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A polarized variety is K-stable if, for any test configuration, the Donaldson–Futaki invariant is positive. In this paper, inspired by classical geometric invariant theory, we describe the space of test configurations as a limit of a direct system of Tits buildings. We show that the Donaldson–Futaki invariant, conveniently normalized, is a continuous function on this space. We also introduce a pseudo-metric on the space of test configurations. Recall that K-stability can be enhanced by requiring that the Donaldson–Futaki invariant is positive on any admissible filtration of the co-ordinate ring. We show that admissible filtrations give rise to Cauchy sequences of test configurations with respect to the above mentioned pseudo-metric.
We show that the anti-canonical volume of an
-Fano variety is bounded from above by certain invariants of the local singularities, namely
for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.
be a reductive group over an algebraically closed subfield
of characteristic zero,
an observable subgroup normalised by a maximal torus of
-variety acted on by
. Popov and Pommerening conjectured in the late 1970s that the invariant algebra
is finitely generated. We prove the conjecture for: (1) subgroups of
closed under left (or right) Borel action and for: (2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of
be a unipotent group which is graded in the sense that it has an extension
by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of
are strictly positive. We study embeddings of
in a general linear group
which possess Grosshans-like properties. More precisely, suppose
acts on a projective variety
and its action extends to an action of
which is linear with respect to an ample line bundle on
. Then, provided that we are willing to twist the linearization of the action of
by a suitable (rational) character of
, we find that the
-invariants form a finitely generated algebra and hence define a projective variety
; moreover, the natural morphism from the semistable locus in
is surjective, and semistable points in
are identified in
if and only if the closures of their
-orbits meet in the semistable locus. A similar result applies when we replace
by its product with the projective line; this gives us a projective completion of a geometric quotient of a
-invariant open subset of
by the action of the unipotent group
We show that any
-dimensional Fano manifold
is K-stable, where
is the alpha invariant of
introduced by Tian. In particular, any such
admits Kähler–Einstein metrics and the holomorphic automorphism group
be a finite extension of
be an absolutely simple split reductive group over
, and let
be a maximal unramified extension of
. To each point in the Bruhat–Tits building of
, Moy and Prasad have attached a filtration of
by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy–Prasad filtration quotient to contain stable vectors for the action of the reductive quotient. Our work extends earlier results by Reeder and Yu, who gave a classification in the case when
is sufficiently large. By passing to a finite unramified extension of
if necessary, we obtain new supercuspidal representations of
Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight, there is an associated quasi-projective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many d-dimensional toric quiver varieties. A procedure for their classification is outlined.
We give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components: if a polarized projective variety X is a chain of subvarieties Xi satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of Xi translated by a vector τ, which can be readily computed from the ideal of Xi. The decomposition is in the strongest sense in that the vertices of the state polytope of X are precisely the sum of vertices of the state polytopes of Xi translated by τ. We also give a similar decomposition formula for the Hilbert–Mumford index of the Hilbert points of X. We give a few examples of the state polytope and the Hilbert–Mumford index computation of reducible curves, which are interesting in the context of the log minimal model program for the moduli space of stable curves.
The notion of Berman–Gibbs stability was originally introduced by Berman for
. We show that the pair
is K-stable (respectively K-semistable) provided that
is Berman–Gibbs stable (respectively semistable).
We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated ‘Latin square conjecture’ due to Alon and Tarsi or, more precisely, on the validity of an equivalent ‘column Latin square conjecture’ due to Huang and Rota.
We introduce an analogue in hyperkähler geometry of the symplectic implosion, in the case of
actions. Our space is a stratified hyperkähler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.
Let G be a complex connected reductive group. The Parthasarathy–Ranga Rao–Varadarajan (PRV) conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for ‘the same reason’ as the PRV ones.
We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the (n−1)-sphere.
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