On the Cone of Kählerian Infinitesimal Deformations for Complex Tori

For any compact Kähler manifold M , the cone of Kählerian in-finitesimal deformations of M , say KID(M), is by definition the subset of H 1 (M, Θ) (the 1 st cohomology of the holomorphic tangent sheaf of M) consisting of those elements in the images of Kodaira-Spencer maps for the germs of complex analytic families (X, M) → (S, 0) such that X carries a Kähler metric. If M is a complex torus of dimension ≥ 2, KID(M) will be described in terms of the canonical matrix representation of H 1 (M, Θ). Introduction. In the theory of deformations of complex structures initiated by Kodaira and Spencer, a basic fact established by Kuranishi [Kur62] says that, given any compact complex manifold M , there exist a complex analytic subset S of a neighborhood of the origin 0 of H 1 (M, Θ) and a complex analytic family π : M → S with π −1 (0) = M , such that it is complete in the sense that any complex structure sufficiently close to M on the fixed underlying differentiable manifold of M arises as a fiber of π, and effective in the sense that the Kodaira-Spencer map is injective at 0. It was first noted by Schumacher [Sch84] that, given any Kähler metric say g 0 on M , there exists uniquely a maximal analytic subgerm of (S, 0), say (S ′ , 0), such that the preimage of S ′ by π carries a Kähler metric g with g| M = g 0. (See also [Fuj84] and [FS90].) By the cone of Kählerian infinitesimal deformations for M , denoted by KID(M) in short, we shall mean the collection of the images in H 1 (M, Θ) of the (Zariski) tangent vectors of S ′ by Kodaira-Spencer maps for all possible S ′ , as g 0 runs through the set of all Kähler metrics on M. A recent paper of Berndtsson [Ber09] gives some information on the geometry of KID(M). Namely, given any complex analytic family π ′ : X → S ′′ such that X is Kähler and S ′′ is smooth, he has shown that the direct image by π ′ of the relative canonical sheaf ω X/S ′′ (:= ω X ⊗ (π ′ * ω S ′′) *) is Nakano semipositive with respect to a canonically defined fiber metric, the so-called L 2 …