In this paper, the author proposes a “fundamental question” about pseudoeffective arithmetic \(\mathbb R\)-Cartier divisors. Dirichlet’s unit theorem is interpreted as the one-dimensional case of this question, and the main theorems give affirmative answers in some higher-dimensional cases. More specifically, let \(X\) be a generically smooth projective arithmetic variety of dimension \(d\) (relative dimension \(d-1\) to \(\mathrm{Spec}\, \mathbb Z\)), and let \(\overline D = (D, g)\) be an arithmetic \(\mathbb R\)-divisor, i.e., \(D = \sum a_i D_i\) for \(a_i\in \mathbb R\) and \(D_i\) a divisor and \(g\) is a Green function associated to \(D\) which is invariant under complex conjugation on \(X(\mathbb C)\). A nonzero rational function \(\varphi\) on \(X\) defines the arithmetic principal divisor \(\widehat{(\varphi)} = ((\varphi), -\log |\varphi|^2)\). As usual, the cohomology is defined to be \[ \widehat H^0(X,\overline D) = \{\varphi\in \mathrm{Rat}(X)^*: \overline D + \widehat{(\varphi)} \text{ is effective}\} \cup \{0\}, \] and \(\overline D\) is big when \(\log \# \widehat H^0(X,n\overline D) \sim n^d\). The fundamental question asks whether the following two conditions are equivalent:

(1)

\(\overline D\) is pseudoeffective (i.e., \(\overline D + \overline A\) is big for any big arithmetic \(\mathbb R\)-divisor \(\overline A\))

(2)

\(\overline D + \widehat{(\varphi)}\) is effective (i.e., \(\geq (0,0)\)) for some \(\varphi \in \mathrm{Rat}(X)^*\otimes \mathbb R\).

(2) readily implies (1), and Dirichlet’s unit theorem follows from showing (1) implies (2) for \(d=1\). The key technical result of the paper is the compactness theorem (Corollary 3.3.2): if \(\overline{D_1}, \dots, \overline{D_m}\) are arithmetic \(\mathbb R\)-divisors satisfying an arithmetic-degree condition and a Chern-form condition with respect to an ample arithmetic divisor, then \(\{(a_1,\ldots, a_m)\in {\mathbb R}^m: \overline D + \sum a_i \overline{D_i} \text{ is effective}\}\) is convex and compact for any \(\overline D\). The author derives Dirichlet’s unit theorem in this Arakelov-theory framework, using arithmetic Riemann-Roch theorem and the compactness theorem. For higher-dimensions, the author also needs an arithmetic \(\mathbb R\)-divisor version of the Hodge index theorem (Theorem 2.2.5); a negative-definite form over \(\mathbb Q\) might not stay negative-definite over \(\mathbb R\), so one needs to know \(\mathbb R\)-divisors for which the intersection with an ample arithmetic divisor is zero. Together with the compactness theorem, the author then proves (1) implies (2) in two cases: when \(D\) is numerically trivial on \(X_{\mathbb Q}\) (Theorem 3.5.3), and when \(\overline D\) has what the author calls multiplicative generators of approximately smallest sections (Definition 3.6.1) and \(D\) is big on the generic fiber of \(X\to \mathrm{Spec}\, \mathbb Z\) (Corollary 3.6.4).