Building on the work of Nogin, we prove that the braid group $B_4$ acts transitively on full exceptional collections of vector bundles on Fano threefolds with $b_2=1$ and $b_3=0$. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds with $b_2=1$ and very ample anticanonical class, every exceptional coherent sheaf is locally free.

Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.

In \emph{Connections on a parabolic principal bundle over a curve, I} we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.

Let $X$ be a smooth complex projective curve of genus $g\geq 1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let $W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$ by $\xi^{-1}$. There is a rational map $D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$, where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$, then $D_{\xi}$ is everywhere defined and is injective.

In this note we consider $\overline\partial$-problem in line bundles over complex projective space $\mathbb{CP}^1$ and prove that the equation can be solved for $(0,1)$ forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{CP}^2$ since by removing a point from it we get a line bundle over $\mathbb{CP}^1$.

On a smooth $n$-dimensional complete variety $X$ over ${\mathbb C}$ we show that the trace map ${\tilde\theta}_X \colon\break H^n (X,\Omega_X^n) \to {\mathbb C}$ arising from Lipman's version of Grothendieck duality in \cite{ast-117} agrees with $$ (2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb C}) \to {\mathbb C} $$ under the Dolbeault isomorphism.

Let $\ce$ be an ample vector bundle of rank $r$ on a projective variety $X$ with only log-terminal singularities. We consider the nefness of adjoint divisors $K_X + (t-r) \det \ce$ when $t \ge \dim X$ and $t>r$. As an application, we classify pairs $(X,\ce)$ with $c_r$-sectional genus zero.

We investigate the pairs $(X,\cE)$ consisting of a smooth complex projective variety $X$ of dimension $n$ and an ample vector bundle $\cE$ of rank $n-1$ on $X$ such that $\cE$ has a section whose zero locus is a smooth elliptic curve.