Compact Calabi–Yau manifold conventions

The inverse of a holomorphic isomorphism $\tau :K_X\to {\mathcal{O}}_X$ sends the constant section $1$ to a nowhere-zero holomorphic section $\mathrm{\Omega }={\tau }^{-1}(1)$. Conversely, for a nowhere-zero holomorphic section $\mathrm{\Omega }$ of $K_X$, the map $f\mapsto f\mathrm{\Omega }$ is a fiberwise holomorphic line-bundle isomorphism ${\mathcal{O}}_X\to K_X$; its local inverse is holomorphic because the coefficient of $\mathrm{\Omega }$ in every local frame is holomorphic and nowhere zero.

Since $X$ is a connected manifold, it is locally path-connected and hence path-connected. By [F3], choose a path-connected CW complex $Y$ and a homotopy equivalence $f:X\to Y$. The trivial line ${\epsilon }_Y^1$ is numerable by its single global trivialization, and [F2] gives $c_1({\epsilon }_Y^1)=0$. Its pullback is the trivial line ${\epsilon }_X^1$, so naturality in [F2], whose target is the CW complex $Y$ and whose source is the CW-type space $X$, gives $c_1({\epsilon }_X^1)=f^{\ast }{c}_1({\epsilon }_Y^1)=0$. The specified holomorphic isomorphism identifies $K_X$ with ${\epsilon }_X^1$ as a complex bundle; by [F6], Chern classes on this CW-type base depend only on the bundle isomorphism class. Thus the integral class $c_1(K_X)$ vanishes.

For $n\ge 2$, use the full lattice $\mathrm{\Lambda }={\sum }_{a=1}^{2n}\mathbb{Z}{\lambda }_a$ generated by a real basis $({\lambda }_a)$ of ${\mathbb{C}}^n\cong {\mathbb{R}}^{2n}$. The real-linear map sending the standard basis to $({\lambda }_a)$ sends ${\mathbb{Z}}^{2n}$ onto $\mathrm{\Lambda }$, so $\mathrm{\Lambda }$ is discrete and bounded sets meet it finitely. If a compact set $K$ meets a translate $K+\lambda$, then $\lambda \in K-K$, a bounded set with only finitely many lattice points; this gives proper discontinuity, while freeness follows from the translation action. Distinct orbits have positive distance because only finitely many lattice translates of a fixed representative lie within any fixed bounded distance of another; small disjoint balls therefore give a Hausdorff quotient. A ball smaller than half the shortest nonzero lattice length has disjoint nonidentity translates and gives a quotient chart; the transition maps are translations, hence holomorphic. The closed fundamental parallelepiped is compact and surjects onto $X_0={\mathbb{C}}^n\mathrm{/}\mathrm{\Lambda }$. The standard Euclidean Hermitian metric and ${\omega }_0={\sum }_{j}d{x}_j\wedge d{y}_j$ descend; ${\omega }_0$ is positive and closed. The translation-invariant holomorphic form $d{z}^1\wedge \cdots \wedge d{z}^n$ descends nowhere zero, so $X_0$ satisfies the broad convention.

For any piecewise smooth curve $\gamma :[0,1]\to X$, let $V_v$ be its parallel section with initial vector $v$. For real scalars $a,b$, the section $a{V}_v+b{V}_w$ is continuous across corners and solves the homogeneous linear equation in [F5] on each smooth piece, with initial value $av+bw$. Uniqueness gives $V_{av+bw}=a{V}_v+b{V}_w$, hence $P_{\gamma }$ is real-linear. Define $Q:T_{\gamma (1)}X\to T_{\gamma (0)}X$ by solving the same equation with prescribed value at $1$ and evaluating at $0$, as permitted by [F5]. Uniqueness at either endpoint gives $Q{P}_{\gamma }=I$ and $P_{\gamma }Q=I$. Thus every loop transport belongs to $\mathrm{GL}(T_{x}X)$, so the subgroup generated by these maps is well-defined.

Let $H=(h_{j\bar{k}})$ be the metric matrix from [F1], and write the chosen nowhere-zero form in a holomorphic chart as $\mathrm{\Omega }=f(z)d{z}^1\wedge \cdots \wedge d{z}^n$. The induced determinant metric on $K_X$ gives $\mathrm{\parallel }\mathrm{\Omega }{\mathrm{\parallel }}^2=\mathrm{\mid }f{\mathrm{\mid }}^2\mathrm{/}\det H$. The globally defined real function $u=\log \mathrm{\parallel }\mathrm{\Omega }{\mathrm{\parallel }}^2$ satisfies $\mathrm{\partial }\bar{\mathrm{\partial }}\log \mathrm{\mid }f{\mathrm{\mid }}^2=0$ locally, so [F1] gives $\rho (\omega )=i\mathrm{\partial }\bar{\mathrm{\partial }}u=d(i\bar{\mathrm{\partial }}u)$. Because $\rho$ is real, $\mathrm{Re}(i\bar{\mathrm{\partial }}u)$ is a real primitive. Thus $[\rho ]=0$ in real de Rham cohomology, and the Chern–Weil identity in [F1] yields $c_1(T_X^{1,0}{)}_{\mathbb{R}}=0$.

On ${\mathbb{C}}^n$, define $D_X({\sum }_a{f}_a{E}_a)={\sum }_{a}X(f_a)E_a$ for the constant real frame $(E_a)=({\mathrm{\partial }}_{x_1},{\mathrm{\partial }}_{y_1},\dots ,{\mathrm{\partial }}_{x_n},{\mathrm{\partial }}_{y_n})$. Translation chart changes have identity differential, so $D$ descends to $X_0$. The fields $E_a$ commute, giving zero torsion on the frame; since torsion is tensorial it vanishes. The metric coefficients $g_0(E_a,E_b)={\delta }_{ab}$ are constant and $D{E}_a=0$, so the connection is metric-compatible. By uniqueness in [F4], $D$ is the Levi-Civita connection. A parallel section has constant coefficients in the global parallel frame, so [F5] makes each loop transport the identity. For $n\ge 2$, $SU(n)$ contains the nonidentity matrix $\mathrm{diag}(e^{it},e^{-it},1,\dots ,1)$ for $t\mathrm{\notin }2\pi \mathbb{Z}$, so the holonomy is a proper subgroup of $SU(n)$. When $n=1$, $SU(1)$ is trivial and this example does not separate the two groups.