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Applying the techniques developed in [AGG], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov-Lawson-Thurston conjecture [GLT], such bundles $M\to…

It is known that to every 1|1 dimensional supercurve X there is associated a dual supercurve \hat{X}, and a superdiagonal \Delta in their product. We establish that the categories of D-modules on X, \hat{X}, and \Delta are equivalent. This…

Let E be the restriction of the null-correlation bundle on $\mathbb{P}^{3}$ to a hyperplane. In this article, we show that the projective bundle $\mathbb{P}(E)$ is isomorphic to a blow-up of a non-singular quadric in $\mathbb{P}^{4}$ along…

Over a scheme with 2 invertible, we show that a vector bundle of rank four has a sub or quotient line bundle if and only if the canonical symmetric bilinear form on its exterior square has a lagrangian subspace. For this, we exploit a…

Let $X$ be a projective smooth geometrically connected curve defined over a finite field $\mathbb{F}_q$ of cardinality $q$. Let $S$ be a finite set of closed points of $X$. Let $\bar{X}$ and $\bar{S}$ be the base change of $X$, $S$ to an…

We provide a splitting criterion for supervector bundles over the projective superspaces $\mathbb{P}^{n|m}$. More precisely, we prove that a rank $p|q$ supervector bundle on $\mathbb{P}^{n|m}$ with vanishing intermediate cohomology is…

Let $k$ be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve $X_1$ defined over $k$, there exists a lifting $X$ of the…

We discuss topological rigidity of vector bundles with asymptotically conical (AC) total spaces of rank greater than 1 with a sufficiently connected link; our focus will mainly be on ALE (asymptotically locally Euclidean) bundles. Within…

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a $2$-punctured elliptic curve $C$. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to $\mathbb{P}^1 \times…

Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is…

Let X be a smooth hypersurface in projective space. We discuss in this paper when X can be defined by an equation det M = 0 (resp. pf M = 0), where M is a matrix (resp. a skew-symmetric matrix) with homogeneous entries. Standard homological…

We study the existence of asymptotically $Z$-stable (a.Z stable) bundles over polycyclic surfaces. Our choice of polynomial central charge is related to the existence of solutions of the deformed Hermitian--Yang--Mills equations, with…

We study full exceptional collections of line bundles on surfaces. We prove that any full strong exceptional collection of line bundles on a weak del Pezzo surface of degree $\ge 3$ is an augmentation in the sense of L.Hille and M.Perling,…

For a smooth subvariety $X\subset\Bbb P^N$, consider (analogously to projective normality) the vanishing condition $H^1(\Bbb P^N,\Cal I^2_X(k))=0$, $k\ge3$. This condition is shown to be satisfied for all sufficiently large embeddings of a…

A plane curve C defined by a homogeneous polynomial satisfying Laplace's equation appears canonically as the vanishing of the Pfaffian of a skew-symmetric matrix of linear forms. As a consequence there is a natural semi-stable rank two…

Consider the moduli space $M_C(r; K_C)$ of stable rank r vector bundles on a curve $C$ with canonical determinant, and let $h$ be the maximum number of linearly independent global sections of these bundles. If $C$ embeds in a K3 surface $X$…

We give examples of derived schemes $X$ and a line bundle $\Ls$ on the truncation $tX$ so that $\Ls$ does not extend to the original derived scheme $X$. In other words the pullback map $\Pic(X) \to \Pic(tX)$ is not surjective. Our examples…

All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the…

Computation of parallel lines (envelopes) to parabolas, ellipses, and hyperbolas is of importance in structure engineering and theory of mechanisms. Homogeneous polynomials that implicitly define parallel lines for the given offset to a…

I provide a construction of intrinsic weakly Ulrich bundles of large rank on any smooth complete surface in ${\bf P}^3$ over fields of characteristic $p>0$ and also for some classes of surfaces of general type in ${\bf P}^n$. I also…