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Register a new userFormally integrable complex structures on higher dimensional knot spaces
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Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let ${rm Imm}_f(S,M)$ the space of all free immersions $varphi:S to M$ and let $B^+_{i,f}(S,M)$ the quotient space ${rm Imm}_f(S,M)/{rm Diff}^+(S)$, where ${rm Diff}^+(S)$ denotes the group of orientation preserving diffeomorphisms of $S$. In this paper we prove that if $M$ admits a parallel $r$-fold vector cross product $varphi in Omega ^r(M, TM)$ and $dim S = r-1$ then $B^+_{i,f}(S,M)$ is a formally Kahler manifold. This generalizes Brylinskis, LeBruns and Verbitskys results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively.
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We introduce a notion of compatibility between (almost) Dirac structures and (1,1)-tensor fields extending that of Poisson-Nijenhuis structures. We study several properties of the Dirac-Nijenhuis structures thus obtained, including their connection with holomorphic Dirac structures, the geometry of their leaves and quotients, as well as the presence of hierarchies. We also consider their integration to Lie groupoids, which includes the integration of holomorphic Dirac structures as a special case.
We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. $mathbb C^n$, it is necessary to study the family of all complex space forms, indexed by the holomorphic curvature $4lambda$, and the dependence of the formulas on the parameter $lambda$. We will also describe Wannerers recent proof of local additive kinematic formulas for unitarily invariant area measures.
Frobenius manifold structures on the spaces of abelian integrals were constructed by I. Krichever. We use D-modules, deformation theory, and homological algebra to give a coordinate-free description of these structures. It turns out that the tangent sheaf multiplication has a cohomological origin, while the Levi-Civita connection is related to 1-dimensional isomonodromic deformations.
In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold $M^{2m}$. Such objects satisfy the elliptic system weakly $[J, Delta^m J]=0$. We prove a very general regularity theorem for semilinear systems in critical dimensions (with emph{critical growth nonlinearities}). In particular we prove that weakly biharmonic almost complex structures are smooth in dimension four.
Let $X$ be a compact manifold, $G$ a Lie group, $P to X$ a principal $G$-bundle, and $mathcal{B}_P$ the infinite-dimensional moduli space of connections on $P$ modulo gauge. For a real elliptic operator $E_bullet$ we previously studied orientations on the real determinant line bundle over $mathcal{B}_P$. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson. Here we consider complex elliptic operators $F_bullet$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_bullet$. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on $X$ with orientations on $X times S^1$. Thus, if $P to X$ and $Q to X times S^1$ are principal $G$-bundles with $Q|_{Xtimes{1}} cong P$, we relate spin structures on $(mathcal{B}_P,F_bullet)$ to orientations on $(mathcal{B}_Q,E_bullet)$ for a certain class of operators $F_bullet$ on $X$ and $E_bullet$ on $Xtimes S^1$. Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=U(m), SU(m)$. In a sequel arXiv:2001.00113 we apply this to define canonical orientation data for all Calabi-Yau 3-folds $X$ over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.
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