Related papers: Smoothing surfaces on fourfolds

We give the classification of globally generated vector bundles of rank $2$ on a smooth quadric surface with $c_1\le (2,2)$ in terms of the indices of the bundles, and extend the result to arbitrary higher rank case. We also investigate…

Let $F\subseteq\mathbb{P}^3$ be a smooth determinantal quartic surface which is general in the N\"other-Lefschetz sense. In the present paper we give a complete classification of locally free sheaves $\mathcal{E}$ of rank $2$ on $F$ such…

This is the revised version of our previous preprint. In this paper, we establish a generic smoothness result for moduli space of semistable sheaves of arbitrary rank over surfaces provided that the second Chern class of the sheaves is…

Let $X\subseteq{\mathbb P}^{n+1}$ be an integral hypersurface of degree $d$. We show that each locally Cohen-Macaulay instanton sheaf $\mathcal E$ on $X$ with respect to $\mathcal O_X\otimes\mathcal O_{\mathbb P^{n+1}}(1)$ in the sense of…

Let $X$ be a smooth projective variety defined over a finite field. We show that any algebraic $1$-cycle on $X$ is rationally equivalent to a smooth $1$-cycle, which is a $\mathbb{Z}$-linear combination of smooth curves on $X$. We also…

We prove that a connected subcartesian space admits embedding in a Euclidean space. The Whitney Embedding Theorem is then stated as a corollary of our result. Based on the above result together with the theory of distribution on smooth…

We study smooth proper embeddings of compact orientable surfaces in compact orientable $4$-manifolds and elements in the mapping class group of that surface which are induced by diffeomorphisms of the ambient $4$-manifolds. We call such…

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to…

Let $\pi: X \longrightarrow C$ be a fibration with reduced fibers over a curve $C$ and consider a polarization $H$ on the surface $X$. Let $E$ be a stable vector bundle of rank $2$ on $C$. We prove that the pullback $\pi^*E$ is a $H-$stable…

The aim of this note is to exhibit explicit sufficient criteria ensuring bigness of globally generated, rank-$r$ vector bundles, $r \geqslant 2$, on smooth, projective varieties of even dimension $d \leqslant 4$. We also discuss connections…

Consider a very ample line bundle $ E \to X$ over a compact complex manifold, endowed with a hermitian metric of curvature $-i \omega $, and the space $\mathcal{O}(E)$ of its holomorphic sections. The Fubini--Study map associates with…

Given a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $\varphi^* E$ is trivial for some surjective holomorphic map $\varphi$, to $M$, from some compact complex manifold. We prove that these…

We study smooth quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$ parameterising $6 \times 6$ skew-symmetric matrices of rank at most 4, not intersecting the Grassmannian $\mathbb{G}(1,5)$. Such surfaces correspond to…

In a previous work, the authors introduced the notion of `coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss-Bonnet formulas on coherent…

Let $(M^{n+1},g,e^{-f}d\mu)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-\'{E}mery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded…

Let $X$ be a projective scheme over a noetherian base scheme $S$, and let $F$ be a coherent sheaf on $X$. For any coherent sheaf $E$ on $X$, consider the set-valued contravariant functor $Hom_{E,F}$ on $S$-schemes, defined by $Hom_{E,F}(T)…

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, $\xi \colon P \to \Sigma$ a principal $G$-bundle, let $N(\xi)$ denote the moduli space of central Yang-Mills connections on $\xi$, for suitably chosen…

We argue that for a smooth surface S, considered as a ramified cover over the projective plane branched over a nodal-cuspidal curve B one could use the structure of the fundamental group of the complement of the branch curve to understand…

We construct a rank-$2$ indecomposable vector bundle on $\mathbb P^2\times\mathbb P^2$ in characteristic $2$ that does not come from a bundle on $\mathbb P^2$ by factor projection nor from a bundle on $\mathbb P^{m} $ by central projection.…

Let $X$ and $Y$ be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let $f : Y \rightarrow X$ be a finite generically smooth morphism such that the corresponding homomorphism…