nLab
Bondal-Orlov reconstruction theorem

Bondal-Orlov reconstruction theorem

Idea

Thomason and Balmer showed that the derived category of coherent sheaves on a smooth variety, when considered as a monoidal category (i.e. with the tensor product) in addition to its triangulated category structure (i.e. as a tensor triangulated category), completely determines the variety uniquely; see spectrum of a triangulated category. However, the derived category still turns out to be an interesting invariant when considered without the monoidal category structure; moreover triangulated equivalences and autoequivalences are also important in relation to the homological mirror symmetry and similar phenomena. In fact, Bondal and Orlov show how to reconstruct a smooth variety from its derived category of coherent sheaves when its canonical sheaf is ample or anti-ample, using only the graded structure (i.e. the translation functor).

Statement

Let XX be a smooth projective variety and suppose that its canonical sheaf ω X\omega_X is ample or anti-ample. If YY is another smooth projective variety such that there is an equivalence F:D b(X)D b(Y)F : D^b(X) \stackrel{\sim}{\to} D^b(Y) between the bounded derived categories of coherent sheaves on XX and YY, then there is an isomorphism XYX \simeq Y.

Sketch of proof

The idea of the proof is to characterize “cohomologically” the closed points, invertible sheaves, and Zariski topology.

The main tool we use is the Serre functor. Due to Grothendieck-Serre duality?, both D b(X)D^b(X) and D b(Y)D^b(Y) have Serre functors S X=(ω X)[n]S_X = (\cdot \otimes \omega_X)[n] and S Y=(ω Y)[m]S_Y = (\cdot \otimes \omega_Y)[m], nn and mm being the dimensions of XX and YY respectively.

For simplicity assume ω X\omega_X is ample; the anti-ample case is analogous.

Step 1. First we characterize the closed points of XX and YY.

Definition

Let 𝒞\mathcal{C} be a k-linear graded category?. A point object of 𝒞\mathcal{C} is an object PP satisfying

  • (PO-1) S(P)P[n]S(P) \simeq P[n],
  • (PO-2) Hom i(P,P)=0\Hom^i(P, P) = 0 for i<0i \lt 0,
  • (PO-3) k(P):=Hom(P,P)k(P) := \Hom(P, P) is a field.

It is clear that the skyscraper sheaf 𝒪 x=Sky x(k(x))\mathcal{O}_x = \mathrm{Sky}_x(k(x)) of the residue field at any closed point xXx \in X is a point object of D b(X)D^b(X), as is any translation 𝒪 x[r]\mathcal{O}_x[r]. In fact, when we impose the ampleness condition on ω X\omega_X, it turns out that all point objects are of this form.

Proposition

If ω X\omega_X is ample, then a complex \mathcal{F}^\bullet in D b(X)D^b(X) is a point object iff 𝒪 x[r]\mathcal{F}^\bullet \simeq \mathcal{O}_x[r] for some closed point xXx \in X, rr \in \mathbb{Z}.

In fact, even though we don’t know that ω Y\omega_Y is also ample, we get the same result for D b(Y)D^b(Y). Let GG be the inverse equivalence to FF and note that FF and GG preserve point objects. Suppose there was a point object PP in D b(Y)D^b(Y) not isomorphic to any 𝒪 y[s]\mathcal{O}_y[s], and note G(P)𝒪 x 0[r]G(P) \simeq \mathcal{O}_{x_0}[r] for some closed x 0x_0. Now for any yYy \in Y, Hom(P,𝒪 y)=Hom(G(P),G(𝒪 y))=Hom(𝒪 x 0[r],𝒪 x[r])=0\Hom(P, \mathcal{O}_y) = \Hom(G(P), G(\mathcal{O}_y)) = \Hom(\mathcal{O}_{x_0}[r], \mathcal{O}_x[r']) = 0 since x 0xx_0 \ne x. But {𝒪 y:yY}\{\mathcal{O}_y : y \in Y\} form a spanning class? of D b(Y)D^b(Y) which implies that P=0P = 0.

Step 2. Next we characterize the invertible sheaves on both varieties.

Definition

Let 𝒞\mathcal{C} be a kk-linear graded category. An invertible object of 𝒞\mathcal{C} is an object LL satisfying for some ss \in \mathbb{Z}

  • (IO-1) Hom s(L,P)=k(P)\Hom^s(L, P) = k(P) for all point objects PP,
  • (IO-2) Hom i(L,P)=0\Hom^i(L, P) = 0 for isi \ne s.
Proposition

If all point objects of D b(X)D^b(X), for a smooth projective variety XX, are translations of skyscraper sheaves of closed points, then all invertible objects of D b(X)D^b(X) are translations of invertible sheaves on XX.

This means that the invertible objects of both D b(X)D^b(X) and D b(Y)D^b(Y) are translations of invertible sheaves, by step 1.

