# Information from the derived categories of coherent sheaves

Recently I read some results about derived categories of coherent sheaves, and see one use Fourier-Mukai transforms to prove that the derived categories of coherent sheaves of a scheme, under some conditions, determine the scheme. More precisely, I have seen

1. (Bondal and Orlov) Let $X$ and $Y$ be smooth projective varieties. If there is an exact equivalence $\text{D}^{\text{b}}(X)\simeq\text{D}^{\text{b}}(Y)$ and the canonical sheaf of $X$ is ample or anti-ample, then $X\simeq Y.$

2. (corollary of above plus some discussions on elliptic curves) Let $C$ be a smooth projective curve and $Y$ be a smooth projective variety. Then $\text{D}^{\text{b}}(X)\simeq\text{D}^{\text{b}}(Y)$ if and only if $X\simeq Y.$

3. (Gabriel) Let $X$ and $Y$ be smooth projective varieties. Then $\text{Coh}(X)\simeq\text{Coh}(Y)$ if and only if $X\simeq Y.$

My question is, do these results really help determine a variety? I mean, passing the question to the level of derived version will make it easier? Is there any concrete applications?

Thanks!!

If I understand the question correctly, it is about what information about a smooth projective variety you can extract from its derived category of coherent sheaves. Some result in this direction (besides those mentioned in the question statement)

• the dimension of a variety is uniquely determined by its derived category
• the canonical ring (and thus Kodaira dimension) is uniquely determined by the derived category.
• a smooth projective surface which is not elliptic, K3 or an abelian variety, is determined up to isomorphism by the derived category.
• For K3 surfaces, the Hodge structure on $H^*(X, \mathbb{Z})$ is determined up to isometry by the derived category (note that $H^2(X, \mathbb{Z})$ is NOT determined up to Hodge isometry by the derived category; if it were, then the derived category would be a perfect invariant for K3 surfaces by Global Torelli).

However, the derived category is not a perfect invariant. For example:

• abelian variety and its dual have equivalent derived categories.