Applications of derived categories to "Traditional Algebraic Geometry"

Question

I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical problems. For example: moduli spaces problems, automorphism groups of varieties, birational classification of varieties, minimal model program and so on.

Furthermore, I would also be interested to know about theorems that are more aesthetically pleasing being stated in the derived category language. For instance, I think Serre duality can be generalised to singular varieties through derived categories.

Any reference about results is very welcome.

Answer 1

I think that a good example of the usefulness of the Derived Category of coherent sheaves for studying classical questions is the recent preprint by Soheyla Feyzbakhsh

Mukai's program (reconstructing a K3 surface from a curve) via wall-crossing,

where the author uses wall-crossing with respect to Bridgeland stability conditions in order to solve the classical Mukai problem of reconstructing a K3 surface from its hyperplane section.

Answer 2

The global Torelli Theorem for cubic fourfolds says the following. Let $X_1 \subset \mathbb{P}^5$ and $X_2 \subset \mathbb{P}^5$ be smooth cubic fourfolds. The fourfolds $X_1$ and $X_2$ are isomorphic if and only if there exists a Hodge isometry between $H^4(X_1,\mathbb{Z})_{pr}$ and $H^4(X_2, \mathbb{Z})_{pr}$. This was originally proved by Voisin (voir Théorèmes de Torelli pour les cubiques de $\mathbb{P}^5$) using a subtle and delicate analysis of the restriction of the period mapping to the family of cubics containing a plane.

In 2019, Huybrechts and Rennemo gave a new proof of the Torelli Theorem for cubic fourfolds using derived categories (see Hochschild cohomology versus the Jacobian ring and the Torelli theorem for cubic fourfolds). The strategy of their proof is the following:

$\bullet$ We first assume that $X_1$ and $X_2$ are very general in the moduli spaces of cubic fourfolds.

$\bullet$ The derived category $D^b(X_1)$ and $D^b(X_2)$ contains $K3$-categories, say $\mathcal{A}_1$ and $\mathcal{A}_2$.

$\bullet$ Since $X_1$ and $X_2$ are very general, a result by Addington and Thomas ensures that there exists actual $K3$ surfaces $S_1$ and $S_2$ such that $\mathcal{A}_i \simeq D^b(S_i)$.

$\bullet$ The very specific Hodge structure of a cubic fourfolds show that any isometry between $H^4(X_1,\mathbb{Z})_{pr}$ and $H^4(X_2,\mathbb{Z})_{pr}$ induces an isometry between the spaces $H^{\bullet}(S_1,\mathbb{Z})$ and $H^{\bullet}(S_2,\mathbb{Z})$ (endowed with the Mukai pairing). Orlov's derived Torelli Theorem then implies that $D^b(S_1) \simeq D^b(S_2)$.

$\bullet$ Huybrechts and Rennemo show that this equivalence commutes with the auto-equivalence given by twisting sheaves by $\mathcal{O}(1)$.

$\bullet$ They deduce that there is an isomorphism $\operatorname{HH}(\mathcal{A}_1, \mathcal{O}(1)) \simeq \operatorname{HH}(\mathcal{A}_2, \mathcal{O}(1))$, where $\operatorname{HH}(\mathcal{A}_i, \mathcal{O}(1))$ is a variant of the Hochschild-cohomology ring of $\mathcal{A}_i$ with coefficients in $\mathcal{O}(1)$.

$\bullet$ They show that there is a ring isomorphism $\operatorname{HH}(\mathcal{A}_i, \mathcal{O}(1)) \simeq J(X_i)$, where $J(X_i)$ is the Jacobian ring of $X_i$.

$\bullet$ The Mather-Yau Theorem allows to conclude that $X_1$ and $X_2$ are isomorphic.

This proof is not necessarily way shorter than the original one by Voisin, but it suggests a shift on perspective that has been very fruitful since. Torelli type Theorems for hypersurfaces of dimension $n$ which degree $d$ divides $n+2$ should be stated using some variant of the Hochshild cohomology (with coefficients) of the Kuznetsov component of the derived category of the hypersurface. These categorical Torelli type results/conjectures have been studied quite carefully recently.