Peng Zhou

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====== What can you say about toric CY variety?  ======

Let $X$ be a toric CY variety, by which we mean $X$ has a toric fan $\Sigma \In N_\R$, with ray generators $v_\rho$ for each ray $\rho$, lying on a affine hyperplane of distance $1$. That is, there is $m \in M$, such that $\la v_\rho, m \ra = 1$ for all $\rho$. That $m$ gives me a distinguished function $W: X \to \C$ (modulo a constant factor).

However, it is wrong to say $Coh(X)$ is a d-CY category, for one thing, this category is not hom finite. If given $E, F$ two coherent sheaves, we don't have $Hom(E, F) \cong Hom(F, E[d])^\vee$. (Just take the example of $X = \C$). One cure is to fix some compact subset (maybe torus invariant) $Z \In X$, and require that $Coh_{Z}(X)$, just sheaves supported along $Z$, this hom-finite sub-category would be OK. Or we can just blindly say, look at the category of 'proper' module? 

What does homological mirror symmetry say? Given a Lagrangian skeleton $\La$, we can ask for those 'infinitesimally wrapped Lagrangians' or sheaves, they are the sheaves with finite rank stalks (hence microlocal stalks) everywhere. 

The question now is: why Fukaya category with such skeleton gives CY property? For example, for $Tot(O_{P^1}(n))$, only for $n=-2$ it is CY. But it is kinda hard to see from the fan. 

===== Example of $\C$, $\P^1$, $\C/\mu_2$ =====
Let's consider the basic example of $\C$. If we look at $\Coh(\C / \C^*)$, the equivariant category,  then $O_0(0)$, endomorphism is only $\C$ in degree $0$. It is not $1$-CY. 

However, if we do the non-equivariant category, we get $End(O_0) = \C \oplus \C[-1]$.

Similarly, if we do mirror to $\C/\mu_2$, it is not CY. 

Well, a necessary and sufficient condition would be, the Serre functor, which is wrap past the stop, then shift down by $d$, is an equivalence.