Let A, B be dual abelian varieties. Let $B^{\natural }$ be the universal vectorial extension of B. Laumon and Rothstein independently lift the Fourier–Mukai transform to $D_A$-modules. They prove that this defines an equivalence of triangulated categories from the derived category $D_{\mathrm {coh}}^b(D_A)$ of coherent $D_A$-modules to the derived category $D_{\mathrm {coh}}^b(O_{B^{\natural }})$ of coherent sheaves on $B^{\natural }$. We extend their results to complex tori. As a replacement of coherent algebraic D-modules, we use Kashiwara’s good analytic D-modules. Moreover, to get an equivalence, we have to replace $O_{B^{\natural }}$ by a commutative $O_B$-algebra, which is locally a polynomial algebra over $O_B$. As an application, we recover the Matsushima–Morimoto theorem that on a complex torus, a vector bundle admits a connection if and only if it is translation invariant, and in this case, it admits an integrable connection. Moreover, we use the Laumon–Rothstein transform to show that the derived category $D_h^b(D_A)$ of holonomic D-modules is a rigid symmetric monoidal triangulated category under the convolution.

Let $X$ be a connected complex manifold and let $Z$ be a compact complex subspace of $X$. Assume that ${\rm Aut}(Z)$ is strongly Jordan. In this paper, we show that the automorphism group ${\rm Aut}(X,\, Z)$ of all biholomorphisms of $X$ preserving $Z$ is strongly Jordan. A similar result has been proved by Meng et al. for a compact Kähler submanifold $Z$ of $X$ instead of a compact complex subspace $Z$ of $X$. In addition, we also show some rigidity result for free actions of large groups on complex manifolds.

Given a compact Kähler manifold X, it is shown that pairs of the form $(E,\, D)$, where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on E, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

Given a torus action (T2, M) on a smooth manifold, the orbit map evx(t)=t·x for each x∈M induces a homomorphism ev*: $\mathbb Z$2→H1(M;$\mathbb Z$). The action is said to be Rank-k if the image of ev* has rank k ([les ]2) for each point of M. In particular, if ev* is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kähler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kähler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kähler if a lift of the metric to the universal covering space is conformal to a Kähler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kähler manifold M is Rank-1 provided that M has no Kähler structure.