arXiv research

New proof for curved 3-cohom manifold rational ellipticity.

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

Classifies rationally elliptic manifolds and proves geometric formality properties.

Study shows conditions for rational ellipticity of manifolds with symmetries.

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…

Upper bound found for torus actions on certain manifolds.

The study explores geometric formality in specific dimensions of rationally elliptic manifolds.

Researchers prove a complex geometric conjecture about certain manifolds.

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 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…

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…

Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipt…

The paper proves properties of manifolds under certain conditions.

The article proves estimates on homotopy and cohomology dimensions in fibrations.

Study eigenvalues of curvature operators to annihilate cobordism invariants.

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber cla…

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

The study classifies manifolds that can be split into two disk bundles.

Study shows unbounded Pontryagin numbers on curved manifolds.

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of …

First time projective elliptic genera constructed for oriented manifolds.

Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface $E(1)_{2,3}$ requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as $E(1)_{2,3}$ and admits neither 1- nor 3-handles, by using rational blow-downs and K…

Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive hom…

Study trisections on rational elliptic surfaces to find new Zariski pairs.

Study properties of algebraic threefolds with specific singularities.

We give new rational blowdown constructions of exotic CP^2#n(-CP^2) (5\leq n\leq 9) without using elliptic fibrations. We also show that our 4-manifolds admit handle decompositions without 1- and 3-handles, for 7\leq n\leq 9. A strategy for rational blowdown constructions of exotic CP^2#n(-CP^2) (1\leq n\leq 4) is also…

The study finds infinitely many special Lagrangian submanifolds in certain complex manifolds.

Defines complex manifolds for Khovanov homology computation.

New quotient spaces with boundaries are constructed using curvature properties.

Study non-negatively curved GKM orbifolds and their cohomology.

We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.

We show that there is a complex structure on the symplectic 4-manifold $W_{4, k}$ obtained from the elliptic surface E(4) by rationally blowing down $k$ sections for $2\le k\le 9$. And we interpret it via ${\mathbb Q}$-Gorenstein smoothing. This answers affirmatively to a question raised by R. Gompf.

Study describes Abel-Jacobi map for elliptic surfaces, refining cubic-line arrangements topology.

Study finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.

Quantum field theory methods yield a physical interpretation of elliptic cohomology.

Study real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.

Classifies rationally 4-periodic biquotients and their cohomology.

New exotic 4-manifolds with even $b_2^+$ and $\mathbb{Z}/2\mathbb{Z}$ fundamental group.

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors …

In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank $\geq 2$ resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic …

The paper proves properties of image structures of fibrations with semi-positive curvature.

We study algebraic solutions of the Riccati equation over the field of rational functions $\mathbb C(t)$, and over the elliptic function field $\mathbb C(\wp,\wp^\prime)$.

An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deform…

The study constructs symplectic 4-manifolds with exotic structures.

Authors prove Torelli theorem for a specific type of gravitational instantons.

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connect…

Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.