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The valuative tree. (English) Zbl 1064.14024

Lecture Notes in Mathematics 1853. Berlin: Springer (ISBN 3-540-22984-1/pbk). xv, 234 p. (2004).
From the introduction: “The purpose of this monograph is to give a new approach to singularities in a local, two-dimensional setting. Our method enables us to study curves, analytic ideals in \(R=\mathbb C[[x,y]]\), and plurisubharmonic functions in a unified way. It is also general and powerful enough so that it can be applied to other situations, such as dynamics of fixed point germs \(f:(\mathbb C^2,0) \circlearrowleft\).
Singularities of curves, ideals and plurisubharmonic functions can be analyzed by performing a composition of point blowups and considering the order of vanishing of the pullbacks along irreducible components of the exceptional divisor. These orders of vanishing define real-valued functions on the ring \(R\) called divisorial valuations. It is a classical fact that the singularity of a curve or ideal is completely determined by the values of all divisorial valuations.
This naturally leads us to look at the set of all divisorial valuations. Indeed our aim is to describe in detail the structure of a slightly larger set \({\mathcal V}\) that we call valuation space.”
“The central theme of our work is that valuation space has a natural structure of a tree modeled on the real line, and that this structure can be used to efficiently encode singularities of various kinds.”
“The valuative tree is a beautiful object which may be viewed in a number of different ways. Each corresponds to a particular interpretation of a valuation, and each gives a new insight into it. Some of them will hopefully lead to generalizations in a broader context.”
Let us consider in more detail the results of this monograph. A valuation on \(R\) is a non-constant real-valued function \(\nu: R\to \overline {\mathbb R}_+:=\mathbb R_+\cup\{\infty\}\) satisfying the usual conditions. The set \(\nu^{-1}(\infty)\) is a prime ideal \(\mathfrak p_\nu\) contained in the maximal ideal \(\mathfrak m\) of \(R\); the valuation \(\nu\) can be extended to a valuation of the field \(K\) of quotients of \(R\) if \(\mathfrak p_\nu\) is the zero ideal. The valuation \(\nu \) is called proper if \(\mathfrak p_\nu\subsetneq \mathfrak m\). It is centered if it is proper and \(\nu(\mathfrak m)>0\). Two centered valuations \(\nu_1\), \(\nu_2\) are called equivalent if there exists a constant \(C>0\) with \(\nu_1=C\nu_2\). The set of all centered valuations on \(R\) is denoted by \(\widetilde{\mathcal V}\). The set \({\mathcal V}:=\{\nu\in {\mathcal V}\mid \nu(\mathfrak m)=1\}\) is a system of representatives of \(\widetilde{\mathcal V}/\sim\) as well is \({\mathcal V}_z=\{\nu\in{\mathcal V}\mid \nu(z)=1\}\) where \(z\in R\) has order \(1\). Clearly \(\widetilde{\mathcal V}\), \({\mathcal V}\) and \({\mathcal V}_z\) carry a natural partial ordering: \(\nu_1\leq \nu_2\) if \(\nu_1(\phi)\leq\nu_2(\phi)\) for every \(\phi\in R\). The valuation defined by the order function of \(\mathfrak m\) is called the multiplicity valuation \(\nu_{\mathfrak m}\); the valuations of the field of quotients of \(R\) which are of the second kind with respect to \(R\) are called divisorial valuations. Furthermore, the authors introduce some other types of valuations, e.g., monomial valuations [cf. 1.5.2] and quasimonomial valuations [cf. 1.5.4]; for another definition cf. 2.23 and a characterization cf. Prop. 6.41. A first classification of all these valuations is given in 2.23.
