## Heights of projective varieties and positive Green forms.(English)Zbl 0973.14013

From the introduction: The purpose of this paper is to study analogs of some basic concepts and results of projective geometry in the context of Arakelov geometry [S. J. Arakelov, Proc. int. Congr. Math., Vancouver 1974, Vol 1, 405-408 (1975; Zbl 0351.14003); see also H. Gillet and Ch. Soulé, Publ. Math., Inst. Hautes Étud. Sci. 72, 93-174 (1990; Zbl 0741.14012)]. As was first noticed by G. Faltings in his work on diophantine approximation for abelian varieties [Ann. Math. (2) 133, 549-576 (1991; Zbl 0734.14007)], higher dimensional arithmetic intersection theory can be used to define the height of any (closed integral) projective subscheme $$X\subset \mathbb{P}^N$$, where $$\mathbb{P}^N$$ is the $$N$$-dimensional projective space over $$\mathbb{Z}$$ (or more generally over the integers in a number field). The Faltings height $$h_F(X)$$, which is a nonnegative real number, is defined in a similar fashion to the degree of a projective variety over a field. That is, $$h_F(X)$$ is the intersection, in the sense of Gillet and Soulé [loc. cit.], of the fundamental class of $$X$$ with the first Chern class of the canonical hermitian line bundle on $$\mathbb{P}^N$$, raised to the power $$d=\dim(X)$$.
In this paper we propose a slightly different definition of the height of $$X$$. Namely we denote by $$h(X)$$ the intersection of the fundamental class of $$X$$ with the $$d$$-th Chern class of the canonical quotient hermitian bundle on $$\mathbb{P}^N$$. We prove that $$h(X)$$ is nonnegative and smaller than $$h_F(X)$$ (except when $$d\leq 1$$ or when the generic fiber of $$X$$ is empty, in which case $$h(X)=h_F(X))$$. Furthermore $$h(X)=0$$ if and only if $$X$$ is a linear subspace $$\mathbb{P}^{d-1}\subset \mathbb{P}^N$$ defined by the vanishing of $$N+1-d$$ standard coordinates (theorem 5.2.3).
We obtain several results on the heights of projective varieties, which are inspired by the analogy between heights and degrees. For instance we compute the height of the join of two varieties (proposition 4.2.2) and the behavior of the height under linear projection (3.3.2). We give several proofs of the following arithmetic Bézout theorem. Assume $$X \subset \mathbb{P}^N$$ and $$Y\subset\mathbb{P}^N$$ are integral projective varieties which meet properly on the generic fiber of $$\mathbb{P}^N$$. Their intersection cycle $$X.Y$$ can then be defined using W. Fulton’s method [“Intersection theory”, 2nd ed. (Berlin 1998; Zbl 0885.14002)]. It is well defined up to the addition of a cycle linearly equivalent to zero in the closed fibers of $$\mathbb{P}^N$$ over $$\mathbb{Z}$$, and its height $$h(X.Y)$$ (defined by extending by linearity the definition for integral subschemes) does not depend on the choice of representative for $$X.Y$$. Denote by $$\deg_\mathbb{Q} (X)$$ and $$\deg_\mathbb{Q} (Y)$$ the degrees in $$\mathbb{P}^N_\mathbb{Q}$$ of the generic fibers of $$X$$ and $$Y$$ respectively. Then we have $h(X.Y)\leq h(X)\deg_\mathbb{Q} (Y)+\deg_\mathbb{Q} (X)h(Y)+ c\deg_\mathbb{Q}(X)\deg_\mathbb{Q} (Y),$ where the constant $$c$$ depends only on $$N$$, $$\dim(X)$$, and $$\dim(Y)$$. We give three different proofs of this inequality (theorems 4.2.3, 5.4.4, and 6.1.1), the smallest value of $$c$$ being the one in theorem 5.4.4 (we believe that $$c$$ can be taken equal to zero, but we cannot prove it except when $$X$$ or $$Y$$ is a linear subspace).
In transcendental number theory, especially in the work of Nesterenko, and Philippon, another definition of height has been known for some time, which does not use Arakelov theory, and cases of the Bezout theorem have been proved in that context. Namely the height of $$X\subset\mathbb{P}^N$$ is defined to be the height of its Chow form, which is a point in a large projective space. The comparison between this definition and $$h_F(X)$$ was made by Ch. Soulé [Astérisque 198-200, 355-371 (1991; Zbl 0756.14014)] and P. Philippon [Publ. Math., Inst. Hautes Étud. Sci. 64, 5-52 (1986; Zbl 0615.10044)]. We extend their result to more general Chow forms and not necessarily standard metrics (theorem 4.3.2). As a byproduct we get the following result. Let $$R$$ be the resultant of $$N+1$$ homogeneous polynomials of degrees $$d_0,\dots,d_N$$ in $$N+1$$ variables. This is a multihomogeneous polynomial with integral coefficients of multidegree $$(\delta_0, \dots,\delta_N)$$, where $$\delta_i= \sum^N_{j=0 \atop j\neq i}d_j$$. Its variables are the coefficients of the “generic” homogeneous polynomials of degrees $$d_0,\dots, d_N$$ in $$N+1$$ variables. So $$R$$ can be viewed as an element of $$\bigotimes^r_{i=0} S^{\delta_i} (S^{d_i} \mathbb{C}^{\check N+1})^\vee$$. Equip this vector space with the hermitian norm $$\|\cdot \|_{\text{Herm}}$$ induced by the standard hermitian structure on $$\mathbb{C}^{N+1}$$. We prove in lemma 4.3.4 that $\log\|R\|_{\text{Herm}}= {1\over 2}\left( \prod^N_{i=0} d_i \right) \cdot\left( (N+1)\left( 1+{1\over 2}+ \cdots+ {1\over N}\right)-N\right) +\varepsilon (d_0,\dots, d_N),$ where $\bigl|\varepsilon (d_0, \dots,d_N) \bigr|\leq {1\over 2}N\left( \prod^N_{i=0} d_i\right) \cdot\sum^N_{i=0} {1 \over d_i}\log(d_i+1).$ We also evaluate the size of $$R$$ for other norms (theorem 4.3.8). Our main analytic tool is the existence of “positive Green forms” for effective cycles $$Z$$ on a complex manifold $$X$$. By this we mean a positive $$C^\infty$$ form $$\eta$$ on $$X-|Z|$$ which is locally $$L^1$$ on $$X$$ and such that the corresponding current $$g=[\eta]$$ on $$X$$ is a Green current for $$Z$$, i.e., such that $$dd^c g+\delta_Z$$ is $$C^\infty$$ on $$X$$ (where $$\delta_Z$$ the current given by integration on $$Z$$). An example of such a positive Green form is the Levine form, familiar to Nevanlinna theory, when $$X$$ is a complex projective space and $$Z$$ a linear subspace. The positivity of these Levine forms has several interesting consequences (proposition 1.4.2, proposition 4.1.3). More generally we give conditions for a given effective cycle (respectively all effective cycles) on $$X$$ to have a positive Green form (propositions 6.2.1, 6.2.2, and 6.2.3), and a counterexample showing that some complex manifolds admit effective cycles with no positive Green forms (6.3).

### MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 11G50 Heights 14G25 Global ground fields in algebraic geometry
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### References:

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