Real rectifiable currents, holomorphic chains and algebraic cycles

We study some fundamental properties of real rectifiable currents and give a generalization of King’s theorem to characterize currents defined by positive real holomorphic chains. Our main tool is Siu’s semi-continuity theorem and our proof largely simplifies King’s proof. A consequence of this result is a sufficient condition for the Hodge conjecture.

Strong approximation in h-mass of rectifiable currents under homological constraint

Let {h:\mathbb{R}\to\mathbb{R}_{+}} be a lower semicontinuous subbadditive and even function such that {h(0)=0} and {h(\theta)\geq\alpha|\theta|} for some {\alpha>0} . If {T=\tau(M,\theta,\xi)} is a k-rectifiable chain, its h-mass is defined as \mathbb{M}_{h}(T):=\int_{M}h(\theta)\,d\mathcal{H}^{k}. Given such a rectifiable flat chain T with {\mathbb{M}_{h}(T)<\infty} and {\partial T} polyhedral, we prove that for every {\eta>0} , it decomposes as {T=P+\partial V} with P polyhedral, V rectifiable, {\mathbb{M}_{h}(V)<\eta} and {\mathbb{M}_{h}(P)<\mathbb{M}_{h}(T)+\eta} . In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint {\partial P=\partial T} . When {h^{\prime}(0^{+})} is well defined and finite, the definition of the h-mass extends as a finite functional on the space of finite mass k-chains (not necessarily rectifiable). We prove in this case a similar approximation result for finite mass k-chains with polyhedral boundary. These results are motivated by the study of approximations of {\mathbb{M}_{h}} by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of {T\mapsto\mathbb{M}_{h}(T)+\mathbb{I}_{\partial S}(\partial T)} with respect to the topology of the flat norm.

Regularity results for solutions to the c-Plateau problem with free boundary having small singular set, and generally in codimension zero

We prove two results for the c-Plateau problem, introduced in [17], which is a minimization problem for integer rectifiable currents. First, we prove there is no solution to the c-Plateau problem with free boundary having singular set of finite Hausdorff codimension two measure and with regular part having constant mean curvature. Second, we prove regularity up to Hausdorff codimension seven of the free boundary of top-dimensional solutions to the c-Plateau problem.