Related papers: Dolbeault-type complexes on $G_2$- and $\mathrm{Sp…

There are two abelian groups which can naturally be associated to an additive category A: the split Grothendieck group of A and the triangulated Grothendieck group of the homotopy category of (bounded) complexes in A. We prove that these…

The elliptic genus for arbitrary two dimensional $N=2$ Landau-Ginzburg orbifolds is computed. This is used to search for possible mirror pairs of such models. An important aspect of this work is that there is no restriction to theories for…

Dualities are hidden symmetries that map seemingly unrelated physical systems onto each other. The goal of this work is to systematically construct families of Hamiltonians endowed with a given duality and to provide a universal description…

We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds…

Interpreting the chiral de Rham complex (CDR) as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for the study of CDR on manifolds with special holonomy. We show how to systematically…

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a…

Let $(M,J,g,\omega)$ be a K\"ahler manifold. We prove a $W^{1,2}$ weak Bott-Chern decomposition and a $W^{1,2}$ weak Dolbeault decomposition, following the $L^2$ weak Kodaira decomposition on Riemannian manifolds. Moreover, if the K\"ahler…

We construct 4-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension 2. We show that these foliations are holomorphic with respect to an (integrable) Hermitian structure which is…

We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie…

We study the special algebraic properties of alternating 3-forms in 6 and 7 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed manifold M in these dimensions. Restricting the…

The Dolbeault complex of a quantized compact Hermitian symmetric space is expressed in terms of the Koszul complex of a braided symmetric algebra of Berenstein and Zwicknagl. This defines a spectral triple quantizing the Dolbeault-Dirac…

We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical systems on Riemann surfaces subjected to an external magnetic field. The realization is shown to be possible only for Riemann surfaces with constant curvature…

In this short paper we prove a derived version of the Riemann-Hilbert correspondence of Deligne and Simpson. Our generalization is twofold: on one side we consider families of representations of the full homotopy type of a smooth analytic…

In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its…

In this article we survey recent results on rigid dualizing complexes over commutative algebras. We begin by recalling what are dualizing complexes. Next we define rigid complexes, and explain their functorial properties. Due to the…

We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex…

We classify closed, conformally flat Lorentzian manifolds of dimension $n \geq 3$ with unipotent holonomy in PO(2,n). They are all Kleinian and fall into four different geometric types according to the intersection of the image of the…

A characterization of maximal domains of existence of adapted complex structures for Riemannian homogeneous manifolds under certain extensibility assumptions on their geodesic flow is given. This is applied to generalized Heisenberg groups…

For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with…

We give a topological interpretation of the space of $L^2$-harmonic forms on Manifold with flat ends. It is an answer to an old question of J. Dodziuk. We also give a Chern-Gauss-Bonnet formula for the $L^2$-Euler characteristic of some of…