## A tropical approach to a generalized Hodge conjecture for positive currents.(English)Zbl 1396.14064

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.

### MSC:

 14T05 Tropical geometry (MSC2010) 32U40 Currents 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 42B05 Fourier series and coefficients in several variables 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)

### Citations:

Zbl 0488.58001; Zbl 1271.14001
Full Text: