| This thesis introduces a system of differential equations associated to the following data: a smooth algebraic variety X acted by a complex connected Lie group G, a line bundle L on X with a linearization of the G-action, and a character chi : G x C*→C * . Let beta : g⊕Z→Z be the infinitesimal character. We call the system a tautological system, denoted by tau(X, L, G, beta), and show that this system is holonomic assuming G acts on X with finitely many orbits.;This construction recovers GKZ systems [GKZ1], [GKZ2] when X ⊂ PCA is an n-dimensional projective toric variety associated to a finite set of integral points A ⊂ Zn , L is the pullback of O (1) and G = ( C* )n. If we extend the group G to be the full automorphism group Aut( X), this system is the extended GKZ system introduced in [HLY1].;In general, if X is a Fano variety, L = K-1X , G is the automorphism group Aut( X), the tautological system tau(X, L, G, beta) provides an approach to study the Picard-Fuchs system of Calabi-Yau hypersurfaces in X. This can also be generalized to the study of Calabi-Yau complete intersections in X. |
