Showing papers on "Calabi conjecture published in 2002"

Reflection of sheaves on a Calabi–Yau variety

Abstract: In this paper we study the reflection of stable sheaves on Calabi-Yau varieties and its effect on the moduli space. It is shown that the reflection defines isomorphisms between the Brill-Noether loci of moduli spaces. Introduction. Let E be a torsion-free sheaf on a smooth projective K3 surface X and let tp : H(X, E) 0 Ox —> E denote the natural evaluation map. If (p is either injective or surjective, then its cokernel or kernel is called the reflection of E. The reflection functor was first introduced by Mukai ([Mu]) and since then it has been exploited for the study of the moduli space of stable sheaves on K3 surfaces ([Ma], [N],[Y]). It seems significant to consider the reflection functor on higher dimensional CalabiYau variety X, in view of Kontsevich's homological mirror conjecture which predicts the existence of equivalence of the derived category D(X) of coherent sheaves on X and the derived Fukaya category of its mirror. Inspired by the conjecture, Seidel and Thomas recently introduced an auto equivalence T? : D(X) —> D(X) called the twist functor with respect to a spherical object f ([ST]). For J £ D(X), Te(!F) is defined to L be the cone of the map Hom^,.?) 0 £ —> J, which coincides with Mukai's reflection in case £ = OxHowever, the problem how this functor is related to the stability of sheaves has not been addressed. In this paper we study the effect of the reflection functor on the moduli space of stable sheaves on higher dimensional Calabi-Yau varieties instead of the derived category. We shall show that under suitable minimality assumption on the first Chern classes, the reflection preserves the stability of sheaves on arbitrary smooth projective varieties. Further we define the Brill-Noether locus of the moduli space of sheaves on Calabi-Yau varieties and prove that the reflection induces isomorphisms between the Brill-Noether loci for different Mukai vectors. This is a higher dimensional generalization of the results in [Ma],[Y] obtained for K3 surfaces. We also consider examples of reflections on a certain Calabi-Yau threefold which appears in string theory([COFKM]). Finally we would like to express our gratitude to the referee for giving valuable suggestions and correcting mistakes in the original manuscript. 1. Reflection of sheaves. Let X be a smooth projective variety of dimension d defined over the complex number field C and let H be an ample line bundle on X. For a line bundle L E PicX, let degL = L • H~ denote its degree. The minimal H-degree dm[n(H) is defined to be the following positive integer dmin(tf) = min{degM | M e Pic(X), deg M > 0}. A line bundle C on X is said to be H-minimal if deg£ = dm\ (H). For example, £ is iJ-minimal in one of the following cases: (1) PicX^Z[H} &nd£ = H] (2) degC = l. * Received August 30, 2001; accepted for publication July 26, 2002. t Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachiojishi, Tokyo 192-0397, Japan (nakasima@comp.metro-u.ac.jp).

Metric pairs and the futaki character

Abstract: The non-vanishing of the Futaki character gives an obstruction to the existence of Kahler metrics of constant scalar curvature, having a Kahler form belonging to a fixed Kahler class [4, 6]. It is shown that, in combination with the resolution of the Calabi conjecture [18], one has an analogous obstruction on pairs of metrics having Kahler forms belonging to a fixed pair of Kahler classes. If the difference of the Futaki characters on two classes of fixed total volume does not vanish identically, there cannot exist a pair of metrics, with Kahler forms in these classes, having the same Ricci form and the same harmonic Ricci form. When the obstruction vanishes, results in [8] are used to construct non-trivial examples of such pairs, which are also extremal in the sense of Calabi [3].