Invariants of big classes from asymptotic total Betti numbers

We introduce and study a new numerical invariant of big line bundles on normal complex projective varieties, defined in terms of the total Betti number of hypersurfaces associated with high tensor powers of the line bundle. We prove that this invariant is birational and provides an upper bound for the volume of the line bundle. When the big line bundle admits a rational Zariski decomposition, we show that the invariant coincides with the volume. In particular, this holds when the line bundle is ample or is defined on a projective surface. We also investigate the analogous problem in the real algebraic setting, proving the existence of real algebraic hypersurfaces associated with high tensor powers of a big line bundle whose real vanishing loci have arbitrarily large total Betti numbers.