20.500.12556/RUL-174320 K3 surfaces from a derived categorical viewpoint magistrsko delo K3 ploskve z vidika izpeljanih kategorij We construct the derived category of an abelian category, equip it with a triangulated structure and define derived functors. In particular we address the bounded derived category of coherent sheaves on a smooth projective variety and introduce derived functors of geometric origin. Utilizing them we define and study Fourier–Mukai transforms first at the level of derived categories, then at the level of K-groups and lastly at the level of rational cohomology. Focusing on complex K3 surfaces we describe some of their invariants, most notably their intersection form and associated Hodge structure. Lastly we present Orlov's proof of the derived Torelli theorem, which characterizes derived equivalent K3 surfaces in terms of their Mukai lattices and the associated moduli spaces of stable sheaves. Konstruiramo izpeljano kategorijo abelove kategorije, jo opremimo s strukturo triangulirane kategorije in definiramo izpeljane funktorje. Posebej se posvetimo omejeni izpeljani kategoriji koherentnih snopov na gladki projektivni raznoterosti in vpeljemo izpeljane funktorje, ki izhajajo iz geometrije. Preko njih definiramo in obravnavamo Fourier–Mukaijeve transformacije, najprej v okviru omejenih izpeljanih kategorij, nato na ravni K-grup in nazadnje na ravni racionalne kohomologije. Zatem se osredotočimo na kompleksne K3 ploskve in opišemo nekaj njihovih invariant, še zlasti presečno formo in Hodgevo strukturo. Nazadnje predstavimo Orlovov dokaz izpeljanega Torellijevega izreka, ki karakterizira izpeljano ekvivalentne K3 ploskve preko Mukaijevih mrež in pridruženih prostorov modulov stabilnih snopov. triangulated category derived category Fourier–Mukai transform K3 surface derived Torelli theorem triangulirana kategorija izpeljana kategorija Fourier–Mukaijeva transformacija K3 ploskev izpeljani Torellijev izrek true false false Angleški jezik Slovenski jezik Magistrsko delo/naloga 2025-10-01 08:15:42 2025-10-01 08:15:44 2025-11-20 03:50:24 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 155099 251010051 19962.pdf 19962.pdf 1 6BD1FD7E2F670D1CF865C48B1580D342 433954bbb46a46a39b0a00969997d1bf0f3fce47ea8471a56a8b78748bf09f08 db98aa6f-9e8d-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=219109 Fakulteta za matematiko in fiziko 0 0 0