Regular points in affine Springer fibers

We introduce a new kind of regularity in the Dirichlet problem for the heat equation, called mild regularity. The mildly regular boundary points include all types of regular points previously considered. Just as regular boundary points have barriers associated with them, the mildly regular boundary points have mild barriers, which are just as important. We consider whether the mild regularity of a boundary point for an open set is inherited by an open subset or superset. We detail the connection of mild regularity with the Green function. We consider, in particular, subdomains of an arbitrary open set that satisfy a certain connectivity condition which takes account of the special nature of the temporal variable, called zones of influence. Our main theorem states that, for any zone of influence, the set of boundary points that are not mildly regular is caloric measure null.

We consider the following type of minimization problem. Given a real valued function f on an open nonempty subset U of a real Banach space E we are looking for a minimum of f on the subset of U which is determined by the constraint condition \(g(x)=y\) where \(g: U \longrightarrow F\) is a given function on U with values in some Banach space F and \(y\in F\) is some selected point, i.e., we are looking for a minimum of f on the level set \([g=y]=\left\{x \in U; g(x)=y\right\}\) of g. If g is of class \(\mathcal{C}^1\) the level surfaces of g have a proper tangent space at regular points x 0 (i.e., \(g^{\prime}(x_0)\) is surjective) at which the nullspace of \(g^{\prime}(x_0)\) has a topological complement. This can be used to introduce local coordinates on the level surface \([g=y]\) in the neighborhood of such a regular point and the constrained minimization problem becomes a minimization problem on an open subset (in terms of the local coordinates) and the characterization of extremal points as derived earlier applies. The conclusion is that a Lagrange multiplier exists, i.e., a continuous linear mapping \(\ell: F \longrightarrow \mathbb{R}\) such that \(f^{\prime}(x_0)= \ell \circ g^{\prime}(x_0)\). The geometrical meaning of this condition is first explained heuristically in a two dimensional setting. In practical cases of interest the condition of the existence of a topological compliment is always met. The well known finite dimensional case is easily obtained. We conclude by some comments on the constrained minimization problem of Dido.

An endomorphism f : ℙ k → ℙ k of degree d ≥ 2 is said to be postcritically finite (or PCF) if its critical set Crit ( f ) is preperiodic, i.e. if there are integers m > n ≥ 0 such that f m ( Crit ( f ) ) ⊆ f n ( Crit ( f ) ) . When k ≥ 2 , it was conjectured in [61] that, in the space End d k of all endomorphisms of degree d of ℙ k , such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space Poly d 2 of all regular polynomial endomorphisms of degree d ≥ 2 of the affine plane 𝔸 2 , we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set. The key object in the article are the complex bifurcation measure and its properties. The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation.

Regular elements in bases of Hecke algebras

We develop a theory of cyclic elements in semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. In particular, we classfiy all nilpotents giving rise to semisimple and regular semisimple cyclic elements. As an application, we obtain an explicit construction of all regular elements in Weyl groups.

Let $\mathfrak{g}$ be a simple finite-dimensional Lie superalgebra with a non-degenerate supersymmetric even invariant bilinear form, $f$ a nilpotent element in the even part of $\mathfrak{g}$, $\Gamma$ a good grading of $\mathfrak{g}$ for $f$ and $\mathcal{W}^{k}(\mathfrak{g},f;\Gamma)$ the $\mathcal{W}$-algebra associated with $\mathfrak{g},f,k,\Gamma$ defined by the generalized Drinfeld-Sokolov reduction. In this paper, we present each $\mathcal{W}$-algebra as the intersection of kernels of the screening operators, acting on the tensor vertex superalgebra of an affine vertex superalgebra and a neutral free superfermion vertex superalgebra. As applications, we prove that the $\mathcal{W}$-algebra associated with a regular nilpotent element in $\mathfrak{osp}(1,2n)$ is isomorphic to the $\mathcal{W}B_{n}$-algebra introduced by Fateev and Lukyanov, and that the $\mathcal{W}$-algebra associated with a subregular nilpotent element in $\mathfrak{sl}_{n}$ is isomorphic to the $\mathcal{W}^{(2)}_{n}$-algebra introduced by Feigin and Semikhatov.

