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**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.

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.

Reviewer: Keyvan Yaghmayi (Salt Lake City)