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Benjamin McKay

Publications and source records attributed to Benjamin McKay.

At least 37 records · Page 2Linked to original sources
arXiv2015

Complex homogeneous surfaces

We classify the transitive, effective, holomorphic actions of connected complex Lie groups on complex surfaces.

math.DG↗
arXiv2015

Invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds

Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $π\colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.

math.CV↗
arXiv2013

Holomorphic geometric structures on Kaehler-Einstein manifolds

We prove that the compact Kaehler manifolds with first Chern class nonnegative that admit holomorphic parabolic geometries are the flat bundles of rational homogeneous varieties over complex tori. We also prove that the compact Kaehler manifolds with negative first Chern class that admit holomorphic cominiscule geometries are the locally Hermitian symmetric varieties.

math.DG↗
arXiv2012

Exotic geometric structures on Kodaira surfaces

On all compact complex surfaces (modulo finite unramified coverings), we classify all of the locally homogeneous geometric structures which are locally isomorphic to the exotic homogeneous surfaces of Lie.

math.DG↗
arXiv2012

Soliton solutions for the Laplacian coflow of some $G_2$-structures with symmetry

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} ψ= - Δ_d ψ$ where $ψ$ is the dual 4-form of a $G_2$-structure $ϕ$ and $Δ_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_2$-structure being coclosed ($dψ=0$). We study this flow for two explicit examples of coclosed $G_2$-structures with symmetry. These are given by warped products of an interval or a circle with a compact 6-manifold $N$ which is taken to be either a nearly Kähler manifold or a Calabi-Yau manifold. In both cases, we derive the flow equations and also the equations for soliton solutions. In the Calabi-Yau case, we find all the soliton solutions explicitly. In the nearly Kähler case, we find several special soliton solutions, and reduce the general problem to a single \emph{third order} highly nonlinear ordinary differential equation.

math.DG↗
arXiv2011

Complete complex parabolic geometries

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

math.DG↗
arXiv2010

Holomorphic Cartan geometries, Calabi--Yau manifolds and rational curves

We prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact Kähler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.

math.AG↗
arXiv2010

Morphisms of Cartan connections

We define what we call morphisms of Cartan connections. We generalize the main theorems on Cartan connections to theorems on morphisms. Many of the known constructions involving Cartan connections turn out to be examples of morphisms. We prove some basic results concerning completeness of Cartan connections. We provide a new method to prove completeness of Cartan connections using families of morphisms.

math.DG↗
arXiv2009

Locally homogeneous structures on Hopf surfaces

We study holomorphic locally homogeneous geometric structures modelled on line bundles over the projective line. We classify these structures on primary Hopf surfaces. We write out the developing map and holonomy morphism of each of these structures explicitly on each primary Hopf surface.

math.DG↗
arXiv2008

Holomorphic Cartan geometries and Calabi--Yau manifolds

We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.)

math.AG↗
arXiv2008

Characteristic forms of complex Cartan geometries

We calculate relations on characteristic classes which are obstructions preventing closed Kähler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations.

math.DG↗
arXiv2008

Smooth projective planes

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory of classical projective planes.

math.DG↗
arXiv2007

Rational curves and ordinary differential equations

The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.

math.DG↗
arXiv2006

Complete projective connections

The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature.

math.DG↗