An illustrative example: the moduli space $M$ of regular pentagons with edges of unit length. This embeds as an open, dense subset of a compact complex surface $\bar{M}$ with a canonical Kaehler form. This surface, a 4-fold blow-up of $\mathbb{CP}^2$, is not Calabi-Yau, however.

The compactified regular pentagon space $\bar{M}$ is the space of 5-tuples of unit vectors in $\mathbb{R}^3$ with centre of mass zero, modulo the diagonal action of $SO(3)$. Since we remember the order of the factors, typical points represent regular pentagons with labelled vertices. There are also points which represent an equilateral triangle together with a pair of antipodal points, and these non-pentagon points form ten 2-spheres in $\bar{M}$.

$\bar{M}$ has a natural symplectic structure, for which the ten 2-spheres are Lagrangian. Up to scale, there's a unique area-form on $S^2$ invariant under $SO(3)$, and the moment map $S^2\to \mathfrak{so}(3)^\ast \cong \mathbb{R}^3$ for this $SO(3)$-action is just the inclusion of $S^2$ into $\mathbb{R}^3$. The product $(S^2)^5$ carries a product symplectic form, again $SO(3)$-invariant, with moment map $\mu(x_1,\dots,x_5)=x_1+\dots + x_5\in \mathbb{R}^3$. The symplectic quotient $\mu^{-1}(0)/SO(3)$ is just $\bar{M}$.

The action of $SO(3)=PU(2)$ respects the complex structure, and $\bar{M}$ inherits a complex structure (by Kaehler reduction). It turns out to be isomorphic as a complex surface to a blow up of $\mathbb{CP}^2$ at four special points. See Seidel's Lectures on 4-dimensional Dehn twists, ex. 1.10. There's a natural action of the icosahedral group.

If one wanted pentagons defined by some other linear equation, say $a_1x_1+\dots + a_5x_5=0$, one would give the $S^2$-factors areas $a_i$.

One can also interpret $\bar{M}$ as an algebro-geometric (GIT) quotient of $(\mathbb{CP}^1)^5$ by $PSL_2(\mathbb{C})$. The quotient happens to be the Deligne-Mumford compactification $\bar{M}_{0,5}(\mathbb{C})$ of configurations of five points on $\mathbb{CP}^1$. The real points $\bar{M}_{0,5}(\mathbb{R})$ (the fixed points of an anti-holomorphic involution of $\bar{M}$) are also interesting: their connected components are polyhedral and are copies of the 2-dimensional Stasheff associahedron (a.k.a. pentagon).

Reference: F. Kirwan, "Cohomology of quotients in symplectic and algebraic geometry".

An illustrative example: the moduli space $M$ of regular pentagons with edges of unit length. This embeds as an open, dense subset of a compact complex surface $\bar{M}$ with a canonical Kaehler form. This surface, a 4-fold blow-up of $\mathbb{CP}^2$, is not Calabi-Yau (trivial canonical bundle) but Fano (ample anticanonical bundle).

The compactified regular pentagon space $\bar{M}$ is the space of 5-tuples of unit vectors in $\mathbb{R}^3$ with centre of mass zero, modulo the diagonal action of $SO(3)$. Since we remember the order of these vectors, typical points represent regular pentagons with a distinguished vertex ("start here") and adjacent edge ("go this way"). There are also points which represent an equilateral triangle together with a pair of antipodal points, and these non-pentagon points form ten 2-spheres in $\bar{M}$.

$\bar{M}$ has a natural symplectic structure, for which the ten 2-spheres are Lagrangian. Take the unique area-form on $S^2$, invariant under $SO(3)$, of total area 1 and inducing the complex orientation of $S^2=\mathbb{CP}^1$. The moment map $S^2\to \mathfrak{so}(3)^\ast \cong \mathbb{R}^3$ for the $SO(3)$-action is just the inclusion of $S^2$ into $\mathbb{R}^3$. The product $(S^2)^5$ carries the product symplectic form, again $SO(3)$-invariant, with moment map $\mu(x_1,\dots,x_5)=x_1+\dots + x_5\in \mathbb{R}^3$. The symplectic quotient $\mu^{-1}(0)/SO(3)$ is just $\bar{M}$.

The action of $SO(3)=PU(2)$ respects the complex structure of $(\mathbb{CP}^1)^5$, and $\bar{M}$ inherits a complex structure by Kaehler reduction. It turns out to be isomorphic as a Kaehler surface to a blow up of $\mathbb{CP}^2$ at four special points with a Fano Kaehler form (but I haven't thought through which points). See Seidel's Lectures on 4-dimensional Dehn twists, ex. 1.10. There's a natural action of the icosahedral group, permuting the $x_i$.

If one wanted pentagons defined by some other linear equation, say $a_1x_1+\dots + a_5x_5=0$, one would give the $S^2$-factors areas $a_i$.

One can also interpret $\bar{M}$ as an algebro-geometric (GIT) quotient of $(\mathbb{CP}^1)^5$ by $PSL_2(\mathbb{C})$. The quotient happens to be the Deligne-Mumford (or Grothendieck-Knudsen) compactification $\bar{M}_{0,5}(\mathbb{C})$ of configurations of five points on $\mathbb{CP}^1$. The real points $\bar{M}_{0,5}(\mathbb{R})$, the fixed points of an anti-holomorphic involution of $\bar{M}$, are also interesting: their connected components are polyhedral and are copies of the 2-dimensional Stasheff associahedron (a.k.a. pentagon).

