On the algebraic dimension of twistor spaces over the connected sum of four complex projective planes

We study the algebraic dimension of twistor spaces of positive type over 4\bbfP^2. We show that such a twistor space is Moishezon if and only if its anticanonical class is not nef. More precisely, we show the equivalence of being Moishezon with the existence of a smooth rational curve having negative intersection number with the anticanonical class. Furthermore, we give precise information on the dimension and base locus of the fundamental linear system $|{-1/2}K|$. This implies, for example, $\dim|{-1/2}K|\leq a(Z)$. We characterize those twistor spaces over 4\bbfP^2, which contain a pencil of divisors of degree one by the property $\dim|{-1/2}K| = 3$.Comment: 23 pages LaTeX 2

Homological Mirror Symmetry in Dimension One

In this paper we complete the proof began by A. Polishchuk and E. Zaslow (math.AG/9801119) of a weak version of Kontsevich's homological mirror symmetry conjecture for elliptic curves. The main difference to the work of Polishchuk and Zaslow is that we consider morphisms between any pair of objects, not only in the transversal case. This enables us to show the conjectured equivalence of categories.Comment: 20 pages, LaTeX2

Derived categories of irreducible projective curves of arithmetic genus one

We investigate the bounded derived category of coherent sheaves on irreducible singular projective curves of arithmetic genus one. A description of the group of exact auto-equivalences and the set of all t-structures of this category is given. We describe the moduli space of stability conditions, obtain a complete classification of all spherical objects in this category and show that the group of exact auto-equivalences acts transitively on them. Harder-Narasimhan filtrations in the sense of Bridgeland are used as our main technical tool.Comment: 44 pages; minor modifications; to appear in Compositio Mat

Homological Mirror Symmetry in Dimension One

In this paper we complete the proof began by A. Polishchuk and E. Zaslow of a weak version of Kontsevich's symmetry conjecture for elliptic curves