch(iZ!F) = jt! j ∗t ch(G) = ch(G) [Mt]
= ch(G) [M0] = ch(G) [E] + ch(G) [Y˜ ]
= ch(G) [E] = k! k ∗ ch(G) = k! ch(µ!F)
= k! µ!(ch(F) td(NZ/E)−1)
= k! µ!(ch(F) td(NZ/X)−1)
in
⊕
i≥0
HiZ×P1(M,Ω
i
M ).
The map q is proper on Z×P1, q◦jt = id and q◦k◦µ = iZ . Applying
q!, we get
ch(iZ!F) = iZ!(ch(F) td(NZ/X)−1)
in
⊕
i≥0
HiZ(Y,Ω
i
Y ). q.e.d.
Definition 3.2. For any complex manifold X, let α(X) be the co-
homology class in
⊕
i≥0
Hi(X,ΩiX) defined by α(X) = eu(OX) td(X)−1.
Lemma 3.3. Let Y and Z be complex manifolds such that Z is a
closed complex submanifold of Y, and let iZ be the corresponding injec-
tion. Assume that there exists a holomorphic retraction R of iZ . Then
we have α(Z) = i∗Z α(Y ).
Proof. By Theorem 2.2 (ii), eu(iZ∗OZ) = iZ! eu(OZ). By Proposition
3.1 and Theorem 2.2 (iii),
eu(iZ∗OZ) = ch(iZ∗OZ)eu(OY ) = (iZ! td(Z)) td(Y )−1eu(OY ),
so that we obtain in
⊕
i≥0
HiZ(Y,Ω
i
Y ) the formula
iZ!
[
eu(OZ)− td(Z) i∗Z(eu(OY ) td(Y )−1)
]
= 0.
Since R is proper on Z, we can apply R! and we get the result. q.e.d.
Lemma 3.4. The class α(X) satisfies α(X)2 = α(X).
PROOF COPY NOT FOR DISTRIBUTION
8 JULIEN GRIVAUX
Proof. We apply Lemma 3.3 with Z = X and Y = X ×X, where X
is diagonally embedded in X × X. Then α(X) = i∗∆ α(X × X). The
Euler class commutes with external products so that
eu(OX×X) = eu(OX) eu(OX)
Thus α(X ×X) = α(X) α(X) and we obtain
α(X) = i∗∆[α(X) α(X) ] = α(X)2.
q.e.d.
Proof of Theorem 1.2. There is a natural isomorphism φ in D bcoh(X)
between δ∗δ∗OX and δ!δ!ωX given by the chain
δ∗δ∗OX ' OX
L⊗OX δ
∗δ∗OX ' δ!(ωX  OX)
L⊗OX δ
∗δ∗OX ' δ!δ!ωX .
Besides, after applying the two HKR isomorphisms (2), φ is given by
derived cup-product with the Euler class of OX (see [6]). Therefore, the
class eu(OX) is invertible in the Hodge cohomology ring of X, and so
is α(X). Lemma 3.4 implies that α(X) = 1.
q.e.d.
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ON A CONJECTURE OF KASHIWARA 9
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CMI, UMR CNRS 6632 (LATP)
Universite´ de Provence
39, rue Fre´de´ric Joliot-Curie
13453 Marseille Cedex 13
France
E-mail address: jgrivaux@cmi.univ-mrs.fr
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On a conjecture of Kashiwara relating Chern and Euler classes of O-modules

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