Sparsity of postcritically finite maps of ℙ k and beyond: A complex analytic approach

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → DerK(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.

We introduce the Hofer–Zehnder Γ-semicapacity [Formula: see text] of a symplectic manifold (M,ω) (or Γ-sensitive Hofer–Zehnder capacity) with respect to a subset Γ⊂π1(M)[Formula: see text] and prove that given a geometrically bounded symplectic manifold (M,ω) and an open subset N⊂M admitting a Hamiltonian free circle action with order greater than two then N has bounded Hofer–Zehnder Γ-semicapacity, where Γ⊂π1(N) is the subgroup generated by the orbits of the action. We give several applications of this result. Using Biran's decomposition theorem, we prove the following: let (M2n,Ω) be a closed Kähler manifold (n>2) with [Ω]∈H2(M,ℤ) and Σ a complex hypersurface representing the Poincaré dual of k[Ω], for some k∈ℕ. Suppose either that Ω vanishes on π2(M) or that k>2. Then there exists a decomposition of M into an open dense subset E such that E\Σ has finite Hofer–Zehnder Γ-semicapacity and an isotropic CW-complex, where Γ⊂π1(E\Σ) is the subgroup generated by the obvious circle action on the normal bundle of Σ. Moreover, we prove that if (M,Σ) is subcritical then M\Σ has finite Hofer–Zehnder Γ-semicapacity. We also show that given a hyperbolic surface M and TM endowed with the twisted symplectic form ω0+π*Ω, where Ω is the Kähler form on M, then the Hofer–Zehnder Γ-semicapacity of the domain Uk bounded by the hypersurface of kinetic energy k minus the zero section M0 is finite if k<1/2, where Γ⊂π1(Uk) is the subgroup generated by the fibers of SM. Finally, we consider the problem of the existence of periodic orbits on prescribed energy levels for magnetic flows. We prove that given any weakly exact magnetic field Ω on any compact Riemannian manifold M there exists a sequence of contractible periodic orbits of arbitrarily small energy, extending a previous result of Polterovich.

studied by Kazhdan and Lusztig in [KL]. For x = gG(O) ∈ X the G(O)-orbit (for the adjoint action) of Ad(g−1)(u) in g(O) depends only on x, and its image under g(O) g(C) is a well-defined G(C)-orbit in g(C). We say that x ∈Xu is regular if the associated orbit is regular in g(C). (Recall that an element of g(C) is regular if the nilpotent part of its Jordan decomposition is a principal nilpotent element in the centralizer of the semisimple part of its Jordan decomposition.) We write X reg for the (Zariski open) subset of regular elements in X. From now on we assume that u is regular semisimple with centralizer T, a maximal torus in G over F. Assume further that u is integral, by which we mean that X is nonempty. Kazhdan and Lusztig [KL] show that X is then a locally finite union of projective algebraic varieties, and in [KL, Sec. 4, Cor. 1] they show that the open subset X reg of X u is nonempty (and hence dense in at least one irreducible component of X). The action of T(F ) on X clearly preserves the subsets X andX reg. Bezrukavnikov [B] proved thatX u reg forms a single orbit under T(F ). (Actually Kazhdan–Lusztig and Bezrukavnikov consider only topologically nilpotent elements u, but the general case can be reduced to their special case by using the topological Jordan decomposition of u.) The goal of this paper is to characterize regular elements in X (for integral regular semisimple u as just described). When T is elliptic (in other words, F anisotropic modulo the center of G), the characterization gives no new information. At the other extreme, in the split case, the characterization gives a clear picture of what it means for a point in X to be regular. We will now state our characterization in the split case, leaving the more technical general statement to the next section (see Theorem 1). Fix a split maximal torus

Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals

In this research, we study the geometry of the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface through the analysis of singular spectral curves and the triality automorphism of Spin(8,C). We establish a characterization of triality invariance, proving that a Spin(8,C)-Higgs bundle admits a reduction to the exceptional group G2 if and only if its spectral curve is invariant under the induced triality action. This transforms the problem of detecting G2-structures into a question about spectral data. We decompose the discriminant locus of the Hitchin fibration into two disjoint strata: a fixed stratum arising from G2-Higgs bundles with singular spectral curves and a free stratum consisting of orbits of size three under triality. We prove the existence of non-abelian spectral data compatible with triality symmetry, showing that non-abelian phenomena persist in free triality orbits. To quantify symmetry breaking, we introduce a triality defect invariant, which measures the dimension of the quotient of the Prym variety by its triality-invariant sublocus, and we prove that Higgs bundles with positive defects form a Zariski open dense subset.

Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$.

On integrals of a class of measurable functions

In this paper, we prove that, if the data of a linear multiobjectiveprogramming problem are smooth functions of a parameter, then in theparameter space there is an open dense subset where the efficient solutionset of the problem can be locally represented as a union of some faces whosevertices and directions are smooth functions of the parameter.

