---
address: Steklov Mathematical Institute RAN
author:
- Dmitri Orlov
title: 'Derived categories of coherent sheaves and motives.'
---

The bounded derived category of coherent sheaves $\db{X}$ is a natural triangulated category which can be associated with an algebraic variety $X.$ It happens sometimes that two different varieties have equivalent derived categories of coherent sheaves $\db{X}\simeq\db{Y}.$ There arises a natural question: can one say anything about motives of $X$ and $Y$ in that case? The first such example (see [@Mu]) – abelian variety $A$ and its dual $\wh{A}$ – shows us that the motives of such varieties are not necessary isomorphic. However, it seems that the motives with rational coefficients are isomorphic in all known cases.

Recall a definition of the category of effective Chow motives $\CH$ over a field $\kk.$ The category $\CH$ can be obtained as the pseudo-abelian envelope (i.e. as formal adding of cokernels of all projectors) of a category, whose objects are smooth projective schemes over $\kk,$ and the group of morphisms from $X$ to $Y$ is the sum $\oplus_{X_i}A^m(X_i\times Y)$ (over all connected components $X_i$) of the groups of cycles of codimension $m=\dim Y$ on $X_i\times Y$ modulo rational equivalence (see [@Ma; @Bl]). In [@Vo] Voevodsky introduced a triangulated category of geometric motives $\DM.$ He started with an additive category $\SmCor,$ objects of which are smooth schemes of finite type over $\kk,$ and the group of morphisms from $X$ to $Y$ is the free abelian group generated by integral closed subschemes $Z\subset X\times Y$ such that the projection on $X$ is finite and surjective onto a connected component of $X.$ There is a natural embedding $[-]:\Sm\to \SmCor$ of the category $\Sm$ of smooth schemes of finete type over $\kk.$ The category $\SmCor$ is additive and one has $[X\coprod Y]=[X]\oplus[Y].$ Further, he considered the quotient of the homotopy category $\H^b(\SmCor)$ of bounded complexes by minimal thick triangulated subcategory $T,$ which contains all objects of the form $[X\times\AA^1]\to
[X]$ and $[U\cap V]\to[U]\oplus[V]\to [X]$ for any open covering $U\cup V=X.$ Triangulated category $\DM$ is defined as the pseudo-abelian envelope of the quotient category $\H^b(\SmCor)/T$(see [@Vo; @Bl]).

There exists a canonical functor $\CH\to\DM,$ which is a full embedding if $\kk$ admits resolution of singularities ([@Vo 4.2.6]). Thus, it doesn’t matter in which category (in $\CH$ or in $\DM$) motives of smooth projective varieties are considered. Denote the motive of a variety $X$ by $\Mg(X),$ and its motive in the category of motives with rational coefficients $\DM\otimes\QQ$(and in $\CH\otimes\QQ$) by $\Mg(X)_{\QQ}.$

\[G1\] Let $X$ and $Y$ be smooth projective varieties, and let $\db{X}{\simeq}\db{Y}.$ Then the motives $\Mg(X)_{\QQ}$ and $\Mg(Y)_{\QQ}$ are isomorphic in $\CH\otimes\QQ$ (and in $\DM\otimes\QQ$)

Let $X$ and $Y$ be smooth projective varieties and let $F: \db{X}\stackrel{}{\to}\db{Y}$ be a fully faithful functor. Then the motive $\Mg(X)_{\QQ}$ is a direct summand of the motive $\Mg(Y)_{\QQ}.$

The category $\DM$ has a tensor structure, and $\Mg(X)\otimes\Mg(Y)=\Mg(X\times Y).$ One defines the Tate object $\ZZ(1)$ to be the image of the complex $[\PP^1]\to[\Spec(\kk)]$ placed in degree 2 and 3 and put $M(p)=M\otimes\ZZ(1)^{\otimes p}$ for any motive $M\in\DM$ and $p\in\NN.$ The triangulated category of geometric motives $\Dm$ is defined by formally inverting the functor $-\otimes\ZZ(1)$ on $\DM.$ The important and nontrivial fact here is the statement that the canonical functor $\DM\to\Dm$ is a full embedding [@Vo 4.3.1]. Therefore, we can work in the category $\Dm.$ Moreover (see [@Vo]), for any smooth projective varieties $X, Y$ and for any integer $i$ there is an isomorphism $$\Hom_{\Dm}(\Mg(X), \Mg(Y)(i)[2i])\cong A^{m+i}(X\times
Y),\quad\text{where}\quad m=\dim Y.$$

