K3 surfaces from a derived categorical viewpoint : magistrsko delo

Jenko, Izak (author), Filip, Matej (mentor), Meinsma, Reinder (comentor)

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.

• Language: English
• Keywords: triangulated category, derived category, Fourier–Mukai transform, K3 surface, derived Torelli theorem
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2025
• PID: 20.500.12556/RUL-174320
• UDC: 512
• COBISS.SI-ID: 251010051
• Publication date in RUL: 01.10.2025

Secondary language

• Language: Slovenian
• Title: K3 ploskve z vidika izpeljanih kategorij

Abstract

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.

• Keywords: triangulirana kategorija, izpeljana kategorija, Fourier–Mukaijeva transformacija, K3 ploskev, izpeljani Torellijev izrek