A study of the Hilbert-Mumford criterion for the stability of projective varieties

By J. Ross, Richard P. Thomas. 2004

188 citations recorded by the source corpus.

We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties $(X,L)$; in particular for K- and Chow stability. For each type of stability this leads to a concept of slope $\mu$ for varieties and their subschemes; if $(X,L)$ is semistable then $\mu(Z)\le\mu(X)$ for all $Z\subset X$. We give examples such as curves, canonical models and Calabi-Yaus. We prove various foundational technical results towards understanding the converse, leading to partial results; in particular this gives a geometric (rather than combinatorial) proof of the stability of smooth curves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7f9b730ab7ead013e289c61c2d59aba8e32b1889

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Citations in the corpus

Extremal metrics and K-stability (PhD thesis) (2006, background): “The destabilising test-configuration is an example of deformation to the normal cone of a subvariety, studied by Ross and Thomas [29] (see also Section 5.2), except we need to take into account the extremal C∗ -action as well.”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, background): “Most arguments are inspired by those in [29, 59, 62, 73, 82].”
Test configurations and Okounkov bodies (2010, methodology): “nds to inﬁnity to the measure G[T ]∗dλ|∆(L). We embed our test conﬁguration into Ctimes a projective space PN,so that the associated action comes from a C×-action onPN.This we can always do (see e.g. [18]). The manifoldXlies embeddedin PN,and we thus via the action get a family Xτ of subamnifolds. A”
K-stability of constant scalar curvature polarization (2008, unknown): “2 ω(j) = | ∂η¯ (j)|2 R ω(j) by (5.15), the sequence of integrals M |∂η¯ (j)|2 ω∞ω n ∞, j= 1,2,..., is bounded, so that η(j), j= 1,2,..., is a bounded sequence in the Sobolev space L1,2(M,ω ∞). Now by [21], we may assume that n≥ 2. Then replacing η(j), j= 1,2,..., by its subsequence if necessary, we ”
Optimal bounds for the volumes of Kähler-Einstein Fano manifolds (2015, unknown): “n enables us to calculate (a kind of) the limit of those Ding invariants via the saturation of the given ﬁltration. See Theorem 4.9 in detail. In Section 4.3, motivated by the work of Ross and Thomas [RT07], we consider speciﬁc test conﬁgurations obtained by the natural ﬁltered linear series coming ”
A generalization of the Ross--Thomas slope theory (2009, methodology): “dson-Futaki invariant) is positive (resp. nonnegative, positive if X 6∼= X× A1 and nonnegative otherwise). We should note that the original K-stability of [Don02] is what is called K-polystability in [RT07]. We follow the convention of [RT07]. These are related as follows. Asymptotically Chow stable”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, background): “There are also other related notion of stability; see, for example, [40], [38], etc.”
Alpha invariant and K-stability of Q-Fano varieties (2010, background): “ an arbitrary semi test conﬁguration (X,L) of order r, we can also deﬁne the Donaldson-Futaki invariant as well by setting w(Kr) as the total weight of the induced action on H0(X,L⊗K)/tH0(X,L⊗K) (cf. [28]). We say that (X,L) is Ksemistable if and only if DF ≥ 0 for any non-trivial test conﬁguration.”
Testing log K-stability by blowing up formalism (2011, background): “It is a natural extension of the pioneering [23], which treats N = 1 case in the notation below.”
K-stability of Fano varieties: an algebro-geometric approach (2020, background): “ymptotic Chow stability. See [RT07, Section 3] for a detailed discussion, especially for the implications among diﬀerent notions of stability. Nevertheless, the directions which were not addressed in [RT07] were more subtle. While it is proved in [Don02] that any polarized manifold (X,L) with a cscK”
Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics (2008, background): “the limit cycle is a smooth (or normal) scheme. This is the deﬁnition adopted by most researchers in the ﬁeld today . As a small sample the reader should consult the following interesting papers ([3],[36],[38],[39],[2],[22],[52] and [21]) . In 2004 the CM 3In this paper Galways denotes the special l”
On K‐stability and the volume functions of ℚ‐Fano varieties (2015, background): “(For the theory of slope stability, we refer the readers to [21, 47].”
A primer on Seshadri constants (2008, background): “In [ RosTho07 ], Ross and Thomas studied various notions of stability for polarized varieties, each of which leads to a concept of slope for varieties and subschemes.”
Lower bounds on the Calabi functional (2005, background): “mbedded as a C∗-invariant subscheme in the product P(U∗ k ) × C, extending the embedding of the central ﬁbre X 0 ⊂ X by the complete linear system U k in P(U∗ k ) = P(U∗ k )×{0}. This is explained in [17]. In essence we consider the C∗-equivariant bundle U = π ∗L k over C and pick an equivariant tri”