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Hard Lefschetz properties, complete intersections and numerical dimensions
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We study the positivity of complete intersections of nef classes. We first give a sufficient and necessary characterization on the complete intersection classes which have hard Lefschetz property on a compact complex torus, equivalently, in the linear case. In turn, this provides us new kinds of cohomology classes which have Hodge-Riemann property or hard Lefschetz property on an arbitrary compact K\"ahler manifold. We also give a complete characterization on when the complete intersection classes are non-vanishing on an arbitrary compact K\"ahler manifold. Both characterizations are given by the numerical dimensions of various partial summations of the given nef classes. As an interesting byproduct, we show that the numerical dimension endows any finite set of nef classes with a loopless polymatroid structure.
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Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes
For a supercritical collection of free divisor classes on a smooth projective variety, the kernel of the product operator on (1,1)-classes is exactly the span of the prime divisors annihilated by the collection.
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