| We consider relationships between the derived categories of twisted coherent sheaves on various families of K3 surfaces, in the context of string theory. An important ingredient of string theory also of interest in algebraic geometry is T-duality; see for example the works of Morrison [Mo], Katz [K], Thomas and Yau [TY], Donagi and Pantev [DP]. Donagi and Pantev [DP] have extended the original duality on genus one fibred K3 surfaces with a section, to the case of any genus one fibration, via a Fourier-Mukai transform. We investigate possibilities of extending this result to the more general case of non-fibered K3s.; In Chapter 5 we show the existence of a 19 dimensional family of pairs of isogenous K3 surfaces ((M, alpha), (Y, beta)) where M is a double cover of P2, Y is a degree 8 surface in P5 and alpha, beta are nontrivial elements of Br(X), Br(Y) respectively. Note that our family is not the same as Mukai's example. Mukai's family consists of pairs ((M, alpha), (Y, beta)), where M is the moduli space of semistable sheaves on Y with prescribed Chern classes (see [Mu1], [Mu2]). The results in [Mu1] imply that TY embeds Hodge isometrically in TM as a sublattice of index 2. This forces beta to be trivial. On the other hand for generic pairs ((M, alpha), (Y, beta)) in our family, only a proper sublattice of TY embeds in TM and vice versa, which gives rise to a nonzero alpha and beta. Hence we can consider nontrivial gerbes on both M and Y as opposed to just on M which is the case in Mukai's papers. The locus we constructed is a natural candidate for extending the Donagi-Pantev duality and an equivalence D (M, alpha) ∼ D (Y, beta) is expected.; In addition for each integer n we construct an 18 dimensional family of genus one fibered K3 surfaces of index n (see Chapter 6). These families intersect the family of double covers of P 2 and deformation theory arguments should prove similar equivalences.; These results also offer possibilities of further exploring Caldararu's conjecture [Cal1], genus one curves over arbitrary number fields, and matrix models in string theory [BL]. |