A centered Krull valuation \(\nu\) on \(R\) is a valuation of \(K\) having values in a totally ordered abelian group \(\Gamma\) which is non-negative on \(R\) and positive on \(\mathfrak m\). For a Krull valuation on \(R\) one has the invariants rank \(\text{rk}\,(\nu)\), i.e., the rank of the ordered group \(\nu(K^\times)\), rational rank \(\text{rat.rk}\,(\nu)\), i.e., \(\dim_{\,\mathbb Q}(\nu(K^\times)\otimes _{\mathbb Z}\mathbb Q)\) and the transcendence degree \(\text{tr.d}\,\nu\) of the residue field of \(\nu\) over the field of complex numbers \(\mathbb C\); they satisfy Abhyankar’s inequality. Also, one has the semigroup \(\nu(R^\times)\) and the graded ring \(\text{gr}_{\nu}(K)=\bigoplus_{\gamma\in \Gamma}\mathfrak n_\gamma/\mathfrak n^+_\gamma\) where, for \(\gamma\in\Gamma\), \(\mathfrak n_\gamma\) (resp. \(\mathfrak n^+_\gamma\)) is the additive subgroup of \(K\) having values \(\geq \gamma\) (resp. \(>\gamma\)) (and multiplication is defined in the canonical way). To any subring \(S\) of \(K\) one can define, in a similar way, the graded ring \(\text{gr}_{\nu}(S)\). In particular, let \(\nu: R\to \overline{\mathbb R}_+\) be a valuation which is also a Krull valuation; then its ideal \(\mathfrak p_\nu\) is the zero ideal. If \(\mathfrak p_\nu\) is not the zero ideal, then we have \(\mathfrak p_\nu=(\phi)\) where \(\phi\in R\) is an irreducible power series; one can associate to \(\nu\) a Krull valuation \(\text{krull} [\nu]: R^\times\to \mathbb Z\times \mathbb R\). Therefore, for any valuation on \(R\), one can define all the invariants which we defined for Krull valuations.
One method to classify valuations on \(R\) is by using sequences of key polynomials (SKP for short); these are finite (resp.infinite) sequences of polynomials in \(\mathbb C[x,y]\) together with a finite (resp. infinite) sequence in \(\overline{\mathbb R}_+\). The notion of SKP is introduced in Def. 2.1; it is shown in Th. 2.8 (resp. Th. 2.22) how to associate to a finite (resp.infinite) SKP a valuation on \(R\). According to the properties of the SKP one gets six different types of valuations [cf.Def. 2.23]. Conversely, any valuation on \(R\) determines uniquely a SKP.
In chapter 3 the authors introduce three different kind of trees: (rooted) nonmetric trees, parameterized trees and metric trees (which are often called \(\mathbb R\)-trees in the literature). The first important result – which gives rise to the name of this monograph – is Th. 3.14 which states that the valuation space \({\mathcal V}\) is a complete nonmetric tree rooted at \(\nu_{\mathfrak m}\); a finer study of the properties of the tree \({\mathcal V}\) is given in Prop. 3.20. The identification of valuations with SKP’s gives an explicit model for the tree structure on \({\mathcal V}\). In section 3.3. the authors introduce a new invariant for a valuation \(\nu\) which measures its distance from the valuation \(\nu_{\mathfrak m}\), namely its skewness \(\alpha(\nu)\in [1,\infty]\), and they study in Th. 3.26 properties of the map \(\alpha:{\mathcal V}\to [1,\infty]\). As one typical result let us mention the following: if \(\nu\) is a divisorial valuation, then \(\alpha(\nu)\) is rational. They also show that \({\mathcal V}\) and \({\mathcal V}_{\text{qm}}\), the set of quasimonomial valuations, can be equipped with a metric which gives the structure of a metric tree; \({\mathcal V}\) is even complete. The semigroup and approximating sequences of a valuation are the contents of sections 3.5 and 3.7 [approximating sequences were also studied by M. Spivakovsky, Am. J. Math. 12, 107–156 (1990; Zbl 0716.13003)].
