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This isn't a terribly refined question, but more broad-brush: are there nice results on explicit mirror pairs of certain Calabi-Yau surfaces? In particular, I'm curious if we know the mirror partners of the smooth, non-compact resolutions of the singular surfaces $\mathbb{C}^{2}/\mathbb{Z}_{N+1}$, as well as $\mathbb{C}^{2}$ itself. I was hoping someone could sort of briefly summarize any big results in this area, maybe also with sources. Also, I can see that there's an extensive literature on mirror symmetry of K3 surfaces, but I'm having a tough time navigating it, for now.

(What got me thinking about this is the fact that the elliptic genus apparently satisfies

$$\text{Ell}_{y,q}(X) = (-1)^{\text{dim}X}\text{Ell}_{y,q}(\tilde{X})$$

where $X$ and $\tilde{X}$ are mirror pairs. Hence, for surfaces the elliptic genus coincides for the mirror partners. By the Dijkgraaf-Moore-Verlinde-Verlinde formula, this would seem to imply that the generating function of the elliptic genera of the Hilbert Schemes of points also coincides for both partners, which seems very non-trivial to me.)

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A2A.

I'm going to focus on the more straightforward mirror symmetry picture here: that of K3 surfaces. Recall that mirror symmetry is mathematically an exchange of symplectic geometry and algebraic geometry. The original paper of Greene-Plesser Duality in Moduli Spaces noted that the deformations of complex structure are related to the deformations of Kähler structure of the mirror. Cohomologically, this was noted to be for the mirror pair $X$ and $X^\vee$: $$ \dim H^1(X, T_X) = \dim H^1(X^\vee, \Omega_{X^\vee}) $$ where $H^1(X, T_X)$ is the space of complex deformations and $H^1(X^\vee, \Omega_{X^\vee})$ contains the Kähler cone.

In general this is then generalized to a flip in Hodge numbers: $h^{p,q}(X)=h^{n-p,q}(X^\vee)$ for $n$-dimensional Calabi-Yau varieties.

In surfaces, this is more subtle since all K3's have the same Hodge diamond and $h^{1,1}(X) = h^{2-1,1}(X^\vee)$ for all K3s. Dolgachev-Nikulin mirror symmetry gives an understanding of this. Dolgachev proposed that using the Picard lattice $\text{Pic}(X)$ and embed that into the integral cohomology of a K3 $H^2(X, \mathbb{Z})$. We have a canonical lattice $H^2(X, \mathbb{Z}) = E_8(-1) \oplus E_8(-1) \oplus U^3$ for the integral cohomology for all K3 surfaces [see Huybrechts' online notes on K3 surfaces for details], where $E_8(-1)$ is the $E_8$ lattice with negative signature and $U$ is the hyperbolic lattice. One has a primitive embedding: $$ M:=\text{Pic}(X) \hookrightarrow H^2(X, \mathbb{Z}). $$ Take the mirror lattice $M^\perp$ that is defined to be the orthogonal complement of $\text{Pic}(X)\oplus U$ in $E_8(-1) \oplus E_8(-1) \oplus U^3$, where $U$ is the hyperbolic lattice associated to $H^{2,0} \oplus H^{0,2}$ for the K3.

As you may know, mirror symmetry happens in families. Here, we define a moduli space $\mathcal{F}_M$ to be the family of all $M$-polarized K3 surfaces, i.e., all K3 surfaces $X$ so that there is a primitive embedding of lattices $\iota: \text{Pic}(X) \hookrightarrow M$. Analogously we can define $\mathcal{F}_{M^\perp}$ to be the family polarized by the mirror lattice. These are the mirror families that Dolgachev proposed in his original paper (http://www.math.lsa.umich.edu/~idolga/mirror96.pdf). The view here is that the orthogonal to the Picard lattice is the transcendental lattice which should be viewed as more complex algebro-geometric data where the Picard lattice is associated to Kähler forms.

This is just one viewpoint. A whole introductory roadmap was written by Ueda that you might enjoy (https://arxiv.org/pdf/1407.1566.pdf).

This is the compact case in the classical mathematical sense of mirror symmetry. Various results have been found in the line of homological mirror symmetry, a categorical interpretation using derived categories and Fukaya categories (Seidel proved this in https://arxiv.org/abs/math/0310414 for quartic surfaces).

As for for open Calabi-Yau varieties as you started to talk about, I think the most current viewpoint of those is that of the log Calabi-Yau pair. This viewpoint has been developed by Gross-Hacking-Keel here: https://arxiv.org/pdf/1106.4977.pdf but this is a bit of a bear to read for the uninitiated.

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