Related papers: On the non-existence of almost complex structures …

We answer an open question concerning the boundedness of canonical fiber spaces in high dimensions and prove the following: for any set of integers $n\geq 3$, $0<d<n$ and $N>0$, there exists a nonsingular projective $n$-fold $X$ of general…

For non-abelian simple objects in a unitary modular category, the density of their braid group representations, the #P-hard evaluation of their associated link invariants, and the BQP-completeness of their anyonic quantum computing models…

Given a quasi-projective scheme M over complex numbers equipped with a perfect obstruction theory and a morphism to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M'/ M that admits a perfect…

Let $k$ be a number field and let $\pi \colon X \rightarrow \mathbb{P}_k^1$ be a smooth conic bundle. We show that if $X/k$ has four geometric singular fibers with $X(\mathbb{A}_k)\neq \emptyset$ or non-trivial Brauer group, then $X$…

In characteristic zero, semistable principal bundles on a nonsingular projective curve with a semisimple structure group form a bounded family, as shown by Ramanathan in 1970's using the Narasimhan-Seshadri theorem. This was the first step…

We use constructions of surfaces as abelian covers to write down exceptional collections of line bundles of maximal length for every surface $X$ in certain families of surfaces of general type with $p_g=0$ and $K_X^2=3,4,5,6,8$. We also…

We consider the possibility of semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. Conditions are given for the existence or nonexistence of coherent associative structures for such fusion rules,…

In this paper, we show that a set of q+a hyperplanes, q>13, a<(q-10)/4, that does not cover PG(n,q), does not cover at least q^(n-1)-aq^(n-2) points, and show that this lower bound is sharp. If the number of non- covered points is at most…

A free action of a finite group on an odd-dimensional sphere is said to be almost linear if the action restricted to each cyclic or 2-hyperelementary subgroup is conjugate to a free linear action. We begin this survey paper by reviewing the…

Consider the blow up $\pi: \widetilde{X} \to X$ of a rational surface $X$ at a point. Let $\widetilde{V}$ be a holomorphic bundle over $\widetilde{X}$ whose restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$ and…

Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…

We study necessary conditions for the existence of lattice tilings of $\R^n$ by quasi-crosses. We prove non-existence results, and focus in particular on the two smallest unclassified shapes, the $(3,1,n)$-quasi-cross and the…

Let $k$ be a number field and let $\pi \colon X \rightarrow\mathbb{P}_k^1$ be a smooth conic bundle. We show that if $X/k$ has four geometric singular fibers and either $X(\mathbb{A}_k)\neq \emptyset$ or $X/k$ has non-trivial Brauer group,…

We show that the relations which define the algebras of the quantum Euclidean planes $\b{R}^N_q$ can be expressed in terms of projections provided that the unique central element, the radial distance from the origin, is fixed. The resulting…

In this article we give a necessary and sufficient condition to characterize projective submanifolds in ${\mathbb P}^N$ with codimensions 2 and 3. The conditions involve the Chern classes of the manifold and a very ample line bundle on the…

We employ combinatorial techniques to present an explicit formula for the coefficients in front of Chern classes involving in the Hattori-Stong integrability conditions. We also give an evenness condition for the signature of stably…

An infinite $(p,q)$-theorem, or an $(\aleph_0,q)$-theorem, involving two families $\mathcal{F}$ and $\mathcal{G}$ of sets, states that if in every infinite subset of $\mathcal{F}$, there are $q$ sets that are intersected by some set in…

From a certain strongly equivariant bundle gerbe with connection and curving over a smooth manifold on which a Lie group acts, we construct under some conditions a bundle gerbe with connection and curving over the quotient space. In…

We establish several non-existence results of positive scalar curvature (PSC) on fiber bundles. We show that under an incompressible condition of the fiber, for $X^m$ a Cartan-Hadamard manifold or an aspherical manifold when $m=3$, the…

We generalise Fl\o{}ystad's theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$…