In chapter 4 the authors use an approach via Puiseux series to get the tree structure on \({\mathcal V}_x\). Let \(z\in R\) be an element of order \(1\), consider \(\mathbb C[[z]]\) and its field of fractions \(k:=\mathbb C((z))\), and let \(\widehat k\) be the algebraic closure of \(k\), i.e., the field of Puiseux series. Let \(\nu_*\) be the valuation of \(\widehat k\) which is trivial on \(\mathbb C\) and satisfies \(\nu_*(z)=1\). Let \(\overline k\) be the completion of \(\widehat k\) with respect to \(\nu_*\). The elements of \(\overline k\) are power series of the form \(\sum_{j\geq1} a_jz^{\widehat\beta_j}\) with non-zero coefficients \(a_j\in \mathbb C\) and rational numbers \(\widehat\beta_{j+1}>\widehat\beta _j\) with the restriction that \(\lim\widehat\beta_j=\infty\) but where the denominators \(\widehat \beta_j\) must not be bounded. Now choose \(z=x\), and consider the ring of formal power series \(\overline k[[y]]\); the authors show that any valuation \(\widehat\nu: \overline k[[y]]\to\overline {\mathbb R}_+\) which extends \(\nu_*\) and satisfies \(\widehat\nu(y)>0\) can be uniquely represented by a number \(\widehat\beta\in\overline{\mathbb R}_+\) and a series \(\sum_{j\geq1} a_jx^{\widehat\beta_j}\) as above with \(\widehat\beta>\widehat\beta_j\) for all \(j\) and with \(\widehat\beta=\lim\widehat\beta_j\) if the series is not finite. Accordingly, the authors classify the set \(\widehat{\mathcal V}_x\) of such valuations \(\widehat \nu\) in Def. 4.2, and they show in Prop. 4.4 that the partial ordering on \(\widehat{\mathcal V}_x\cup\{\widehat\nu_*\}\) where \(\widehat\nu_*\) is defined by \(\widehat\nu_*((y-\widehat\phi)=0\) for all \(\widehat\phi\in \overline k\) with \(\nu_*(\widehat\phi)\geq0\) gives this set a structure of a complete nonmetric tree rooted at \(\widehat\nu_*\). They show also in section 4.5 that \(\widehat{\mathcal V}_x\) is embedded as nonmetric tree is the Berkovich projective line over the local field \(\overline k\) [cf. V. G. Berkovich, “Spectral theory and analytic geometry over non-Archimedean fields”, Math. Surv. Monogr. 33 (1990; Zbl 0715.14013)].
In chapter 5 the authors introduce various topologies on the set \({\mathcal V}\) by using the tree structure of \({\mathcal V}\) and the fact that \({\mathcal V}\) is a set of function \(R\to \overline {\mathbb R}_+\). We just cite one result: The strong topology on \({\mathcal V}\) is strictly stronger than the weak topology. It is not locally compact. The three subsets of \({\mathcal V}\) consisting, respectively, of the divisorial, the irrational and the infinitely singular valuations are all strongly dense in \({\mathcal V}\) . They also introduce topologies on the set \({\mathcal V}_K\) of all equivalence classes of centered Krull valuations. One of them is the well-known Zariski topology.
With regard to chapter 6, we refer to the authors’ introduction to chapter 6: “We have already described several different approaches to the valuative tree. They all fundamentally derive from the definition of a valuation as a function on the ring \(R\). On the other hand, valuations can also be viewed geometrically as sequences of infinitely near points. It is therefore natural to ask whether the valuative tree can be recovered through a purely geometric construction.
In this chapter we show that this is indeed the case. The construction goes as follows. To any composition of point blowups we associate the dual graph of the exceptional divisor. The set of vertices of this graph is naturally a poset and the collection of all such posets form an injective system whose limit is a nonmetric tree \(\Gamma^*\) modeled on the rational numbers. By filling in the irrational points and adding all the ends we obtain a nonmetric tree \(\Gamma\) modeled on the real line. We call \(\Gamma\) the universal dual graph. Its points are encoded by sequences of infinitely near points above the origin of \(\mathbb C^2\). We show how to equip \(\Gamma\) with a natural Farey parameterization as well as with an integer valued multiplicity function.
The main result is then that there exists a natural isomorphism from the universal dual graph \(\Gamma\) to the valuative tree \({\mathcal V}\).”
In chapter 7 the ground is laid for applications of the theory to singularities. This chapter contains a detailed analysis of measures on trees. We only mention some of the results. Let \({\mathcal T}\) be a complete, parameterizable, rooted nonmetric tree, equipped with the weak topology, let \({\mathcal T}^o\) be the set on non-ends in \({\mathcal T}\), and let \({\mathcal M}\) be the set of complex Borel measures on \({\mathcal T}\). Now \({\mathcal M}\) can be considered as the dual of \(C({\mathcal T})\), hence \({\mathcal M}\) comes with a norm, turning it into a Banach space. To every measure \(\rho\in{\mathcal M}\) the authors associate a function \(f_\rho\) and a function \(g_\rho\) on \({\mathcal T}^o\); the collection of the functions \(f_\rho\) is denoted by \({\mathcal N}\) – these function are left continuous and of bounded variation – and the collection of the functions \({g_\rho}\) is denoted by \({\mathcal P}\) – these functions are called complex tree potentials. In section 7.2 they study the weak topology on \({\mathcal T}\). In section 7.4 the authors treat the space \({\mathcal N}\) of functions on \({\mathcal T}\); this space can be naturally identified with \({\mathcal M}\) [cf. Th. 7.38]. In section 7.6 the space \({\mathcal P}\) is studied; the identification of \({\mathcal M}\) and \({\mathcal P}\) is in Th. 7.50. Finally, in section 7.8, atomic measures in \({\mathcal M}\), which play an important role in the last chapter, are studied, and they characterize the subsets of \({\mathcal N}\) and \({\mathcal T}\) which correspond to atomic measures.
