Table of Contents
Notes
Batyrev-Borisov Construction
This note is my attempt to understand how the mirror construction works, and how it compares with the monomial-divisor toric correspondence.
Hitchin Fibration – Learning Roadmap
The Hitchin fibration is one of the central structures in modern geometry. It connects:
- Higgs bundles
- integrable systems
- non-abelian Hodge theory
- geometric Langlands
- mirror symmetry
This page collects a recommended path to learn the subject, with references.
Some prediction about mirror symmetry
Forget about 3d MS for a second. Toric HMS suppose to be easy.
More about CY completion
In this note, I want to record how to do the CY completion for a smooth and proper category $C$. Let $S: C \to C$ denote the Serre functor, which has the property that $$ Hom(x, y) = Hom(y, Sx)^\vee. $$
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.