The structure of $\sigma$-ideals of compact sets

Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where $\sigma$-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of $\sigma$-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a $\sigma$-ideal imposes severe definability restrictions. A typical instance is the dichotomy theorem, which states that $\sigma$-ideals which are analytic or coanalytic must be actually either complete coanalytic or else ${G_\delta}$. In the second part we discuss (generators or as we call them here) bases for $\sigma$-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel $\sigma$-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the third part we develop the connections of the definability properties of $\sigma$-ideals with other structural properties, like the countable chain condition, etc.