A numerical characterization of polarized manifolds (X,L) with K_{X}=-(n-i)L by the ith sectional geometric genus and the ith \Delta-genus

TL;DR

This paper provides a numerical characterization of polarized manifolds with specific canonical bundle relations using sectional geometric genus and elta-genus, focusing on particular cases of the integer i.

Contribution

It introduces a new approach to characterize polarized manifolds with K_X=-(n-i)L through sectional invariants for specific values of i.

Findings

Numerical criteria for i=2 and i=3 cases.

Characterization for cases with i between max{2, dim Bs|L|+2} and n-1.

Extension of previous classifications using sectional invariants.

Abstract

Let (X,L) be a polarized manifold of dimension n. In this paper, by using the ith sectional geometric genus and the ith \Delta-genus, we will give a numerical characterization of (X,L) with K_{X}=-(n-i)L for the following cases (i) i=2, (ii) i=3 and n \geq 5, (iii) max{2, dim Bs|L|+2} \leq i \leq n-1.