Step 3. Now we establish a bijection between the sets of points of the varieties.

Since 𝒪 X\mathcal{O}_X is an invertible object, we can assume without loss of generality that F(𝒪 X)F(\mathcal{O}_X) is isomorphic to some invertible sheaf 0\mathcal{L}_0 on YY. For any xXx \in X,

(1)k(x)=Hom(𝒪 X,𝒪 x)=Hom( 0,𝒪 y[s]) k(x) = \Hom(\mathcal{O}_X, \mathcal{O}_x) = \Hom(\mathcal{L}_0, \mathcal{O}_y[s])

for some yYy \in Y, which implies s=0s=0 and F(𝒪 x)𝒪 yF(\mathcal{O}_x) \simeq \mathcal{O}_y, k(x)k(y)k(x) \simeq k(y). We get a bijection of sets ϕ:XY\phi : X \to Y by mapping xyx \mapsto y.

Step 4. Now we can get an isomorphism of the canonical ring?s of the varieties.

Assume WLOG F(𝒪 X)𝒪 YF(\mathcal{O}_X) \simeq \mathcal{O}_Y. Note that S X k(𝒪 X)[nk]ω X nkS_X^k(\mathcal{O}_X)[-nk] \simeq \omega_X^{nk} for any kk. By commutativity of FF with the Serre functor, F(ω X nk)=S Y k(𝒪 Y)[nk]=ω Y nkF(\omega_X^{nk}) = S_Y^k(\mathcal{O}_Y)[-nk] = \omega_Y^{nk}. Therefore

(2)H 0(X,ω X nk)=Hom(𝒪 X,ω X nk)=Hom(𝒪 Y, Y nk)=H 0(Y,ω Y nk) H^0(X, \omega_X^{nk}) = \Hom(\mathcal{O}_X, \omega_X^{nk}) = \Hom(\mathcal{O}_Y, \otimes_Y^{nk}) = H^0(Y, \omega_Y^{nk})

and kH 0(X,ω X nk) kH 0(Y,ω Y nk)\bigoplus_k H^0(X, \omega_X^{nk}) \simeq \bigoplus_k H^0(Y, \omega_Y^{nk}), that is we have an isomorphism of the canonical rings A(X)A(X) and A(Y)A(Y).

Step 5. Since X=Proj(A(X))X = \mathrm{Proj}(A(X)) iff ω X\omega_X is ample, it remains therefore only to show that ω Y\omega_Y is ample. To do this we characterize the Zariski topology “cohomologically”.

For any invertible sheaf \mathcal{L} on XX, αHom(𝒪 X,)\alpha \in \Hom(\mathcal{O}_X, \mathcal{L}), and xXx \in X, let α X *:Hom(,𝒪 x)Hom(𝒪 X,𝒪 x)\alpha_X^* : \Hom(\mathcal{L}, \mathcal{O}_x) \to \Hom(\mathcal{O}_X, \mathcal{O}_x) be the induced map that takes ff to fαf \circ \alpha, and define the subset U α={xX:α x *0}XU_\alpha = \{ x \in X : \alpha_x^* \ne 0 \} \subset X. Now ω X\omega_X being ample is equivalent to the subcollection {U α:αHom(𝒪 X,ω X m),m}\{ U_\alpha : \alpha \in \Hom(\mathcal{O}_X, \omega_X^m), m \in \mathbb{Z} \} forming a basis of the Zariski topology on XX. Now the equivalence FF maps any 𝒪 Xα𝒪 x\mathcal{O}_X \stackrel{\alpha}{\to} \mathcal{L} \to \mathcal{O}_x to

(3)𝒪 YF(α)F()𝒪 ϕ(x)[s]. \mathcal{O}_Y \stackrel{F(\alpha)}{\to} F(\mathcal{L}) \to \mathcal{O}_{\phi(x)}[s].

F(α)F(\alpha) corresponds bijectively to some βHom(𝒪 Y,ω Y m)\beta \in \Hom(\mathcal{O}_Y, \omega_Y^m) and in fact our bijection f:XYf : X \to Y becomes a homeomorphism mapping U αU_\alpha to V F(α)V_{F(\alpha)}. It follows that {V β:βHom(𝒪 Y,ω Y m),m}\{ V_\beta : \beta \in \Hom(\mathcal{O}_Y, \omega_Y^m), m \in \mathbb{Z} \} forms a basis of the Zariski topology on YY, which implies ω Y\omega_Y is ample.

See also

References

The original paper:

A nice exposition of the proof with more details can be found in section 3.1 of the course notes of Igor Dolgachev on derived categories:

The theorem is also discussed in:

Last revised on January 1, 2015 at 15:21:17. See the history of this page for a list of all contributions to it.