Related papers: On the index of harmonic maps from surfaces to com…
We study the classification of affine holomorphic bundles over a compact complex manifold $X$ in general, and we apply the general theory to the case $X=\mathbb{P}^1_\mathbb{C}$. We study the moduli space of framed, non-degenerate rank 2…
We study the moduli spaces which classify smooth surfaces along with a complex line bundle. There are homological stability and Madsen--Weiss type results for these spaces (mostly due to Cohen and Madsen), and we discuss the cohomological…
The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly…
We construct projective asymptotically good moduli spaces parametrizing boundary polarized CY surface pairs, which are projective slc Calabi-Yau pairs $(X,D)$ such that $D$ is ample and $X$ has dimension two. The moduli space provides a…
We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves…
We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…
We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's…
A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean…
We construct examples of non-isomorphic algebraic vector bundles on the punctured affine space with isomorphic pullbacks to the smooth quadric.
The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex…
We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the "sheaf of holomorphic connections" (the sheaf of splittings of the one-jet sequence) for the determinant…
Harmonic analysis is a tool to infer cosmic topology from the measured astrophysical cosmic microwave background CMB radiation. For overall positive curvature, Platonic spherical manifolds are candidates for this analysis. We combine the…
We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the…
For smooth projective curves the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper…
Answering a problem raised by Lazarsfeld, Hwang and Mok proved that a surjective holomorphic map from a rational homogeneous space of Picard number 1 onto projective manifold different from projective space must be a biholomorphism. THe aim…
A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete…
We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic…
We present an algorithm for computing line bundle valued cohomology classes over toric varieties. This is the basic starting point for computing massless modes in both heterotic and Type IIB/F-theory compactifications, where the manifolds…
In the present note we describe geometrically the homology classes in the total space of a surface bundle over a surface in terms of the holonomy map. We treat the cases where the base surface is closed or has one boundary component. We…
Spaces of holomorphic maps from the Riemann sphere to various complex manifolds (holomorphic curves ) have played an important role in several area of mathematics. In a seminal paper G. Segal investigated the homotopy type of holomorphic…