Information from the derived categories of coherent sheaves

Question

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!!

Answer 1

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.

A nice place to read about this stuff is Hyubrecht's "Fourier--Mukai transforms in algebraic geometry".

As for the applications, I do not know much about them but here is one example: Bridgeland had a conjecture describing the group of autoequivalences of the derived category of a K3 surface. This conjecture has been proved in the Picard rank 1 case by Bayer--Bridgeland; Sheridan--Smith inferred from this that the symplectic Torelli group of some K3 surfaces is infinitely generated. In general it is very hard to say anything about symplectic Torelli group. Though one should note that in this example, there were a lot of extra non-trivial ideas (like homological mirror symmetry) so maybe it is not fair to say that this result is an application of theorems about derived categories.