Cohomology groups of sections of homogeneous line bundles over a toroidal group (Q2810583)

Let \(\Lambda\subset\mathbb{C}^n\) be a lattice of rank \((n+m)\) with \(0\leq m\leq n\). Then \(X:= \mathbb{C}^n/\Lambda\) is called a toroidal group if all holomorphic functions on \(X\) are constant. In the special case when \(m=n\), \(X\) is nothing but an \(n\)-dimensional compact complex torus. Also it is known that the toroidal group \(X\) has the structure of a principal \((\mathbb{C}^*)^{n-m}\)-bundle \(\mu: X\to\mathbb{T}\), where \(\mathbb{T}\) is an \(m\)-dimensional compact complex torus.NEWLINENEWLINE Notice that any homogeneous line bundle \({\mathcal L}\) on \(X\) is topologically trivial; furthermore \({\mathcal L}\) is given by a representation \(\varrho: \Lambda\to\mathbb{C}^*\) of \(\Lambda\). In a seminal paper by \textit{Ch. Vogt} [J. Reine Angew. Math. 335, 197--215 (1982; Zbl 0485.32020)], a precise list of equivalent conditions regarding line bundles on \(X\), the cohomology groups of \(X\) and the lattice \(\Lambda\) was established. Among themNEWLINENEWLINENEWLINENEWLINE (a) all topologically trivial line bundles \({\mathcal L}\) on \(X\) are homogeneous;NEWLINENEWLINENEWLINENEWLINE (b) conditions on the approximation of \(\Lambda\) by a rational lattice;NEWLINENEWLINENEWLINENEWLINE (c) \(\dim_{C} H^1(X, O_X)<\infty\).NEWLINENEWLINENEWLINENEWLINE This is motivated by the fact that toroidal groups \(X\) are pseudoconvex manifolds. In contrast to strongly pseudoconvex manifolds \textit{B. Malgrange} [C. R. Acad. Sci., Paris, Sér. A 280, 93--95 (1975; Zbl 0298.32004)] exhibited a two-dimensional toroidal group \(X\) such that condition (c) above is violated.NEWLINENEWLINENEWLINENEWLINE In this paper, the author provides a condition called \((H)_S\) on the lattice \(\Lambda\) analogous to (b) and proves the following main result:NEWLINENEWLINENEWLINENEWLINE Theorem. Let \({\mathcal L}\) be a (non-analytically trivial) homogeneous line bundle on \(X\). Assume that \((H)_S\) is satisfied. Then either:NEWLINENEWLINENEWLINENEWLINE (i) \(H^i(X, O_X({\mathcal L}))= 0\) for \(i>0\), orNEWLINENEWLINENEWLINENEWLINE (ii) \(H^i(X,O({\mathcal L}))\cong H^i(T,O_{{\mathcal T}})\) for \(i>0\).NEWLINENEWLINENEWLINENEWLINE Assume that if \((H)_S\) is not satisfied. Then one hasNEWLINENEWLINENEWLINENEWLINE (iii) \(H^i(X,O({\mathcal L}))\) is a non-Hausdorff Fréchet space for any \(i\) with \(1\leq i\leq m\).NEWLINENEWLINENEWLINENEWLINE Notice that in the special case when \(X\) is a compact \(n\)-dimensional complex torus and the topological trivial line bundle is equivalent to a homogeneous line bundle, then, as pointed out by the author, one recovers the case (i) above.