Degenerations of 2-dimensional Tori

In this paper we classify the possible degenerate fibers which can occur in a semistable degeneration of two-dimensional tori under the assumption that the canonical bundle of the total space of the family is trivial. 1. Preliminaries Let π : X → ∆ be a proper map of a Kähler manifold X onto the unit disk ∆ = {t ∈ C :| t |< 1}, such that the fibers Xt are nonsingular compact complex manifolds for every t 6= 0. We call π a degeneration and the fiber X0 = π (0) the degenerate fiber. Definition 1.1. A map ψ : Y → ∆ is called a modification of a degeneration π if there exists a birational map f : X → Y such that ψ = π ◦ f and ψ is an isomorphism outside of the degenerate fiber. A degeneration is called semistable if the degenerate fiber is a reduced divisor with normal crossings. Not every degeneration can be modified to a semistable one. Nonetheless, it is possible to reduce any degeneration to a semistable one after a base change according to Mumford’s theorem ([1]). Definition 1.2. The polyhedron Π(V ) of a variety with normal crossings V = V1 + · · · + Vn, dimVi = d is the polyhedron whose vertices correspond to the irreducible components Vi and the vertices Vi1 , · · · , Vik form a (k − 1)−simplex if Vi1 ∩ · · · ∩ Vik 6= 0. 2. Basic Tools Let π : X → ∆ be a semistable degeneration of surfaces whose degenerate fiber is X0 = V1 + V2 + · · ·+ Vn. We will state some results from [2]. Lemma 2.1. ([2]) Let C = Vi ∩ Vj be a double curve of a semistable degeneration of surfaces. Then (C2)Vi + (C 2)Vj = −TC, where TC is the number of triple points of the fiber X0 on C. Lemma 2.2. ([2]) Let T be the number of all triple points of π, then