We study the geometry of a general Fano variety of dimension four, genus nine, and Picard number one. We compute its Chow ring and give an explicit description of its variety of lines. We apply these results to study the geometry of non quadratically normal varieties of dimension three in a five dimensional projective space.

Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\subset G$ be a maximal compact subgroup. Let $X$, $Y$ be irreducible smooth complex projective varieties and $f:X\to Y$ an algebraic morphism, such that ${\pi }_1\left(Y\right)$ is virtually nilpotent and the homomorphism $f_{*}:{\pi }_1\left(X\right)\to {\pi }_1\left(Y\right)$ is surjective. Define $$\begin{array}{cc}\hfill {ℛ}^f({\pi }_1\left(X\right),G)& =\{\rho \in Hom({\pi }_1\left(X\right),G)\mid A\circ \rho \text{factors}\text{through}{f}_{*}\},\hfill \\ \hfill {ℛ}^f({\pi }_1\left(X\right),K)& =\{\rho \in Hom({\pi }_1\left(X\right),K)\mid A\circ \rho \text{factors}\text{through}{f}_{*}\},\hfill \end{array}$$ where $A:G\to GL(Lie(G\left)\right)$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${ℛ}^f({\pi }_1(X,x_0),G)//G$ admits a deformation retraction to ${ℛ}^f({\pi }_1(X,x_0),K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements...

Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct image bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.

In this paper, we consider the problem of determining which topological complex rank-2 vector bundles on non-Kähler elliptic surfaces admit holomorphic structures; in particular, we give necessary and sufficient conditions for the existence of holomorphic rank-2 vector bundles on non-{Kä}hler elliptic surfaces.

Etant donnés $A_1,\cdots ,A_s$ ($s\ge 3$) des $S{L}_2\left(ℂ\right)$-modules non triviaux de dimensions respectives $n_1+1\ge \cdots \ge n_s+1$ (avec $n_1=n_2+\cdots +n_s$) et $\phi \in ℒ(A_2\otimes \cdots \otimes A_s,A_1^{*})$ un $S{L}_2\left(ℂ\right)$-homomorphisme, nous montrons que l’hyperdéterminant de $\phi$ est nul sauf si les modules $A_i$ sont irréductibles et si l’homomorphisme est la multiplication des polynômes homogènes à deux variables.

We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail.

We study a class of torsion-free sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable if its rank is not too large, while semistable torsion-free sheaves satisfying certain cohomological conditions are instanton. We also study a few examples of moduli spaces of instanton sheaves.

We study general elements of moduli spaces ${\mathfrak{M}}_{{\mathbb{P}}^2}\left(2,c_1,c_2\right)$ of rank-2 stable holomorphic vector bundles on ${\mathbb{P}}^2$ and their minimal free resolutions. Incidentally, a quite easy proof of the irreducibility of ${\mathfrak{M}}_{{\mathbb{P}}^2}\left(2,c_1,c_2\right)$ is shown.