Kähler structures on complex torus
Let \(T\) be the \(n\)-dimensional real torus \(\mathbb R^{n}/\mathbb Z^{n}\) and \(\beta T_{\mathbb C}=\mathbb R^{n}\times\mathbb R^{n}/\mathbb Z^{n}=(\mathbb C^{*})^{n}\) its complexification. Take a Kähler form \(\omega\) on \(T_{\mathbb C}\) which is invariant under the natural \(T\)-action on \(T_{\mathbb C}\). A straightforward calculation shows that \(\omega=\sqrt{-1}\partial\overline{\partial}F\) with a \(T\)-invariant function \(F\in C^{\infty}(\mathbb R^{n})\). It is shown that the \(T\)-action on \(T_{\mathbb C}\) preserving \(\omega\) is Hamiltonian and that its \(T\)-invariant moment map, considered as a map \(\Phi:\mathbb R^{n}\to \mathbb R^{n}\), satisfies the equation \(\varphi=\frac{1}{2}\operatorname {grad}F\). The Kähler form \(\omega\) represents the Chern class of the trivial holomorphic line bundle \(L\) on \(T_{\mathbb C}\). Let \(\nabla\) be a connection on \(L\) with curvature form \(\omega\). The \(T\)-action on \(T_{\mathbb C}\) can be lifted to a linear representation of \(T\) on \(H(L)\), the vector space of all holomorphic sections \(s\) in(\(L,\nabla\)), which is in a natural way isomorphic to the vector space \(\mathcal O(T_{\mathbb C})\) of holomorphic functions on \(T_{\mathbb C}\) (\(s\in H(L)\Leftrightarrow s\in C^{\infty}(T_{\mathbb C})\) and \(\nabla_{v}s=0\) for every antiholomorphic vector field \(v\) on \(T_{\mathbb C}\)). Therefore each irreducible representation of \(T\) occurs with multiplicity one in \(H(L)\). Given a \(T\)-invariant Hermitian metric on \(L\), the subspace \(H_{\omega}\) of \(L^{2}\)-sections \(s\in H(L)\) is \(T\)-invariant. The weights of \(T\) can be identified with the elements of the lattice \(\mathbb Z^{n}\subset \mathbb R^{n}\backsimeq {\mathfrak t}^{*}\). It is shown that the eigenspace \(H_{\lambda}(L),\lambda\in\mathbb Z^{n}\), is contained in \(H_{\omega}\) if and only if \(\lambda\in\Phi(\mathbb R^{n})\).
scientific article; zbMATH DE number 1542079