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Register a new userApplications of Foliation Theory to Invariant Theory
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Ricardo Augusto Emmanuel Mendes
Publication date
2020
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English
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We give applications of Foliation Theory to the Classical Invariant Theory of real orthogonal representations, including: The solution of the Inverse Invariant Theory problem for finite groups. An if-and-only-if criterion for when a separating set is a generating set. And the introduction of a class of generalized polarizations which, in the case of representations of finite groups, always generates the algebra of invariants of their diagonal representations.
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Classically, theorems of Fatou and Julia describe the boundary regularity of functions in one complex variable. The former says that a complex analytic function on the disk has non-tangential boundary values almost everywhere, and the latter describes when a function takes an extreme value at a boundary point and is differentiable there non-tangentially. We describe a class of intermediate theorems in terms of averaged Julia-Fatou quotients. Boundary regularity is related to integrability of certain quantities against a special measure, the so-called Nevanlinna measure. Applications are given to spectral theory.
In this paper, we introduce a sub-family of the usual generalized Wronskians, that we call geometric generalized Wronskians. It is well-known that one can test linear dependance of holomorphic functions (of several variables) via the identical vanishing of generalized Wronskians. We show that such a statement remains valid if one tests the identical vanishing only on geometric generalized Wronskians. It turns out that geometric generalized Wronskians allow to define intrinsic objects on projective varieties polarized with an ample line bundle: in this setting, the lack of existence of global functions is compensated by global sections of powers of the fixed ample line bundle. Geometric generalized Wronskians are precisely defined so that their local evaluations on such global sections globalize up to a positive twist by the ample line bundle. We then give three applications of the construction of geometric generalized Wronskians: one in intermediate hyperbolicity, and two in foliation theory. In intermediate hyperbolicity, we show the algebraic degeneracy of holomorphic maps from C p to a Fermat hypersurface in P N of degree $delta$ > (N + 1)(N -- p): this interpolates between two well-known results, namely for p = 1 (first proved via Nevanlinna theory) and p = N -- 1 (in which case the Fermat hypersurface is of general type). The first application in foliation theory provides a criterion for algebraic integrability of leaves of foliations: our criterion is not optimal in view of current knowledges, but has the advantage of having an elementary proof. Our second application deals with positivity properties of adjoint line bundles of the form K F + L, where K F is the canonical bundle of a regular foliation F on a smooth projective variety X, and where L is an ample line bundle on X.
We determine the Waring ranks of all sextic binary forms using a Geometric Invariant Theory approach. In particular, we shed new light on a claim by E. B. Elliott at the end of the 19th century concerning the binary sextics with Waring rank 3.
In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.
In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{Gamma}$ where $Gamma$ is a product of general linear groups over a field $K$ of characteristic zero, and $U$ is a finite dimensional rational representation of $Gamma$. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where $U=text{End}(W^{oplus k})$ and $Gamma = text{GL}(W)$ and on the case where $U=text{End}(Votimes W)$ and $Gamma = text{GL}(V)times text{GL}(W)$, though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series.
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