MathDB

The coupled equations imply alpha-K-polystability for triples

About 15 years old · traced to

Let (X,L)(X,L) be a smooth compact polarized variety and let EE be a holomorphic vector bundle over XX of rank rr and degree zero. For positive coupling constants α0,α1∈R\alpha_0,\alpha_1\in\mathbb R, set α=rπ2α1α0\alpha=\frac{r\pi^2\alpha_1}{\alpha_0}. Let ω∈c1(L)\omega\in c_1(L) be a Kähler metric and HH a Hermitian metric on EE. The coupled-equations conjecture. If ω\omega and HH satisfy the coupled equations, then the triple (X,L,E)(X,L,E) is α\alpha-K-polystable. This conjecture is motivated by the Kempf–Ness framework and by results showing vanishing of the invariant for product test configurations and nonnegativity for base-preserving test configurations. The source notes that the limiting α→0\alpha\to0 cscK case had been verified through work of Tian, Donaldson, Mabuchi and Stoppa.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The coupled equations imply alpha-K-polystability for triples

    Let (X,L,E)(X,L,E) be a smooth polarized variety and a holomorphic vector bundle over XX of rank rr and degree zero, where LL is ample. Let ω∈c1(L)\omega \in c_1(L) be a Kähler metric and let HH be a Hermitian metric on EE, with coupling constants α0,α1\alpha_0,\alpha_1 and α=rπ2α1α0\alpha=\frac{r\pi^2\alpha_1}{\alpha_0}. The coupled-equations conjecture. If ω\omega and HH satisfy the coupled equations, then the triple (X,L,E)(X,L,E) is α\alpha-K-polystable. This conjecture proposes an algebro-geometric obstruction to solutions of the coupled Kähler and Yang–Mills equations; the text gives supporting evidence from vanishing and nonnegativity of the associated invariant for product and base-preserving test configurations.

    source: Mario Garcia-Fernandez, “Coupled equations for Kähler metrics and Yang-Mills connections (Thesis)”, arXiv:1102.0985 (2011).

References

Primary source

Mario Garcia-Fernandez, “Coupled equations for Kähler metrics and Yang-Mills connections (Thesis)”, arXiv:1102.0985 (2011).

kahler-geometrystabilityyang-mills-theory
Posted

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.