On Noether's inequality for threefolds

Abstract

Let $S$ be a complex minimal algebraic surface of general type.Let $K_{S}$ be the canonical bundle of $S$ and $p_{g}(S)$ be the geometric genus of $S$ .Then in general, we have a classical inequality: $K_{s}^{2}\geqq 2p_{g}(S)-4$ , which is Noether's inequality.In this paper, we will study a three-dimensional analogue.Since we have Noether's inequality for minimal surfaces (and also canonical models of surfaces) we expect some inequalities between the geometric genus and the cube of the first Chern class for three dimensional canonical models, which may be singular and not factorial.Very optimistically, we might expect that: for any canonical model $X$ of a threefold of general type, we should have $K_{X}^{3}\geqq 2p_{g}(X)-6$ .But that is not the case in general.MAIN THEOREM (Theorems 2.4, 3.1, 4.1).Let $X$ be a three-dimensional alge- braic vanety defined over C. Assume that $X$ has at most canonical singulanties and that a canonical divisor $K_{X}$ is $nef$ and big.Let $d=\dim\Phi_{K_{X}}(X)$ .(1) If $d=3$ , then $K_{X}^{3}\geqq 2p_{g}-6$ .(2) If $d=2$ and $K_{X}$ is Cartier, then either $2a)$ $K_{X}^{3}\geqq 2p_{g}(X)-4$ or $2b)

Locations

• Journal of the Mathematical Society of Japan - View - PDF