Intersection theory of nefb-divisor classes

Abstract

We prove that any nef $b$ -divisor class on a projective variety defined over an algebraically closed field of characteristic zero is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef $b$ -divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint-free curve class is a power of a nef $b$ -divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of $b$ -divisors which were defined in our previous work.

Chern–Weil and Hilbert–Samuel formulae for singular Hermitian line bundles

“…It is a priori not clear that if a nef Weil R-b-divisor is Cartier, then it is nef as a Cartier R-b-divisor. This is known to be true in the toroidal setting [6,Lemma 4.24] and if we work with algebraic varieties over a countable field (instead of complex manifolds), see [16,Corollary 4].…”

“…Remark 4.9. In [16,Theorem 5] Dang and Favre show that in the case of algebraic varieties over a countable field, any nef Weil R-b-divisor is approximable nef. We will show in Section 5.2 that any b-divisor that comes from a psh metric in a suitable sense is approximable nef.…”

“…This yields a top intersection product on the space of approximable nef b-divisors, which is continuous with respect to approximating sequences. The details can be found in [16,Section 3]. We mention that in [16] the authors work over a countable ground field, but one can either check that this part of their argument does not use the countability assumption, or apply the following lemma:…”

“…The details can be found in [16,Section 3]. We mention that in [16] the authors work over a countable ground field, but one can either check that this part of their argument does not use the countability assumption, or apply the following lemma:…”