Riemann-Roch Polynomials, MBM Classes and Poor IHS Manifolds

Abstract

We study poor irreducible holomorphic symplectic manifolds, namely those containing no rational curves and no codimension-one subvarieties. We show that, for such manifolds, several natural cones in \(H^{1,1}(X,\mathbb R)\) coincide, giving strong rigidity consequences. In particular, poor elliptic irreducible holomorphic symplectic manifolds admit no holomorphic foliations. Using this rigidity property, we give a criterion for detecting monodromy birationally minimal classes from the roots of Riemann--Roch polynomials on any irreducible holomorphic symplectic manifold. To each primitive negative class we associate a polynomial depending only on the deformation type and on its Beauville--Bogomolov--Fujiki square. Under a simple-root assumption on the positive real roots, the position of the number \(1\) among these roots implies that the class is MBM whenever it is of type \((1,1)\). This gives a uniform root-theoretic sufficient condition for the existence of rational curves on irreducible holomorphic symplectic manifolds. We then describe poor elliptic irreducible holomorphic symplectic manifolds in terms of their deformation spaces and show that every connected component of the moduli space contains a poor irreducible holomorphic symplectic manifold of positive Picard rank.