International Journal of Mathematics

We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n + 1)-space from ℤ₂ to ℤ. We demonstrate how the ℤ-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian immersions into ℂⁿ and again including orientations strengthens the results. More precisely, we prove that the number of double points of an exact Lagrangian immersion of a closed manifold M whose associated Legendrian embedding has good DGA is at least half of the dimension of the homology of M with coefficients in an arbitrary field if M is spin and in ℤ₂ otherwise.

Let Γ=ℤ_m * ℤ_n or ℤ * ℤ_n, and let Γ(h) be the subtree consisting of all reduced words starting with any reduced word h ∈ Γ\{e}. We prove that the C*-algebra generated by and the projection P_h onto the subspace ℓ²(Γ(h)) has a unique nontrivial closed ideal ℐ, ℐ is *-isomorphic to , and the quotient algebra is *-isomorphic to either or depending on the last letter of h. We also prove that is a purely infinite, simple C*-algebra if the last letter of h is a generator of ℤ, and that has a unique nontrivial closed ideal if the last letter of h is a generator of ℤ_n; furthermore, is *-isomorphic to and is again a purely infinite, simple C*-algebra. As consequences, all the C*-algebras above have real rank zero, and is nuclear for any h ≠ e.

Let T be a positive closed current of bidegree (1,1) on a compact complex surface. We show that for all ε > 0, one can find a finite composition of blow-ups π such that π*T decomposes as the sum of a divisorial part and a positive closed current whose Lelong numbers are all less than ε.

We consider the affine action of a nilpotent Lie group on ℝⁿ. Lipsman (1995) conjectured that such an action is proper in the sense of Palais if and only if the action is (CI) in the sense of Kobayashi. The present paper gives a counterexample to Lipsman's conjecture for n ≥ 5.