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In this thesis we study various aspects of hyper-Kähler manifolds and abelian varieties such as their derived categories, sheaves, cycles, and topology. The thesis consists of six parts.
The first part is mostly a survey of results of Taelman. We show that the LLV algebra is a derived invariant. Therefore, derived equivalences between hyper-Kähler manifolds yield Hodge isometries between their Verbitsky components, which come from isometries between their Mukai lattices. Moreover, derived equivalent hyper-Kähler manifolds have isomorphic rational Hodge structures.
The study of derived categories of hyper-Kähler manifolds is further refined in the next section. We introduce an extended Mukai vector with values in the Mukai lattice. This yields a structural result dividing derived equivalences into three cases with different geometric meaning. Moreover, for hyper-Kähler manifolds deformation-equivalent to Hilbert scheme of points of K3 surfaces we define an integral lattice which is a derived invariant giving a higher-dimensional analogue to Mukai's results for K3 surfaces. This has many consequences such as finiteness of Fourier--Mukai partners of hyper-Kähler manifolds of this deformation type.
The subsequent section conceptualizes the extended Mukai vector by introducing the notion of atomic objects on hyper-Kähler manifolds. We relate the notion of atomicity to different obstruction maps. Stable atomic bundles are shown to be projectively hyperholomorphic, a class of bundles for which we prove formality of the dg algebra of derived endomorphisms. A thorough study of atomic Lagrangian submanifolds yields a structural result and expectations for the general behaviour of atomic objects. Our methods also yield that there do not exist spherical sheaves on any hyper-Kähler manifold of dimension at least four.
The question of topological properties of hyper-Kähler manifolds is discussed in the fourth part, which is a joint work with Jieao Song. The main result is a conditional bound on the second Betti number for hyper-Kähler manifolds in terms of the second and fourth Chern class of its tangent bundle. In the known examples this bound is better than previous ones and remains true for orbifolds of dimension four. We further investigate (conjectural) properties that the generalized Fujiki constants and Riemann--Roch polynomials possess and discuss implications.
The fifth part studies group actions on hyper-Kähler manifolds and derived categories and is joint work with Georg Oberdieck. We show that fixed loci of actions by finite groups on moduli spaces of stable objects on certain smooth projective varieties which are induced by an action of the groups on their derived categories are covered by moduli spaces of semistable objects of the equivariant category. This yields a generalization of the derived McKay correspondence for symplectic surfaces and completely determines the fixed locus of a symplectic automorphism acting on a moduli space of stable objects on a K3 surface.
The final part of this thesis is a joint work with Olivier de Gaay Fortman and studies one-cycles on abelian varieties. We link the integral Hodge conjecture for one-cycles to lifts of the correspondence obtained from the Poincaré bundle. This implies the integral Hodge conjecture for one-cycles for products of Jacobians of curves. The arguments also work over other fields and yield the integral Tate conjecture for one-cycles for Jacobians of curves over the separable closure of a finitely generated field.
