Volume 2016, Issue 716

We introduce a holomorphic sheaf ℰ${\mathcal{E}}$ on a Sasakian manifold S and study two new notions of stability for ℰ${\mathcal{E}}$ along the Sasaki–Ricci flow related to the ‘jumping up’ of the number of global holomorphic sections of ℰ${\mathcal{E}}$ at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bounded in C0${C^{0}}$ along the flow, and the C∞${C^{\infty}}$ closure of the Sasaki structure on S under the diffeomorphism group does not contain a Sasaki structure with strictly more global holomorphic sections of ℰ${\mathcal{E}}$, then the Sasaki–Ricci flow converges exponentially fast to a Sasaki–Einstein metric. Secondly, we show that if the Futaki invariant vanishes, and the lowest positive eigenvalue of the ∂¯${\bar{\partial}}$ Laplacian on global sections of ℰ${\mathcal{E}}$ is bounded away from zero uniformly along the flow, then the Sasaki–Ricci flow converges exponentially fast to a Sasaki–Einstein metric.

We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed Riemannian manifold M carrying an effective and isometric action of a compact, connected Lie group G . Using resolution of singularities, we determine the asymptotic distribution of eigenvalues along the isotypic components, and relate it to the reduction of the corresponding Hamiltonian flow, proving that the reduced spectral counting function satisfies Weyl’s law, together with an estimate for the remainder.

We prove a comparison formula for the Donaldson–Thomas curve-counting invariants of two smooth and projective Calabi–Yau threefolds related by a flop. By results of Bridgeland any two such varieties are derived equivalent. Furthermore there exist pairs of categories of perverse coherent sheaves on both sides which are swapped by this equivalence. Using the theory developed by Joyce we construct the motivic Hall algebras of these categories. These algebras provide a bridge relating the invariants on both sides of the flop.

We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra 𝔤$\mathfrak{g}$. These are sheaves on locally closed subvarieties of the projective variety 𝔼⁢(r,𝔤)$\mathbb{E}(r,\mathfrak{g})$ of elementary subalgebras of 𝔤$\mathfrak{g}$ of dimension r . We show that representations of constant radical or socle rank studied in [5] which generalize modules of constant Jordan type lead to algebraic vector bundles on 𝔼⁢(r,𝔤)$\mathbb{E}(r,\mathfrak{g})$. When 𝔤$\mathfrak{g}$ is the Lie algebra of an algebraic group G , rational representations of G enable us to realize familiar algebraic vector bundles on G -orbits of 𝔼⁢(r,𝔤)$\mathbb{E}(r,\mathfrak{g})$.

In this paper we study a generalization of the Kähler–Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)$(1,1)$-form. We show that when a twisted Kähler–Einstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the Kähler–Ricci flow, and it builds on work of Tian–Zhu.

In this paper we study characters on special linear groups SLn⁢(R)${\mathrm{SL}_{n}(R)}$, where R is either an infinite field or the localization of an order in a number field. We give several applications to the theory of measure-preserving actions, operator-algebraic superrigidity, and almost homomorphisms.

For any n≥3${n\geq 3}$, we prove that there are equivalences between • irreducible n -dimensional non-degenerate complex projective varieties 𝑿⊂ℙ2⁢n+1${{\boldsymbol{X}}\subset\mathbb{P}^{2n+1}}$ different from rational normal scrolls and 3-covered by cubic curves, up to projective equivalence, • n -dimensional complex Jordan algebras 𝑱${{\boldsymbol{J}}}$ of rank 3, up to isotopy, • quadro-quadric Cremona transformations 𝑪:ℙn-1⇢ℙn-1${{\boldsymbol{C}}:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{n-1}}$ of the complex projective space of dimension n-1${n-1}$, up to linear equivalence. These three equivalences form what we call the 𝑋𝐽𝐶${\mathit{XJC}}$ -correspondence . We also provide some applications to the classification of particular types of varieties in the class defined above and of quadro-quadric Cremona transformations.