A variety is the moduli space of structure sheaves of points

Question

In the last paragraph of the first page of this paper, it is mentioned that an $n$-dimensional Calabi-Yau manifold $X$ is the moduli space of structure sheaves of its points and I am not really sure what this exactly means. Is this trying to mention the functor-of-points viewpoint or is there some other result? Thank you in advance.

Answer 1

This line is nothing special to Calabi-Yau manifolds. If $\check{X}$ is any complex manifold, we can say that $\check{X}$ parametrizes the structure sheaves of points of $\check{X}$ in the (tautological) sense that there is a bijection \begin{align*} \{\mathcal O_p: p \in \check{X}\} \leftrightarrow \{p: p \in \check{X}\} = \check{X}. \end{align*} If you want to be extra-formal, you can upgrade this to an isomorphism of functors of points. Namely, for each morphism $f: T \to \check{X}$, we can take its graph $\Gamma(f) \subset \check{X} \times T$; the structure sheaves of these graphs will parametrize the morphisms $f$. (The special case $T = *$ recovers the earlier bijection.)

What does this give you in the context of the paper? We're assuming that there is an equivalence of categories $$ \mathcal{Fuk}(X) \cong D^b(\mathrm{Coh}(\check{X})) $$ identifying certain features on each side with each other (cf. the table in the paper). In particular, the sheaves $\mathcal O_p$ form a family of objects of the right-hand category, parametrized by $\check{X}$. So there must be a corresponding family of objects of the left-hand category, also parametrized by $\check{X}$. This is the family of Lagrangians mentioned in the following sentence of the paper. Then one can identify the Floer cohomology groups of these objects with the Ext groups of the sheaves $\mathcal O_p$.