4.1 Star-products
Let X be a complex manifold (or a smooth variety). We denote by δX : X ,→ X × X the diagonal embedding and we set 4X = δX(X). We denote by OX the structure sheaf on X, by ΩX the sheaf of differential forms of maximal degree and by ΘX the sheaf of vector fields. As usual, we denote by DX the sheaf of rings of differential operators on X. Recall that a bi-differential operator P on X is a C-bilinear morphism OX×OX → OX which is obtained as the composition δX−1 ◦ ˜P where ˜P is a differential operator on X × X defined on a neighborhood of the diagonal and δ−1 is the restriction to the diagonal:
P (f, g)(x) = ( ˜P (x1, x2; ∂x1, ∂x2)(f (x1)g(x2))|x1=x2=x. Hence the sheaf of bi-differential operators is isomorphic to
DX ⊗OX DX,
where the both DX are regarded as OX-modules by the left mul-tiplications.
Definition 4.1. A star algebra on OX[[~]] is a C~-bilinear sheaf morphism
? : OX[[~]] × OX[[~]] → OX[[~]]
satisfies the following conditions:
(i) the star product makes OX[[~]] into a sheaf of associated unital C~-algebra with unit 1 ∈ OX.
(ii) there is a sequence Pi : OX × OX → OX of bi-differential operators, such that for any two local sections f, g ∈ OX
one has
f ? g = f g +
∞
P
i=1
Pi(f, g)~i.
Note that f ? g ≡ f g mod ~, and Pi(f, 1) = Pi(1, f ) = 0 for all f and i > 0. We call (OX[[~]], ?) a star algebra.
4.2 DQ-algebras
Definition 4.2. A DQ-algebra A on X is a C~-algebra locally isomorphic to a star-algebra (OX[[~]], ?) as a C~-algebra.
Clearly, a DQ-algebra is a sheaf of ~-adically complete flat C~-algebra on X satisfying A /~A ' OX. Note also that for an algebraic variety X, a DQ-algebra A is called deformation quantization of OX in [3] and [20].
Remark 4.3. For a smooth projective variety X, there exists a DQ-algebra AX on X. For details, one refers to [3].
4.3 Riemann-Roch theorem for DQ-modules
Let X be a complex manifold. To a DQ-algebra AX on X, we associate its ~-localization, the C~,loc-algebra
AXloc = C~,loc⊗C~AX. Hence we have an exact functor
Mod(AX) ⊗C~C
~,loc
−→ Mod(AXloc).
IfM is an AXloc-module, M0 is an AX-submodule andM0⊗C~
C~,loc −→∼ M , we shall say that M0 generates M .
A coherent AXloc-module M is good if for any open relatively compact subset U of X, there exists a coherent AX|U-module which generates M |U.
We denote by Modgd(AXloc) the thick abelian subcategory of Mod (AXloc) consisting of good AXloc-modules. We denote by Dbgd(AXloc) the full triangulated subcategory of Db(AXloc) consist-ing of objects with cohomology sheaves belongconsist-ing to Modgd(AXloc).
We also denote by K(Modgd(AXloc)) the Grothendieck group of Modgd(AXloc) and Kgd(AXloc) the Grothendieck group of Dbgd(AXloc) and note that
(4.1) K(Modgd(AXloc)) ' Kgd(AXloc) by [2, P. 283 Lemma 1.6].
If (Y,AY) is another complex manifold endowed with a DQ-algebra AY, denote by qi the i-th projection defined on X × Y (i = 1, 2). Let M ∈ Dbcoh(AX) and K ∈ Dbcoh(AXa×Y). Set
M ◦ K := Rq2!(K ⊗LAX q1−1M ).
If q2 is proper on q−11 supp(M )∩supp(K ), then by [13, Theorem 9.1], M ◦ K ∈ Dbcoh(AY).
Similarly, let M1 ∈ Dbgd(AXloc) and K1 ∈ Dbgd(AXloca×Y). Set M1 ◦K1 := Rq2!(K1
⊗LAloc
X q−11 M1).
If q2 is proper on q1−1supp(M1) ∩ supp(K1), then M1 ◦ K1 ∈ Dbgd(AYloc) (see [18, Corollary 3.3.5]).
If X and Y are compact, then a kernel K ∈ Dbcoh(AXa×Y) defines a functor
(4.2) ◦K : Dbcoh(AX) → Dbcoh(AY)
which is called the Fourier-Mukai transform induced by K . Hence the Fourier-Mukai transform of (4.2) defines a group ho-momorphism of Grothendieck groups
◦[K ] : Kcoh(AX) → Kcoh(AY)
where [K ] ∈ Kcoh(AXa×Y). Similarly, a kernel K loc belongs to Dbgd(AXloca×Y) defines a functor
(4.3) ◦K loc : Dbgd(AXloc) → Dbgd(AYloc)
which is called the Fourier-Mukai transofrm induced by K loc and the Fourier-Mukai transform of (4.3) defines a group homo-morphism of Grothendieck groups complex manifolds endowed with DQ-algebras AX and AY and let K ∈ Dbcoh(AXa×Y). Then the following diagram is which induces the following commutative diagram
Kcoh(AX) −◦[−−K ]→ Kcoh(AY)
where the group homomorphisms gr (4.4) are induced by (4.5).
Proof. Using the fact that the functor gr commutes with the
convolution ◦.
