arXiv research

Survey explores interactions between convex and complex geometry.

Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define a…

We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kähler manifolds.

Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.

Authors discuss complex and non-Archimedean geometry, proving a conjecture.

Examines properties of holomorphic fibrations in complex geometry.

We define $(p,q)$ hermitian geometry as the target space geometry of the two dimensional $(p,q)$ supersymmetric sigma model. This includes generalised Kähler geometry for $(2,2)$, generalised hyperkähler geometry for $(4,2)$, strong Kähler with torsion geometry for $(2,1)$ and strong hyperkähler with torsion geometry f…

Survey on holomorphic structures on complex manifolds.

Study on complex tori foliations and flat geometries.

The study sets limits on the complexity of Klein geometries.

Quantum complexity lowerbound proved using differential geometry.

Method resolves 4D symplectic orbifolds using complex geometry.

Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible wit…

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex o…

Complete complex parabolic geometries (including projective connections and conformal connections) are flat and homogeneous. This is the first global theorem on parabolic geometries.

Advances M-polyfolds for complex geometry applications.

Classifies holomorphic parabolic geometries on complex manifolds.

We pursue the study of holomorphic Cartan geometry with singularities. We introduce the notion of logarithmic Cartan geometry on a complex manifold, with polar part supported on a normal crossing divisor. In particular, we show that the push-forward of a Cartan geometry constructed using a finite Galois ramified coveri…

New complex-valued maps found on complex geometries.

Survey of geometry developments, including complex structures on surfaces.

In [DM] it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affimatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type (G,H), with G a complex affine Lie group…

The study identifies two sources of invariants in 2--nondegenerate CR geometries.

A dictionary connects symplectic to contact geometry, with applications to complex and G-structures.

For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein…

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a comp…

The paper extends symplectic techniques to generalized complex geometry.

Book introduces principles of LCK geometry for complex manifold students.

The paper uses complex-valued functions to simplify plane differential geometry and kinematics.

Paper solves open problem in complex Finsler geometry.

We study local automorphisms of holomorphic Cartan geometries. This leads to classification results for compact complex manifolds admitting Cartan geometries. We prove that a compact Calabi-Yau manifold bearing a holomorphic Cartan geometry of algebraic type admits a finite unramified cover which is a complex torus.

Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.

Recent developments in Seiberg-Witten theory and relations with Complex Geometry.

The paper proves complex geometry results for manifolds of the form X × R², answering a 1994 conjecture.

Study of free particle's geometry and its perturbations using complex projective structures.

Unified framework for complex, split-complex, and dual numbers.

Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension $d$ admits, away from a closed analytic subset of positive codimension, …

Diffeology extends differential geometry to complex spaces.

In this survey paper we give a proof of hyperbolicity of the complex of curves for a non-exceptional surface S of finite type combining ideas of Masur/Minsky and Bowditch. We also shortly discuss the relation between the geometry of the complex of curves and the geometry of Teichmueller space.

Study on null submanifolds in indefinite complex contact geometry.

New complexes refine multicomplexes for subRiemannian geometry.

The paper connects fibrations to generalized complex structures in semi-toric geometry.

Survey of combination theorems in geometry and dynamics.

We give sharp $C^{2,α}$ estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application…

We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry. We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the eq…

We classify holomorphic Cartan geometries on every compact complex curve, and on every compact complex surface which contains a rational curve.

Develops orbibundle theory for complex hyperbolic geometry.

Study para-hyperKähler geometry of anti-de Sitter structures.

Physics: Similar long-distance properties can mask vastly different short-distance metrics.