Integrability of rough almost complex structures

Solutions of the Strominger System via Stable Bundles on Calabi-Yau Threefolds

“…Here we identify the codomain in (18) (resp. in (19)) with Ω 6 R (ad 0 E) (resp. Ω 6 R (ad T X)) in (15) via multiplication by the volume form ω 3 0 3!…”

“…where (0 ⊕ Ker T ⊕ Ker ∆ 0 ) ⊥ denotes the orthogonal complement in V 1 of the direct sum of Kernels 0 ⊕ Ker T ⊕ Ker ∆ 0 corresponding to (18), (19) and (20). Note here that Ker ∆ 0 is given by the covariant constant sections in Ω 0 R (ad T X), which correspond to the Lie algebra of the centralizer of the holonomy group of ∇ 0 in U (3).…”

Zoll Manifolds and Complex Surfaces

“…Suppose that ∇ is of differentiability class C k,α , and suppose that f is a holomorphic function on some open subset of N ; we then consider the function Ψ * f on Z = PT C M . Now, by [18], f is a C 1 function with respect to our (original, unchanged) C 1 structure on N , and Ψ * f is therefore a C 1 function, since Ψ was C 1 by construction. Moreover, since Ψ * D ⊂ T 0,1 N , Ψ * f = 0 solves the Cauchy-Riemann equations∂ D (Ψ * f ) = 0 with respect to the C k,α almost-complex structure which D determines on Z − Z.…”

“…On the other hand, since D is involutive and Ψ| Z−Z is a diffeomorphism, τ ≡ 0 on a set of full measure, and therefore vanishes in the distributional sense. But Hill and Taylor [18] have recently shown that the Newlander-Nirenberg theorem holds for Lipschitz almost complex structures for which τ = 0 in just this distributional sense. Thus every point of N has a neighborhood on which we can find a pair (z 1 , z 2 ) of differentiable complex-valued functions with dz k ∈ Λ 1,0 (N , J) and dz 1 ∧ dz 2 = 0.…”

On conformally Kähler, Einstein manifolds

“…The Christoffel symbols ofĝ 0 are thus of class C 0,α and the standard Atiyah-Hitchin-Singer formulation of the twistor construction [7,40] (see Figure 4) gives us an almost-complex structure J on the 6-manifold Z defined as the 2-sphere bundle S(Λ + ) → Y . We now apply the HillTaylor version [29] of the Newlander-Nirenberg theorem for rough almost-complex structures. Since J is an almost-complex structure of class C 0,α , α > 1/2, its Nijenhuis tensor N J is not only well-defined in the distributional sense, but actually Notice that the results of Tian and Viaclovsky [70] tell us that the hypotheses of this proposition hold whenever an anti-self-dual manifold (X, g ∞ ) arises as a bubble.…”