Band 2015, Heft 709

By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kähler manifold X , the Yang–Mills flow converges away from an analytic singular set. If E is semi-stable, then the limiting metric is Hermitian-Einstein and will decompose the limiting bundle E ∞ into a direct sum of stable bundles. Bando and Siu prove E ∞ can be extended to a reflexive sheaf E^∞${\hat{E}_\infty }$ on all of X . In this paper, we construct an isomorphism between E^∞${\hat{E}_\infty }$ and the double dual of the stable quotients of the graded Seshadri filtration Gr s(E)**${{\rm Gr}^{\hspace*{0.56905pt}s}(E)^{**}}$ of E .

We consider positive-( 1 , 1 ) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighborhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be produced already in ℂ n , hence our result is optimal. The key idea is an implementation, for currents in an almost complex setting, of the classical blow-up of curves in algebraic or symplectic geometry. Unlike the classical approach in ℂ n , we cannot rely on plurisubharmonic potentials.

A characterization of vertex operator algebra V + ℤα with (α,α)/2 not being a perfect square is given in terms of dimensions of homogeneous subspaces of small weights. This result contributes to the classification of rational vertex operator algebras of central charge 1.

In earlier work [Nonlinearity 24 (2011), 2265–2280], we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities modeled on each such profile. In particular, we show that for each positive integer k ≥ 3, there exist compact solutions in all dimensions m ≥ 3 that become singular at the rate (T-t)-2+2/k$(T-t)^{-2+2/k}$.

Let F be a p -adic field. If π be an irreducible representation of GL (n,F)${{\rm GL}(n,F)}$, Bump and Friedberg associated to π an Euler factor L(π,𝐵𝐹,s1,s2)${L(\pi ,\mathit {BF},s_1,s_2)}$ in [Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday. Part II, Weizmann, Jerusalem (1990), 47–65], that should be equal to L(φ(π),s1)L(φ(π),Λ2,s2)${L(\phi (\pi ),s_1)L(\phi (\pi ),\Lambda ^2,s_2)}$, where φ(π)${\phi (\pi )}$ is the Langlands' parameter of π. The main result of this paper is to show that this equality is true when (s1,s2)=(s+1/2,2s)$(s_1,s_2)=(s+1/2,2s)$, for s in ℂ. To prove this, we classify in terms of distinguished discrete series, generic representations of GL (n,F)${{\rm GL}(n,\hspace*{-0.56905pt}F)}$ which are χα${\chi _\alpha }$-distinguished by the Levi subgroup GL ([(n+1)/2],F)× GL ([n/2],F)${\rm GL}([(n+1)/2], \hspace*{-0.56905pt}F) \times {\rm GL}([n/2], \hspace*{-0.56905pt}F)$, for χα(g1,g2)=α(det(g1)/det(g2))${\chi _\alpha (g_1,g_2)=\alpha (\det (g_1)/{\det }(g_2))}$, where α is a character of F * of real part between -1/2${-1/2}$ and 1/2. We then adapt the technique of [`Derivatives and L -functions for GL (n)${\rm GL}(n)$', preprint 2011] to reduce the proof of the equality to the case of discrete series. The equality for discrete series is a consequence of the relation between linear periods and Shalika periods for discrete series, and the main result of [Math. Res. Lett. 19 (2012), no. 4, 785–804].

In this paper, we prove that if the area functional of a surface Σ 2 in a symplectic manifold (M2n,ω¯)${(M^{2n},\bar{\omega })}$ has a critical point or has a compatible stable point in the same cohomology class, then it must be J -holomorphic. Inspired by a classical result of Lawson–Simons, we show how various restrictions of the stability assumption to variations of metrics in the space “projectively induced” metrics are enough to give the desired conclusion.

We study the Mahler measure of some three-variable polynomials that are conjectured to be related to L -functions of elliptic curves at s = 3 by Boyd. The connection with L -functions can be explained with the use of a regulator and a result of Goncharov. Finally, we prove a relationship between two formulas.

We give a universal bound for the bounded geodesic image theorem of Masur–Minsky. The proof uses elementary techniques. We also give a universal bound for a stronger version of subsurface projection, this demonstrates good control over many standard subsurface projections simultaneously.

Following [`Gaussian densities and stability for some Ricci solitons', preprint 2004], in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability (also called ν-stability in this paper) restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the ν-entropy on symmetric spaces of compact type. In particular, we exhibit many more ν-stable and ν-unstable examples than previously known and also the first ν-stable examples, other than the standard spheres, whose second variations are negative definite.