The last chapter is devoted to the treatment of singularities of ideals of \(R\) [cf. section 8.1] and cohomology classes of the vou\^te etoilèe in section 8.2. The authors announce that further applications of the results of chapter 7 to singularities of plurisubharmonic functions and to the dynamics of fixed point germs \((\mathbb C^2, 0),\circlearrowleft\) will be explored in three forthcoming papers [Valuative analysis of planar plurisubharmonic functions. Valuations and multiplier ideals. Eigenvaluations].
In section 8.2 they associate to an ideal \(I\) of \(R\) a positive tree potential \(g_I\), called the tree transform of \(I\), define a subset \({\mathcal M}^+_{\mathcal I}\) of \({\mathcal M}^+\) and prove that \(g_I\) corresponds to an element of \({\mathcal M}^+_{\mathcal I}\), and, conversely, every element \( \rho\in {\mathcal M}^+_{\mathcal I}\) gives rise to an ideal \(I_\rho\) of \(R\). The authors use the normalized blowup of an ideal to describe the geometric structure of \(I_\rho\). They can characterize integrally closed (=complete) ideals, prove the factorization theorem of Zariski for complete ideals and the existence of complete ideals satisfying the proximity relations [cf. J. Lipman, Contemp. Math. 159, 293–306 (1994; Zbl 0814.13016)]. They also calculate the mixed multiplicity \(e(I,J)\) of two primary ideals; these where studied by B. Teissier [Astérisque 7–8 (1973), 285–362 (1974; Zbl 0295.14003)].
Denote by \(\mathfrak B\) the set of all blow-ups above the origin in \(\mathbb C^2\); for each \(\pi\in \mathfrak B\) let \(X_\pi\) be the total space of \(\pi\), so that one has a proper morphism \(\pi: X_\pi\to (\mathbb C^2,0)\). Then \(\mathfrak B\) can be considered as an inverse system; the vou\^te etoilèe \(\mathfrak X\) is the projective limit \(\varprojlim_{\pi\in\mathfrak B}X_\pi\) [cf. H. Hironaka, Astérisque 7–8 (1973), 415–440 (1974; Zbl 0287.14006)]. Then one has \(H^2(\mathfrak X,\mathbb C)=\varinjlim H^2(X_\pi,\mathbb C)\); the intersection product on each \(H^2(X_\pi,\mathbb C)\) gives rise to an intersection product on \(H^2(\mathfrak X,\mathbb C)\) which is a negative definite hermitian form. To each cohomology class \(\omega\in H^2(\mathfrak X,\mathbb C)\) the authors associate a function \(g_\omega: {\mathcal V}_{\text{qm}}\to \mathbb C\) whose associated Laplacian is a complex atomic measure \(\rho_\omega\) on \({\mathcal V}\) supported by divisorial valuations, and they prove that the map \(\omega\mapsto \rho_\omega\) induces an isometry between \(H^2(\mathfrak X,\mathbb C)\) and the set of complex atomic measures supported on divisorial valuations.
In the first two appendices the authors study infinitely singular valuations and the tangent space at a divisorial valuation. In appendix C they give on overview of classifications of valuations and Krull valuations on \(R\), and, in appendix D, they show how to interpret some classical invariants of plane algebroid curves over \(\mathbb C\) in terms of the valuative tree, specifically in terms of skewness and multiplicity.

MSC:

14H20 Singularities of curves, local rings
05C05 Trees
13A18 Valuations and their generalizations for commutative rings
13B22 Integral closure of commutative rings and ideals
13F25 Formal power series rings
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