Related papers: Branched SL(r,C)-opers

We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to…

Given a symmetric operad $P$, and a signature (or generating sequence) $\Phi$ for $P$, we define a notion of the "categorification" (or "weakening") of $P$ with respect to $\Phi$. When $P$ is the symmetric operad whose algebras are…

For the Borromean link, we determine its irreducible ${\rm SL}(2,\mathbb{C})$-character variety, and find a formula for the twisted Alexander polynomial as a function on the character variety.

Take a holomorphic Lie algebroid $(V,\phi)$ over a rationally connected smooth complex projective variety $X$. We show that, under certain conditions, a vector bundle $E$ over $X$ admits a $(V,\phi)$-connection if and only if $E$ is…

Connections on a trivial bundle MxG can be identified with their holonomy maps, i.e. with homomorphisms of a groupoid of paths in M into the gauge group G. For a connected compact G, various algebras depending on the set of the smooth…

Let G be a connected reductive group and X be a smooth curve over an algebraically closed field of characteristic zero. We show that every meromorphic G-connection on X admits a possibly degenerate oper structure; in particular, every…

We show, e.g., that a holomorphic Banach vector bundle over a pseudoconvex open subset of, say, Hilbert space is holomorphically trivial if it is continuously trivial. Some applications are also given.

This paper gives an explicit description of the categorical operad whose algebras are precisely symmetric monoidal categories. This allows us to place the operad in a sequence of four, and therefore a sequence of four successively stricter…

We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the…

Let $k$ be a perfect field, and $X$ an irreducible smooth projective curve over $k$. We give a criterion for a vector bundle over $X$ to admit a logarithmic connection singular over a finite subset of $X$ with given residues, where residues…

We introduce the notion of Hitchin variety over $\C$. Let $L$ be a holomorphic line bundle over a Hitchin variety $X$. We investigate the space of all global sections of sheaf of differential operators $\cat{D}^k (L)$ and symmetric powers…

It has recently been observed that, in contrast to the classical case, holomorphic structures on line bundles over the quantum projective line are not uniquely determined by degree. We formulate a fixed-point-theoretic framework for the…

A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if…

Let $X$ be a normal projective variety over an algebraically closed field of characteristic zero. Let $D$ be a reduced Weil divisor on $X$. Let $G$ be a reductive linear algebraic group. We introduce the notion of a logarithmic connection…

Commutative rings of one-dimensional difference operators of rank l>1 and their deformations are effectively constructed. Our analytical constructions are based on the so-called ''Tyurin parameters'' for the stable framed holomorphic vector…

We investigate the analog of holomorphic vector bundles in the context of Sasakian manifolds.

In this paper, we introduce a generalization of G-opers for arbitrary parabolic subgroups P<G. For parabolic subgroups associated to even nilpotents, we parameterize (G,P)-opers by an object generalizing the base of the Hitchin fibration.…

Considering the $B$-branes over a complex manifold as the objects of the bounded derived category of coherent sheaves on that manifold, we extend the definition of holomorphic gauge fields on vector bundles to $B$-branes. We construct a…

We investigate the relative opers over the complex analytic family of compact complex manifolds of relative dimension one. We introduce the notion of relative opers arising from the second fundamental form associated with a relative…

We clarify steps for determining the ${\rm SL}(2,\mathbb{C})$-character variety of any arborescent knot. Interestingly, we show that the `excellent parts' of arborescent knots $K_1,K_2$ are isomorphic if $K_1$ can be related to $K_2$…