On the Geometry of Hyperkähler Manifolds

Explore the intricate world of hyperkähler manifolds in this comprehensive algebraic geometry seminar. Delve into the geometry of these higher-dimensional analogs of K3 surfaces, examining their properties, known examples, and the concept of deformation type. Investigate the moduli space of stable sheaves on K3 surfaces, drawing from the seminal works of Mukai, Göttsche-Huybrechts, O'Grady, and Yoshioka. Advance to the study of hyperkähler manifolds of K3-ntype with symplectic actions, uncovering their birational geometry and isomorphism to certain moduli spaces. Explore lattice theory and Bridgeland stability conditions to gain deeper insights into these fascinating mathematical structures. Conclude by examining wall-crossing techniques for moduli spaces to characterize their Mukai vector, based on collaborative research with D. Mattei and Y. Dutta.