Introduction to the Theory of Analytic Spaces

Narasimhan, Raghavan

doi:10.1007/bfb0077071

Cited by 327 publications

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“…We now recall some facts on analytic sets, see [79,105]. Let X be an arbitrary complex manifold of dimension k 1 .…”

Section: Thereforementioning

confidence: 99%

“…The notions of dimension, of Zariski topology and of holomorphic maps can be extended to analytic spaces. The precise definition uses the local ring of holomorphic functions, see [79,105]. An analytic space is normal if the local ring of holomorphic functions at every point is integrally closed.…”

Section: Thereforementioning

confidence: 99%

“…The reader will find in [30,79,85,96,105] the basics on currents on complex manifolds. We also refer to [30,75,88,125] for the theory of compact Kähler manifolds.…”

Section: Appendix a Currents And Pluripotential Theorymentioning

confidence: 99%

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Dynamics in Several Complex Variables

Abate

2017

Lecture Notes in Mathematics

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The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic (p.s.h.) functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. Nervertheless, we choose to show their effectiveness and to describe the theory for two large families of maps: the endomorphisms of projective spaces and the polynomial-like mappings.The first chapter deals with holomorphic endomorphisms of the projective space P k . We establish the first properties and give several constructions for the Green currents T p and the equilibrium measure µ = T k . The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems. We show the existence of a proper algebraic set E , totally invariant, i.e. f −1 (E ) = f (E ) = E , such that when a ∈ E , the probability measures, equidistributed on the fibers f −n (a), converge towards the equilibrium measure µ, as n goes to infinity. A similar result holds for the restriction of f to invariant subvarieties. We survey the equidistribution problem when points are replaced by varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of µ: Kmixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space DSH(P k ) of differences of quasi-p.s.h. functions. In particular, we show that the measure µ is moderate, i.e. µ, e α|ϕ| ≤ c, on bounded sets of ϕ in DSH(P k ), for suitable positive constants α, c. Finally, we study the entropy, the Lyapounov exponents and the dimension of µ.The second chapter develops the theory of polynomial-like maps, i.e. proper holomorphic maps f : U → V where U, V are open subsets of C k with V convex and U ⋐ V . We introduce the dynamical degrees for such maps and construct the equilibrium measure µ of maximal entropy. Then, under a natural assumption on the dynamical degrees, we prove equidistribution properties of points and various statistical properties of the measure µ. The assumption is stable under small pertubations on the map. We also study the dimension of µ, the Lyapounov exponents and their variation.Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.

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“…We now recall some facts on analytic sets, see [79,105]. Let X be an arbitrary complex manifold of dimension k 1 .…”

Section: Thereforementioning

confidence: 99%

Section: Thereforementioning

confidence: 99%

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Dynamics in Several Complex Variables

Abate

2017

Lecture Notes in Mathematics

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“…After a finite number of steps we conclude that U k ⊂ W , and so x is an interior point of W . According to [10], the usual definition of C-analytic set is equivalent to the following one. …”

Section: Analytic Manifolds and Actionsmentioning

confidence: 99%

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Candel

Quiroga-Barranco

2003

Geometriae Dedicata

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“…complex) analytic sets in Ω properly contained in X. Any germ of a real or complex analytic set X admits decomposition into a finite union of irreducible germs, while a global decomposition into irreducible components in general exists only for complex analytic sets (see, e.g., [12]). …”

Section: Introductionmentioning

confidence: 99%

On the holomorphic closure dimension of real analytic sets

Adamus

Shafikov

2011

Trans. Amer. Math. Soc.

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Abstract. Given a real analytic (or, more generally, semianalytic) set R in C n (viewed as R 2n ), there is, for every p ∈R, a unique smallest complex analytic germ X p that contains the germ R p . We call dim C X p the holomorphic closure dimension of R at p. We show that the holomorphic closure dimension of an irreducible R is constant on the complement of a closed proper analytic subset of R, and we discuss the relationship between this dimension and the CR dimension of R. Main resultsGiven a real analytic set R in C n (we may identify C n with R 2n ), we consider the germ R p at a point p ∈ R and define X p to be the unique smallest complex analytic germ at p that contains R p . We will call dim C X p the holomorphic closure dimension of R at p (denoted by dim HC R p ). It is natural to ask how this dimension varies with p ∈ R. In [15, Thm. 1.1] the second author showed that, if R is irreducible of pure dimension, then dim HC R p is constant on a dense open subset of R, which allows us to speak of the generic holomorphic closure dimension of R in this case. Upper semicontinuity implies that the generic value of dim HC R p is the smallest value of the holomorphic closure dimension on R. It remained an open problem as to whether the jump in the holomorphic closure dimension actually occurs. In the present paper we construct real analytic sets with non-constant holomorphic closure dimension and study the structure of the locus of points where this dimension is not generic. Then there exists a closed real analytic subset S ⊂ R of dimension less than d such that the holomorphic closure dimension of R is constant on R \ S.We note that the holomorphic closure dimension is well defined on all of R, including its singular locus. If R is not of pure dimension, then (see, e.g., [5]) the locus of smaller dimension is contained in a proper real analytic subset of dimension less than d, which can be included into S. In particular, S may have a non-empty interior in R (see Example 6.4). We also note that unless additional conditions are

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Introduction to the Theory of Analytic Spaces

Cited by 327 publications

References 0 publications

Dynamics in Several Complex Variables

Dynamics in Several Complex Variables

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On the holomorphic closure dimension of real analytic sets

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