Complements on Log Canonical Fano Varieties and Index Conjecture of Log Calabi-Yau Varieties

This thesis aims to generalise the theory of complements to log canonical Fano varieties and relate theory of complements to the index conjecture of log Calabi-Yau varieties. We mainly work over an algebraically closed field of characteristic zero, more specifically over C. We will first introduce some basic background theory for birational geometry, including notion of singularities, pairs, complements. We will then cover some backgrounds of (log) Fano and Calabi-Yau varieties. We will also state the main new results in the introduction. The majority of work is then split into the following 4 sections: complements on surfaces, complements on log canonical 3-fold, index conjecture for log Calabi-Yau varieties, relative 3-fold complements. For the section about complements on surfaces, we will firstly cover the known result about the theory for complements and then prove new results about semi-dlt surfaces. We will extend the notion of complements to semidlt surfaces and then prove a result for "gluing" complements for semi-dlt surfaces. Then we will move on to prove results about boundedness of complements for global log canonical Fano 3-fold. In other direction, we will prove the index conjecture for log Calabi-Yau varieties in dimension 3 in full generality and then prove some new inductive results towards the conjecture. In the last chapter, I will include proof of the boundedness of complements for log Fano in the relative case in dimension 3. The last chapter is a part of joint work, with Stefano Filipazzi and Joaquin Moraga, where we proved a general theorem of boundedness of complements in dimension 3.Cambridge Trust Cambridge International Scholarshi