Suppose, one has a fully faithful functor $F:\db{X}\to\db{Y}$ between derived categories of coherent sheaves of two smooth projective varieties $X$ and $Y$ of dimension $n$ and $m$ respectively. Any such functor has a right adjoint $F^*$ by [@BV], and by Theorem 2.2 from [@Or1] (see also [@Or2 3.2.1]) the functor $F$ can be represented by an object on the product $X\times Y,$ i.e. $F\cong\Phi_{\A},$ where $\Phi_{\A}=\bR p_{2*}(p^*_1(-)\stackrel{\bL}{\otimes}\A)$ for some $\A\in\db{X\times Y}.$ With any functor of the form $\Phi_{\A}:{\db{X}}\to{\db{Y}}$ one can associate an element $a\in
A^*(X\times Y, \QQ)$ by the following rule $$\label{epsi}
a= p^*_1\sqrt{ \td_X}\cdot \ch(\A)\cdot p^*_2\sqrt{\td_Y},$$ where $\td_X$ and $\td_Y$ are Todd classes of the varieties $X$ and $Y.$ The cycle $a$ has a mixed type. Let us consider its decomposition on the components $a=a_0+\cdots+a_{n+m},$ where index is the codimension of a cycle on $X\times Y.$ Each component $a_q$ induces a map of motives $$\alpha_q : \Mg(X)_{\QQ}\to \Mg(Y)_{\QQ}(q-m)[2(q-m)].$$ Thus the total cycle $a$ gives a map $ \alpha:
\Mg(X)_{\QQ}\stackrel{}{\to}\bigoplus_{i=-m}^{n}
\Mg(Y)_{\QQ}(i)[2i].$ Now consider the object $\B\in \db{X\times
Y},$ which represents the adjoint functor $F^*,$ i.e. $F^*\cong \Psi_{\B},$ where $\Psi_{\B}=\bR
p_{1*}(p^*_2(-)\stackrel{\bL}{\otimes}\B).$ One attaches to the object $\B$ a cycle $b=b_0+\cdots + b_{n+m}$ defined by the same formula (\[epsi\]). The cycle $b$ induces a map $\beta:
\bigoplus_{i=-m}^{n} \Mg(Y)_{\QQ}(i)[2i] \stackrel{}{\to}
\Mg(X)_{\QQ}.$ Since the functor $\Phi_{\A}$ is fully faithful, the composition $\Psi_{\B}\circ\Phi_{\A}$ is isomorphic to the identity functor. Applying the Riemann-Roch-Grothendieck theorem, we obtain that the composition $$\Mg(X)_{\QQ}\stackrel{\alpha}{\to}\bigoplus_{i=-m}^{n}
\Mg(Y)_{\QQ}(i)[2i] \stackrel{\beta}{\to} \Mg(X)_{\QQ}$$ is the identity map, i.e. $\Mg(X)_{\QQ}$ is a direct summand of $\bigoplus_{i=-m}^{n} \Mg(Y)_{\QQ}(i)[2i].$

Assume now that $\dim X=\dim Y=n$ and, moreover, suppose that the support of the object $A$ also has the dimension $n.$ Therefore, $a_q=0$ when $q=0,\dots,
n-1,$ i.e. $a=a_n+\cdots +a_{2n}.$ It is easily to see that in this case $b=b_{n}+\cdots+b_{2n}$ as well. This implies that the composition $\beta\cdot\alpha: \Mg(X)_{\QQ}\to \Mg(X)_{\QQ},$ which is the identity, coincides with $\beta_n\cdot\alpha_n$. Hence, $\Mg(X)_{\QQ}$ is a direct summand of $\Mg(Y)_{\QQ}.$ Furthermore, since the cycles $a_n$ and $b_n$ are integral in this case we get the same result for integral motives, i.e. the integral motive $\Mg(X)$ is a direct summand of the motive $\Mg(Y)$ as well. Thus, we obtain

Let $X$ and $Y$ be smooth projective varieties of dimension $n,$ and let $F: \db{X}\stackrel{}{\to}\db{Y}$ be a fully faithful functor such that the dimension of the support of an object $\A$ on $X\times Y,$ which represents $F,$ is equal to $n.$ Then the motive $\Mg(X)$ is a direct summand of the motive $\Mg(Y).$ If, in addition, the functor $F$ is an equivalence, then the motives $\Mg(X)$ and $\Mg(Y)$ are isomorphic.

Examples of such functors are known, they come from birational geometry (see e.g. [@Or2]). In these examples one of the connected components of $\supp(\A)$ gives a birational map $X\dasharrow Y.$ Blow ups and antiflips induce fully faithful functors, and flops induce equivalences. Note that an isomorphism of motives implies an isomomorphism of any realization (singular cohomologies, l-adic cohomologies, Hodge structures and so on).

For arbitrary equivalence $\Phi_{\A}: \db{X}\to\db{Y}$ the map of motives $\alpha_n:\Mg(X)_{\QQ}\to \Mg(Y)_{\QQ},$ induced by the cycle $a_n\in A^n(X\times Y, \QQ),$ is not necessary an isomorphism (e.g. Poincare line bundle $\mathcal{P}$ on the product of abelian variety $A$ and its dual $\wh{A}$). However, the following conjecture, which specifies Conjecture \[G1\], may be true.

Let $\A$ be an object of $\db{X\times Y},$ for which $\Phi_{\A}:\db{X}\to\db{Y}$ is an equivalence. Then there exist line bundles $L$ and $M$ on $X$ and on $Y$ respectively such that the component $a'_{n}$ of the object $\A':= p_1^* L\otimes \A\otimes p_2^*
M$ gives an isomorphism between motives $\Mg(X)_{\QQ}$ and $\Mg(Y)_{\QQ}.$

I am grateful to Yu. I. Manin for very useful discussions.

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