Definition 4.5. Let X be a compact complex manifold, and denote by H∗(X, C) = L
i
Hi(X, C). One defines the Mukai vector of an object E ∈ Dbcoh(OX) as the cohomology class
υ : Kcoh(OX) → H∗(X, C) by the formula
υ([E]) = ch([E]).ptd(X)
where ch([E]) is the Chern character of [E] and td(X) is the Todd class of tangent bundle of X.
Let X and Y be compact complex manifolds and let E ∈ Dbcoh (OX×Y). Define the cohomological integral transform associated to E
Φ[E] : H∗(X, C) → H∗(Y, C)
by Φ[E](α) = q2∗(υ([E].q1∗(α)), where qi denotes the i-th projec-tion defined on X × Y (i = 1, 2).
We have the following theorem.
Theorem 4.6.([5, Proposition 3.1.9] or [9, Corollary 5.29]) The following diagram is commutative
Kcoh(OX) −−→ K◦[E] coh(OY)
υ
y
yυ H∗(X, C) −−→ HΦ[E] ∗(Y, C).
Combining Proposition 4.4 with Theorem 4.6, we obtain the following theorem.
Theorem 4.7.(Riemann-Roch for A -modules) Let (X, AX) and (Y,AY) be two compact complex manifolds endowed with DQ-algebras AX and AY and let K ∈ Dbcoh(AXa×Y). Then the fol-lowing diagram is commutative
Kcoh(AX) −◦[−−K ]→ Kcoh(AY)
υ◦gr
y
y
υ◦gr
H∗(X, C) −−−→ HΦ[grK ] ∗(Y, C).
4.2 Riemann-Roch theorem for Aloc-modules
First, we need the following lemma.
Lemma 4.8. Let (X,AX) be a compact complex manifold en-dowed with a DQ-algebra AX. Let M ∈ Modgd(AXloc) and let M0 ⊂ M which generates M . Then [M0/~M0] belongs to K(Modcoh(OX)) depends only on M .
Proof. We consider another generator M00 of M . Since X is compact, there exists m, n ≥ 0 such that M00 ⊂ ~−nM0 and M0 ⊂ ~−mM00. Hence
M00 ⊂ ~−nM0 ⊂ ~−m−nM00. Since our modules have no ~-torsion, we have
~−nM0/~−n+1M0 ' M0/~M0.
Hence we may assume that for N large enough, we have
M00 ⊂M0 ⊂ ~−NM00.
We shall prove [M0/~M0] = [M00/~M00] by induction on N ≥ 0.
When N = 0, it is trivial.
When N = 1, we have the following exact sequences:
0 → ~−1M00/M0 → ~−1M0/M0 → ~−1M0/~−1M00 → 0 0 → M0/M00 → ~−1M00/M00 → ~−1M00/M0 → 0.
Since ~−1M0/~−1M00 'M0/M00, we get [M0/~M0] = [M00/~M00].
For N > 1. Put M000 = M00 + ~M0. Then M000 ⊂M0 ⊂ ~−1M000,
hence, as above, [M000/~M000] = [M0/~M0]. On the other hand, M00 ⊂ M000 ⊂ ~−N +1M00
hence by induction hypothesis, we obtain [M000/~M000] = [M00/~M00].
Hence [M0/~M0] = [M00/~M00], as desired. Theorem 4.9. Let (X,AX) be a compact complex manifold endowed with a DQ-algebra AX, then we have a well defined group homomorphism
(4.6) µ : K(Modgd(AXloc)) → K(Modcoh(OX)).
Proof. Define a function
Γ : Modgd(AXloc) → K(Modcoh(OX))
sending M to [M0/~M0] with M0 a generator of M . We need to show that Γ is additive. For an exact sequence
(?) 0 → N1 ⊗C~ C~,loc →α N ⊗C~C~,loc
→β N2⊗C~C~,loc → 0
in Modgd(AXloc) with N1,N and N2 of no ~-torsions and set M2 := β(N ), M := N and M1 := ker(β|N : N → β(N )).
Then we have the exact sequence
(∗) 0 → M1 → M → M2 → 0
From (1.1) and (4.1), the group homomorphism of (4.6) in-duces the group homomorphism
(4.7) λ : Kgd(AXloc) → Kcoh(OX).
Now let X, Y be compact complex manifolds. From Lemma 4.8, we obtain the following theorem.
Theorem 4.10. Let (X,AX), (Y,AY) be compact complex man-ifolds endowed with DQ-algebras AX and AY and let K loc ∈ Dbgd (AXa×Y). Then we have the following commutative diagram
Kgd(AXloc) ◦[K
where λ0s denote the group homomorphisms of (4.7).
Proof. Let K ∈ Dbcoh(AXa×Y) which generates K loc. Let Mloc ∈ Dbgd(AXloc) and letM ∈ Dbcoh(AX) which generates Mloc. Then M ◦ K generates Mloc ◦ K loc and λ([Mloc ◦ K loc]) =
[gr(M ◦ K )] = [grM ◦ grK ] = [grM ] ◦ [grK ] = λ[Mloc] ◦
λ[K loc] by Lemma 4.8, as desired.
Combining Theorem 4.6 and Theorem 4.10, we obtain the following theorem.
Theorem 4.11.(Riemann-Roch for Aloc-modules) Let (X,AX) and (Y,AY) be two compact complex manifolds endowed with DQ-algebras AX and AY and let K loc ∈ Dbgd(AXa×Y). Then the following diagram is commutative
Kgd(AXloc) ◦[K
loc]
−−−−→ Kgd(AYloc)
υ◦λ
y
yυ◦λ H∗(X, C) −−−−−→ HΦλ([K loc]) ∗(Y, C).