References: 

F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry.

J.-C. Hausmann and A. Knutson, Polygon spaces and Grassmannians.

An illustrative example: the moduli space $M$ of regular pentagons with sides of unit length. This embeds as an open, dense subset of a compact complex surface $\bar{M}$ with a canonical Kaehler form. This surface, a 4-fold blow-up of $\mathbb{CP}^2$, is not Calabi-Yau, however.

The compactified regular pentagon space $\bar{M}$ is the space of 5-tuples of unit vectors in $\mathbb{R}^3$ with centre of mass zero, modulo the diagonal action of $SO(3)$. Since we remember the order of the factors, typical points represent regular pentagons with labelled vertices. There are also points which represent an equilateral triangle together with a pair of antipodal points, and these non-pentagon points form ten 2-spheres in $\bar{M}$.

$\bar{M}$ has a natural symplectic structure, for which the ten 2-spheres are Lagrangian. Up to scale, there's a unique area-form on $S^2$ invariant under $SO(3)$, and the moment map $S^2\to \mathfrak{so}(3)^\ast \cong \mathbb{R}^3$ for this $SO(3)$-action is just the inclusion of $S^2$ into $\mathbb{R}^3$. The product $(S^2)^5$ carries a product symplectic form, again $SO(3)$-invariant, with moment map $\mu(x_1,\dots,x_5)=x_1+\dots + x_5\in \mathbb{R}^3$. The symplectic quotient $\mu^{-1}(0)/SO(3)$ is just $\bar{M}$.

The action of $SO(3)=PU(2)$ respects the complex structure, and $\bar{M}$ inherits a complex structure (by Kaehler reduction). It turns out to be isomorphic as a complex surface to a blow up of $\mathbb{CP}^2$ at four special points. See Seidel's Lectures on 4-dimensional Dehn twists, ex. 1.10. There's a natural action of the icosahedral group.

If one wanted pentagons defined by some other linear equation, say $a_1x_1+\dots + a_5x_5=0$, one would give the $S^2$-factors areas $a_i$.

One can also interpret $\bar{M}$ as an algebro-geometric (GIT) quotient of $(\mathbb{CP}^1)^5$ by $PSL_2(\mathbb{C})$. The quotient happens to be the Deligne-Mumford compactification $\bar{M}_{0,5}(\mathbb{C})$ of configurations of five points on $\mathbb{CP}^1$. The real points $\bar{M}_{0,5}(\mathbb{R})$ (the fixed points of an anti-holomorphic involution of $\bar{M}$) are also interesting: their connected components are polyhedral and are copies of the 2-dimensional Stasheff associahedron (a.k.a. pentagon).

Reference: F. Kirwan, "Cohomology of quotients in symplectic and algebraic geometry".

An illustrative example: the moduli space $M$ of regular pentagons with edges of unit length. This embeds as an open, dense subset of a compact complex surface $\bar{M}$ with a canonical Kaehler form. This surface, a 4-fold blow-up of $\mathbb{CP}^2$, is not Calabi-Yau, however.

The compactified regular pentagon space $\bar{M}$ is the space of 5-tuples of unit vectors in $\mathbb{R}^3$ with centre of mass zero, modulo the diagonal action of $SO(3)$. Since we remember the order of the factors, typical points represent regular pentagons with labelled vertices. There are also points which represent an equilateral triangle together with a pair of antipodal points, and these non-pentagon points form ten 2-spheres in $\bar{M}$.

$\bar{M}$ has a natural symplectic structure, for which the ten 2-spheres are Lagrangian. Up to scale, there's a unique area-form on $S^2$ invariant under $SO(3)$, and the moment map $S^2\to \mathfrak{so}(3)^\ast \cong \mathbb{R}^3$ for this $SO(3)$-action is just the inclusion of $S^2$ into $\mathbb{R}^3$. The product $(S^2)^5$ carries a product symplectic form, again $SO(3)$-invariant, with moment map $\mu(x_1,\dots,x_5)=x_1+\dots + x_5\in \mathbb{R}^3$. The symplectic quotient $\mu^{-1}(0)/SO(3)$ is just $\bar{M}$.

The action of $SO(3)=PU(2)$ respects the complex structure, and $\bar{M}$ inherits a complex structure (by Kaehler reduction). It turns out to be isomorphic as a complex surface to a blow up of $\mathbb{CP}^2$ at four special points. See Seidel's Lectures on 4-dimensional Dehn twists, ex. 1.10. There's a natural action of the icosahedral group.

If one wanted pentagons defined by some other linear equation, say $a_1x_1+\dots + a_5x_5=0$, one would give the $S^2$-factors areas $a_i$.

One can also interpret $\bar{M}$ as an algebro-geometric (GIT) quotient of $(\mathbb{CP}^1)^5$ by $PSL_2(\mathbb{C})$. The quotient happens to be the Deligne-Mumford compactification $\bar{M}_{0,5}(\mathbb{C})$ of configurations of five points on $\mathbb{CP}^1$. The real points $\bar{M}_{0,5}(\mathbb{R})$ (the fixed points of an anti-holomorphic involution of $\bar{M}$) are also interesting: their connected components are polyhedral and are copies of the 2-dimensional Stasheff associahedron (a.k.a. pentagon).

Reference: F. Kirwan, "Cohomology of quotients in symplectic and algebraic geometry".