Volume 2013, Issue 678

In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If S is a K3, v  = 2 w is a Mukai vector on S , where w is primitive and w 2  = 2, and H is a v -generic polarization on S , then the moduli space M v of H -semistable sheaves on S whose Mukai vector is v admits a symplectic resolution M̃ v . A particular case is the 10-dimensional O'Grady example M̃ 10 of an irreducible symplectic manifold. We show that M̃ v is an irreducible symplectic manifold which is deformation equivalent to M̃ 10 and that H 2 ( M v , ℤ) is Hodge isometric to the sublattice v ⊥ of the Mukai lattice of S . Similar results are shown when S is an abelian surface.

In [7], Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C *-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds into a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K -theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two results are natural coarse-geometric analogues of certain well-known implications of a-T-menability and amenability for group C *-algebras. The techniques used are E -theoretic, building on work of Higson, Kasparov and Trout [11], [12] and Yu [28].

In joint work with Chen and Weber, the author has elsewhere shown that ℂℙ 2 ⋕2 ℂℙ 2 admits an Einstein metric. The present paper gives a new and rather different proof of this fact. Our results include new existence theorems for extremal Kähler metrics, and these allow one to prove the above existence statement by deforming the Kähler–Einstein metric on ℂℙ 2 ⋕3 ℂℙ 2 until bubbling-off occurs.

We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum over those n  ≦  x with k distinct prime factors, provided that k  =  o (log  log   x ) as x  → ∞. We estimate the fourth moments of these sums, and use a conditioning argument to show that if k is of the order of magnitude of log  log   x then the analogous normal limit theorem does not hold. The methods extend to treat the sum over those n  ≦  x with at most k distinct prime factors, and in particular the sum over all n  ≦  x . We also treat a substantially generalised notion of random multiplicative function.

We give a local formula for the index of a transverse Dirac-type operator on a compact manifold with a Riemannian foliation, under the assumption that the Molino sheaf is a sheaf of abelian Lie algebras.

For every symmetrically normed ideal ℰ of compact operators, we give a criterion for the existence of a continuous singular trace on ℰ. We also give a criterion for the existence of a continuous singular trace on ℰ which respects Hardy –Littlewood majorization. We prove that the class of all continuous singular traces on ℰ is strictly wider than the class of continuous singular traces which respect Hardy –Littlewood majorization. We establish a canonical bijection between the set of all traces on ℰ and the set of all symmetric functionals on the corresponding sequence ideal. Similar results are also proved in the setting of semi-finite von Neumann algebras.

It has been conjectured, motivated in part by old results of Dixmier and Day on bounded Hilbertian representations of amenable groups, that every norm-closed amenable subalgebra of ℬ (ℋ) is automatically similar to an amenable C*-algebra. Results of Curtis and Loy (1995), Gifford (2006), and Marcoux (2008) give some evidence to support this conjecture, but it remains open even for commutative subalgebras. We present more evidence to support this conjecture, by showing that a closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ℳ must be similar to a selfadjoint-subalgebra. Technical results used include an approximation argument based on Grothendieck's inequality and the Pietsch Domination Theorem, together with an adaptation of a theorem of Gifford to the setting of unbounded operators affiliated to ℳ.

In this article, we give an alternative proof for the convergence of the Kähler –Ricci flow on a Fano manifold ( M ,   J ). The proof differs from the one in our previous paper [J. Amer. Math. Sci. 17 (2006), 675 –699]. Moreover, we generalize the main theorem given there to the case that ( M ,   J ) may not admit any Kähler–Einstein metrics.