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 $X$.

The symmetric algebra g (denoted S(\g)) over a Lie algebra \g (frak g) has the structure of a Poisson algebra. Assume \g is complex semi-simple. Then results of Fomenko- Mischenko (translation of invariants) and A.Tarasev construct a polynomial subalgebra \cal H = \bf C[q_1,...,q_b] of S(\g) which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of \g. Let G be the adjoint group of \g and let \ell = rank \g. Identify \g with its dual so that any G-orbit O in \g has the structure (KKS) of a symplectic manifold and S(\g) can be identified with the affine algebra of \g. An element x \in \g is strongly regular if \{(dq_i)_x\}, i=1,...,b, are linearly independent. Then the set \g^{sreg} of all strongly regular elements is Zariski open and dense in \g, and also \g^{sreg \subset \g^{reg} where \g^{reg} is the set of all regular elements in \g. A Hessenberg variety is the b-dimensional affine plane in \g, obtained by translating a Borel subalgebra by a suitable principal nilpotent element. This variety was introduced in [K2]. Defining Hess to be a particular Hessenberg variety, Tarasev has shown that Hess \subset \g^sreg. Let R be the set of all regular G-orbits in \g. Thus if O \in R, then O is a symplectic manifold of dim 2n where n= b-\ell. For any O\in R let O^{sreg} = \g^{sreg}\cap O. We show that O^{sreg} is Zariski open and dense in O so that O^{sreg} is again a symplectic manifold of dim 2n. For any O \in R let Hess (O) = Hess \cap O. We prove that Hess(O) is a Lagrangian submanifold of O^{sreg} and Hess =\sqcup_{O \in R} Hess(O). The main result here shows that there exists, simultaneously over all O \in R, an explicit polarization (i.e., a "fibration" by Lagrangian submanifolds) of O^{sreg} which makes O^{sreg} simulate, in some sense, the cotangent bundle of Hess(O).

Unique factorization and small regular local noether lattices

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.

Let M be a connected generic real-analytic CR-submanifold of a finite-dimensional complex vector space E. Suppose that for every a∈M the Lie algebra \({\frak{hol}}(M,a)\) of germs of all infinitesimal real-analytic CR-automorphisms of M at a is finite-dimensional and its complexification contains all constant vector fields \(\alpha\frac {\partial}{\partial z} \), α∈E, and the Euler vector field \(z\frac{\partial}{\partial z} \). Under these assumptions we show that: (I) every \({\frak{hol}}(M,a)\) consists of polynomial vector fields, hence coincides with the Lie algebra \({\frak{hol}}(M)\) of all infinitesimal real-analytic CR-automorphisms of M, (II) every local real-analytic CR-automorphism of M extends to a birational transformation of E, and (III) the group Bir (M) generated by such birational transformations is realized as a group of projective transformations upon embedding E as a Zariski open subset into a projective algebraic variety. Under additional assumptions the group Bir (M) is shown to have the structure of a Lie group with at most countably many connected components and Lie algebra \({\frak{hol}}(M)\). All of the above results apply, for instance, to Levi non-degenerate quadrics, as well as a large number of Levi degenerate tube manifolds.

1. We consider a compact, connected metric space AI which we shall call the continuum M. A point p of Al is said to be a regular point if for each E >0 there exists a neighborhood U, of p (i.e., an open subset of M containing p) such that d(U,) 0 there exists a neighborhood Up such that d(Up) <E and the cardinal number of F(Up) <oa, and (2) a is the smallest cardinal number for which (1) is true. Regular points of no finite order are called points of order co. Let Ma denote the set of all points of M of order a. Then