A metric tree is a metric space such that between any two points there is a unique topological segment (i.e., a subset homeomorphic to an interval of the real line), and such that this segment is a geodesic (i.e., the isometric image of an interval). The theory of group actions on metric trees and of the structure of these trees on the boundary at infinity has many different aspects, and it has been made clear that this theory captures a lot of the interesting phenomena of hyperbolic geometry, or, to be more precise, of the theory of groups acting by isometrics on the hyperbolic space I/~ and on its sphere at infinity. One of the particular features, in the case of metric trees, is that the ideas and methods which are used are often of a combinatorial nature, and, probably, simpler and more intuitive than those of classical hyperbolic geometry. In particular, in [2] and [3], we give a series of criteria for the ergodicity of the geodesic flow associated with groups acting properly and isometrically on metric trees, which are discrete analogs of theorems due to Hopf and Sullivan in the case of hyperbolic manifolds. The invariant measures on these flows come from conformal densities on the boundary. This theory of conformal densities on the boundary was developed, in the case of hyperbolic manifolds, by Patterson and Sullivan (see [5] and [6]), and later, in the case of metric trees, in [11 . The result that we prove here concerns a Fatou-type behavior of certain natural functions associated with conformal densities on the boundary of a metric tree. A somehow more special result of this kind, in the case of conformal densities on the boundary of the hyperbolic n-space, is used by Sullivan in [6], in his proof for the probabilistic criterion of the ergodicity of the geodesic flow on a hyperbolic manifold (see also [7, Theorem 2.14]). T h e p r o o f of the corresponding result for metric trees, which we give in [31, involves a different kind of argument, although the Fatou-type behavior is also valid in that case, and, indeed, the aim of the present paper is to prove this result in the case of metric trees. Let us begin by describing the background. Let X be a metric tree which is complete and locally compact. We denote the distance between two points in X by Ix YlA geodesic ray (resp. a geodesic segment) is a map f from [0, oo[ (resp. from a closed interval) into X such that I f(r t ) f(r2)l = Irt r~l for every r, and r2. The boundary OX is the space of the ends of X. Every geodesic ray r defines a point in OX, which is denoted by r(oo). Let us also recall that between any two points x E X and r /E OX, there is a unique image of a geodesic ray, which is denoted by Ix, r/[. There is a natural topology on X U OX, which can easily be defined with the use of the ray structure from any given point in X, and for this topology, the space X U OX is compact and X lies in i t as an open dense subset. Better than that, X tO OX is, in fact, equipped with a natural class of metrics, and the induced class of metrics on OX, called visual metrics, is most useful. Let us recall its definition: For every point z E X, the visual metric I I~ on OX is defined by

Following the notion of stability introduced by T. Kobayashi and S. Nasrin in [14], we show in the context of a threadlike Lie group G that any non-Abelian discrete subgroup is stable. One consequence is that any resulting deformation space ℐ(Γ,G,H) is a Hausdorff space, where Γ acts on the threadlike homogeneous space G/H as a discontinuous subgroup. Whenever k = rank(Γ) > 3, this space is also shown to be endowed with a smooth manifold structure. But if k = 3, then ℐ(Γ,G,H) admits a smooth manifold structure as its open dense subset. These phenomena are strongly linked to the features of adjoint orbits of the basis group G on the parameter space ℛ(Γ,G,H) (which is semi-algebraic in this case) and specifically to their dimensions, as it will be seen throughout the paper. This also allows to provide a proof of the Local Rigidity Conjecture in this setup.

Dimers and cluster integrable systems

In this paper four classes of separable metric infinite-dimensional manifolds are studied; those which are locally the countable infinite product of lines, those which are locally open subsets of the Hubert cube, and those which are locally one of two dense sigma-compact subsets of the Hubert cube.A number of homeomorphism, product, characterization, and embedding theorems are obtained concerning these manifolds.1. Introduction.Let s denote the countable infinite product of open intervals and regard the Hubert cube Ix as the product of the closures of those intervals.A Frchet manifold (or f-manifold) is a separable metric space having an open cover by sets each homeomorphic to an open subset of s.This definition is justified since all separable infinite-dimensional Frchet spaces are known to be homeomorphic to s (see [7] for references).A Hubert cube manifold (or Q-manifold) is a separable metric space having an open cover by sets each homeomorphic to an open subset of/00.Let X be a separable metric space.A closed subset K of X is said to have Property Z in X (more briefly we say that K is a Z-set in X) provided that for any nonnull homotopically trivial open subset U of X, U\K is nonnull and homotopically trivial.Following Anderson [6] we say that a subset M of X has the (finitedimensional) compact absorption property, or (f-d) cap, in X if (1) M =Un=i Mn, where each Mn is a (finite-dimensional) compact Z-set in X such that Mn<=Mn+1, and (2) for each e > 0, each integer m > 0, and each (finite-dimensional) compact subset K of X, there is an integer n>0 and an embedding h: K-+ Mn such that h | K O Am = id and d(h, id) <e.We use the convention that (f-d) cap represents two alternatives in an obvious way, one being cap and the other being f-d cap.In this sense we say that Mis an (f-d) cap-set for A'provided that M has the cap (or f-d cap) in X.In [6] R. D. Anderson has established the following theorem: Let Xbe I or and let M, N be subsets of X, both having the (f-d) cap in X.Then there is a

For some pseudo-Hermitian symmetric manifolds there are Zariski open subsets which are biholomorphically equivalent to tube domains. These tube domains correspond to homogeneous nonconvex cones.

Groupes d'Artin, arrangements projectifs et groupes fondamentaux des variétés complexes algébriques lisses

We show that for every $n\ge 7$ there is a smooth rational affine variety with exotic model. Moreover, we show that for every $n\ge 6$ there are Zariski open subsets $U_1, U_2$ of $\mathbb {C}^n$, such that they are holomorphically but not algebraically equivalent.