Rational curves and bounds on the Picard number of Fano manifolds

TL;DR

This paper proves the Generalized Mukai Conjecture for certain Fano manifolds with high pseudoindex and provides alternative proofs for Fano fourfolds and fivefolds, advancing understanding of their geometric properties.

Contribution

It establishes the conjecture for a broad class of Fano manifolds and offers new proofs for specific low-dimensional cases.

Findings

Proves the Generalized Mukai Conjecture for Fano manifolds with pseudoindex ≥ (dim X + 3)/3

Provides alternative proofs for Fano fourfolds and fivefolds

Enhances understanding of the structure and bounds of Fano manifolds

Abstract

We prove that Generalized Mukai Conjecture holds for Fano manifolds $X$ of pseudoindex $i_X \ge (\dim X +3)/3$. We also give different proofs of the conjecture for Fano fourfolds and fivefolds.