Volume 24, Issue 3

An automorphism of a spherical building is called domestic if it maps no chamber to an opposite chamber. In this paper we classify domestic automorphisms of spherical buildings of classical type.

In this work we deal with complete and stochastically complete gradient conformal solitons, which correspond to natural extensions of gradient Yamabe solitons, gradient k -Yamabe solitons, almost gradient Yamabe solitons and h -almost gradient Yamabe solitons. Under mild hypotheses on the potential function, we obtain new characterization and triviality results via applications of suitable generalized maximum principles.

Building upon our previous joint work with A. Joshi and M. Poddar, we continue our study of abelian covers of rational surfaces which are branched over line arrangements. We use these covers as building blocks to construct new infinite families of closed simply connected nonspin irreducible symplectic 4-manifolds with positive signature.

In contrast to classical knots, the knot type of a genus two handlebody-knot is not determined by its exterior, and it is often a challenging task to distinguish handlebody-knots with homeomorphic exteriors. The present paper considers an invariant (the annulus diagram), defined via Johannson’s characteristic submanifold theory and the Koda–Ozawa classification for essential annuli, and demonstrates its capability to distinguish such handlebody-knots; particularly, the annulus diagram is able to differentiate members in the handlebody-knot families given by Motto and Lee–Lee.

For n , d ∈ ℕ, the cone 𝓟 n +1,2 d of positive semidefinite real forms in n + 1 variables of degree 2 d contains the subcone Σ n +1,2 d of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the Hilbert cases ( n + 1, 2 d ) with n + 1 = 2 or 2 d = 2 or ( n + 1, 2 d ) = (3, 4). In this paper, we induce a filtration of intermediate cones between Σ n +1,2 d and 𝓟 n +1,2 d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties V k − n ⊊ … ⊊ V n ⊊ … ⊊ V 0 containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n + 1 variables of degree d . By showing that V 0 , …, V n (and V n +1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σ n +1,2 d . We moreover prove that, in the non-Hilbert cases of ( n + 1)-ary quartics for n ≥ 3 and ( n + 1)-ary sextics for n ≥ 2, all the remaining cone inclusions are strict.

Given k points ( x i , y i ) ∈ ℙ 2 × ℙ 2 , we characterize rank deficiency of the k × 9 matrix Z k with rows xi⊤ ⊗ yi⊤ $\begin{array}{} \displaystyle x_i^\top \otimes y_i^\top \end{array}$ in terms of the geometry of the point configurations { x i } and { y i }. In this paper we present results for k ≤ 6. For k ≤ 5, the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case k = 6 the rank-drop locus is captured by the classical theory of cubic surfaces. The results for k = 7, 8 and 9 are presented in the sequel [7] to this paper.

Given k ≤ 9 points ( x i , y i ) ∈ ℙ 2 × ℙ 2 , we characterize rank deficiency of the k × 9 matrix Z k with rows xi⊤ ⊗ yi⊤ $\begin{array}{} \displaystyle x_i^\top \otimes y_i^\top \end{array}$ in terms of the geometry of the point configurations { x i } and { y i }. In [3] we presented results for the cases k ≤ 6. In this paper we deal with the remaining cases k = 7, 8 and 9. The results involve the interplay of quadric surfaces, cubic curves and Cremona transformations.

Let X be a compact Riemann surface of genus g ≥ 2 and let G be the group Spin(8, ℂ) or PSO(8, ℂ). Let τ be a choice of a non-trivial outer automorphism of G of order 3, called triality. Suppose that X is equipped with a holomorphic order 3 automorphism α . Then α acts on the moduli space M ( G ) of principal G -bundles over X by taking the pull-back, and τ also acts on M ( G ), by modifying the action of G on the principal bundles through any automorphism of G of order 3 representing τ . The combination of these two actions gives an automorphism of order 3 of the moduli space M ( G ). In this work, the notion of α -trialitarian G -bundle is introduced to describe the fixed points of the automorphism of M ( G ) mentioned above. In particular, an equivalent notion of stability and polystability for such bundles is proved, and an explicit construction of them is provided.