This paper provides a counterexample to the strong version of the Hodge conjecture for positive currents formulated by J.-P. Demailly [Invent. Math. 69, 347–374 (1982; Zbl 0488.58001)].
On smooth complex projective variety \(X\), the pairing between currents and differential forms allows one to define weak limit of currents. Moreover, alongside integration, it allows one to associate a strongly positive closed current \([Z]\) to subvariety \(Z\) of \(X\).
Now let \(p\) and \(q\) be nonnegative integers such that \(p + q = \text{dim} X\). By the Hodge conjecture (HC) the intersection \(H^{2q}(X, \mathbb{Q}) \cap H^{q,q}(X)\) consists of classes of \(p\)-dimensional algebraic cycles with rational coefficients. The Hodge conjecture for currents \((\text{HC}')\) states that if \(\mathscr{T}\) is a \((p,p)\)-dimensional real closed current on \(X\) with cohomology class \(\{\mathscr{T}\} \in \mathbb{R} \otimes_{\mathbb{Z}} \left(H^{2q}(X, \mathbb{Z})/\text{tors} \cap H^{q,q}(X)\right)\) then \(\mathscr{T}\) is a weak limit of the form \(\mathscr{T} = \lim\limits_{i \to \infty} \mathscr{T}_i\) where \(\mathscr{T}_i = \sum_j \lambda_{ij}[Z_{ij}]\) for real numbers \(\lambda_{ij}\) and \(p\)-dimensional subvarieties \(Z_{ij}\) of \(X\). According to the Hodge conjecture for strongly positive currents \((\text{HC}^+)\) if \(\mathscr{T}\) is a \((p, p)\)-dimensional strongly positive closed current on \(X\) with cohomology class \(\{\mathscr{T}\} \in \mathbb{R} \otimes_{\mathbb{Z}} \left(H^{2q}(X, \mathbb{Z})/\text{tors} \cap H^{q,q}(X)\right)\) then \(\mathscr{T}\) is a weak limit of \(\mathscr{T}_i = \sum_j \lambda_{ij}[Z_{ij}]\) for positive real numbers \(\lambda_{ij}\). In 1982, Demailly proved that for smooth projective varieties \((\text{HC}^+) \Longrightarrow\) (HC) [J.-P. Demailly, Invent. Math. 69, 347-374 (1982; Zbl 0488.58001)]. Moreover, in 2012, he showed that (HC) \(\Longleftrightarrow (\text{HC}')\) and asked whether \((\text{HC}')\) implies \((\text{HC}^+)\) [J.-P. Demailly, Analytic methods in algebraic geometry. Somerville, MA: International Press (2012; Zbl 1271.14001)].
In this paper, the authors show that \((\text{HC}^+)\) fails even on toric varieties where the Hodge conjecture readily holds. In fact, they construct a \(4\)-dimensional smooth projective toric variety \(X\) and a \((2, 2\))-dimensional strongly positive closed current \(\mathscr{T}\) on \(X\) such that the cohomology class of \(\mathscr{T}\) belongs to \(H^4(X, \mathbb{Z})/\text{tors} \cap H^{2,2}(X)\) but \(\mathscr{T}\) is not a weak limit of currents \(\mathscr{T}_i = \sum_j \lambda_{ij}[Z_{ij}]\) for positive real numbers \(\lambda_{ij}\). The construction goes through careful investigation of certain \((p, p\))-dimensional closed currents \(\mathscr{T}_\mathscr{C}\) associated to tropical varieties \(\mathscr{C}\) of dimension \(p\) in \(\mathbb{R}^n\). They show, using the Hodge index theorem, that if \(\mathscr{T}_\mathscr{C}\) is a weak limit of integration currents then tropical Laplacian of \(\mathscr{C}\) has at most one negative eigenvalue. Ultimately, they construct a tropical surface in \(\mathbb{R}^4\) whose tropical Laplacian has more than one negative eigenvalue. The recent work of K. Adiprasito and the first author [“Convexity of complements of tropical varieties, and approximations of currents”, arXiv:1711.02045] generalizes the main result of this paper by providing a family of counter-examples for \((\text{HC}^+)\) to any dimension and codimension greater than 1.