arXiv research

We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give par…

Study connects taffy pulling, fractions, and rational tangles.

In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit…

We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.

Compact Kähler manifolds with positive curvature are projective and rationally connected.

Study shows conditions for rational ellipticity of manifolds with symmetries.

For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,…

We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.

In this paper, we prove that if a compact Kähler manifold $X$ has a smooth Hermitian metric $ω$ such that $(T_X,ω)$ is uniformly RC-positive, then $X$ is projective and rationally connected. Conversely, we show that, if a projective manifold $X$ is rationally connected, then the tautological line bundle $\mathscr{O}_{T…

Let M be a compact, connected and simply-connected Riemannian manifold, and suppose that G is a compact, connected Lie group acting on M by isometries. The dimension of the space of orbits is called the cohomogeneity of the action. If the direct sum of the higher homotopy groups of M, tensored with the field of rationa…

We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other …

New metrics compare rational spectra using optimal transport.

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…

The paper connects curvature positivity to rational connectedness in complex geometry.

Study on negative Sasakian structures on specific 5-manifolds.

We classify simply connected rationally elliptic manifolds of dimension five and those of dimension six with small Betti numbers from the point of view of their rational cohomology structure. We also prove that a geometrically formal rationally elliptic six dimensional manifold, whose second Betti number is two, is rat…

Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.

The paper studies Kähler manifolds with partially semi-positive curvature and rational connectedness.

The study finds families of surfaces with specific properties and connects them to rational curve configurations in K3 surfaces.

New invariant P helps classify simply-connected 8-manifolds.

The study connects conic connections and torsion-free principal connections on G-structures.

We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds---all of them being formal. In the course of this examination we classify rationally elliptic homotopy types and characterise 7-di…

In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a rational surface ${\mathbf CP}^2 \sharp 7{\bar{\mathbf CP}^2}$ admits an exotic s…

New proof for curved 3-cohom manifold rational ellipticity.

An $n$-dimensional manifold $M$ is said to be rationally $4$-periodic if there is an element $e\in H^4(M;\mathbb{Q})$ with the property that cupping with $e$, $\cdot \cup e:H^\ast(M;\mathbb{Q})\rightarrow H^{\ast + 4}(M;\mathbb{Q})$ is injective for $0< \ast \leq \dim M-4$ and surjective when $0\leq \ast < \dim M-4$. W…

Classifies torus bundles bounding 4-manifolds with rational homology.

We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.

The quotients $Y=X/conj$ by the complex conjugation $conj\: X\to X$ for complex rational and Enriques surfaces $X$ defined over $\R$ are shown to be diffeomorphic to connected sums of $\barCP2$, whenever $Y$ are simply connected.

Bryant \cite{Br} proved the existence of torsion free connections with exotic holonomy, i.e. with holonomy that does not occur on the classical list of Berger \cite{Ber}. These connections occur on moduli spaces $\Y$ of rational contact curves in a contact threefold $\W$. Therefore, they are naturally contained in the …

The study of higher tangential structures, arising from higher connected covers of Lie groups (String, Fivebrane, Ninebrane structures), require considerable machinery for a full description, especially for connections to geometry and applications. With utility in mind, in this paper we study these structures at the ra…

A conjecture about rational curves' formal principle and convergence proved for Goursat type families.

Overview of dynamics in algebraic correspondences and their connections.

Models for self-equivalences and diffeomorphisms of manifolds.

We classify simply connected, closed cohomogeneity one manifolds with singly generated or 4-periodic rational cohomology and positive Euler characteristic.

The abstract discusses connections between K-stability, heights, and rational points on Fano varieties.

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.

The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.

New 4-manifold accounts for rationally slice knots.

Rational ellipticity proven for $G$-manifolds with specific quotient properties.

We introduce a new class of zero-dimensional weighted complete intersections, by abstracting the essential features of rational cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M whose rational cohomology algebra belongs to this class,…

Study projective connections on surfaces using osculating spaces.

Study 1-flat G-structures on uniruled projective manifolds.

New knot invariant λ bounds rational unknotting.

Solves open problems on curved projective varieties.

In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots in $\mathbb{RP}^3$.

This paper provides a rational model for fiberwise THH transfer using A-infinity algebras.

New dHYM connections found on complex vector bundles.

The paper connects reflection groups to maps with specific dynamical properties.