Topics on Coherent Sheaves for Deligne-Mumford Stacks

Abstract

\par The problems studied in this thesis are problems concerned with Vafa-Witten theory in Physics which gives a relationship between Physics and mathematics through a deep relationship called S-duality and providing new relationships between various invariants. This deep study intertwines number theory, arithmetic geometry and representation theory and algebraic geometry enabling us to recover beautiful interesting formulas. Sheaves on Deligne-Mumford stacks are related to this study \cite{JK} following a large body of work of several people in the related fields \cite{TT1}, \cite{TT2}.\par In this thesis we mainly study two important problems concerned with sheaves on Deligne-Mumford stacks. We mainly study sheaves on smooth toric Deligne-Mumford stacks and Bogomolov-Gieseker inequality for modified semi-stable sheaves on tame smooth Deligne-Mumford projective stacks in any dimension. \par In chapter 2 we recall the definitions and preliminaries on Deligne-Mumford stacks. Then we recall the important notion of semi-stability modified to the setting of projective Deligne-Mumford stacks. We restate the notions of modified slope and define semi-stability analogously. We impose a condition \S\ref{condition_star}, with which we work in the rest of chapter 4 and chapter 5. \par In chapter 3 we first study toric Deligne-Mumford stacks and torsion free toric sheaves on them. We give examples of toric Deligne-Mumford stacks, torus actions on Deligne-Mumford stacks and prove a gluing formula for torsion free toric sheaves on a Toric Deligne-Mumford stack generalising \cite{Kool}, \cite{GJK}, \cite{WW} on any arbitrary toric Deligne-Mumford stack. \par In chapter 4 we generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p0$ and generalize Langer's method to smooth Deligne-Mumford stacks. \par In chapter 5 we generalize the Bogomolov inequality formula to higher dimensions and to Simpson Higgs sheaves on tame Deligne-Mumford stacks.