Related papers: H-Instanton Bundles on Three-Dimensional Smooth To…

We study the scheme of multi-jumping lines of an $n$-instanton bundle mainly for $n\leq 5$. We apply it to prove the irreducibility and smoothness of the moduli space of 5-instanton. Some particular situations with higher $c_2$ are also…

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, and degenerate forms in $\Omega$ belong to a non-degenerate quadric…

We prove that the singular locus of a rank 2 instanton sheaf $E$ on $\mathbb{P}^3$ which is not locally free has pure dimension 1. Moreover, we also show that the dual and double dual of $E$ are isomorphic locally free instanton sheaves,…

The purpose of this paper is to study minimal monads associated to a rank two vector bundle $\mathcal E$ on $\mathbb P^n$. In particular, we study situations where $\mathcal E$ has $H^i_*(\mathcal E) =0$ for $1<i<n-1$, except for one pair…

Let M(v) be the moduli of stable sheaves on K3 surfaces X of Mukai vector v. If v is primitive, than it is expected that M(v) is deformation equivalent to some Hilbert scheme and weight two hogde structure can be described by H^*(X,Z).…

We present a new family of monads whose cohomology is a stable rank two vector bundle on $\PP$. We also study the irreducibility and smoothness together with a geometrical description of some of these families. Such facts are used to prove…

We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic…

In this paper we construct vector bundles associated to monads on $X=\mathbb{P}^n\times\mathbb{P}^n\times\mathbb{P}^m\times\mathbb{P}^m$. We first establish the existence of such monads on $X$. Once the monads exist, the next natural…

We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the…

Let $\mathbb{T}$ be the three dimensional torus acting on $\mathbb{P}^{3}$ and $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ be the fixed locus of the corresponding action on the moduli space of rank $2$ framed instanton sheaves on…

This thesis is an investigation of the moduli spaces of instanton bundles on the Fano threefold $Y_5$ (a linear section of $\mathbb{G}r(2,5)$). It contains new proofs of classical facts about lines, conics and cubics on $Y_5$, and about…

Let $MI_{Simp,P^{2n+1}}(k)$ be the moduli space of stable symplectic instanton bundles on $P^{2n+1}$ with second Chern class $c_2=k$ (it is a closed subscheme of the moduli space $MI_{P^{2n+1}}(k)$), We prove that the dimension of its…

Recently N.Nekrasov and A.Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of the 4-dimensional real affine space. In this paper we study the relation between…

Let $X$ be a projective K3 surfaces. In two examples where there exists a fine moduli space $M$ of stable vector bundles on $X$, isomorphic to a Hilbert scheme of points, we prove that the universal family $\mathcal{E}$ on $X\times M$ can…

We discuss N=2 supersymmetric compactifications to four dimensions from the point of view of F-theory and heterotic theory. In a relatively simple setup, we illustrate the spectral theory for vector bundles on K3xT^2 and discuss the…

We look at the following question raised by Koll\'ar and Peskine. (Actually, it is a slightly weaker version of their question.) Let $V_t$ be a family of rank two vector bundles on $\Bbb P^3$. Assume that the general member of the family is…

The existence of topological invariants analogous to Chern/Pontryagin classes for a standard $SO(D)$ or $SU(N)$ connection, but constructed out of the torsion tensor, is discussed. These invariants exhibit many of the features of the…

We study semi-stable sheaves of rank $2$ with Chern class $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-folds $V_4$ of Picard number $1$, degree $4$ and index $2$. We show the moduli space of such sheaves is isomorphic to the moduli space of…

Adjusting conventional Chern-Simons theory to ${\rm G}_2$-manifolds, one describes ${\rm G}_2$-instantons on bundles over a certain class of $7$-dimensional flat tori which fiber non-trivially over $T^4$, by a pullback argument. Moreover,…

We introduce a method to construct $G_2$-instantons over compact $G_2$-manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing $G_2$-instantons obtained…