In this paper, we study special generalized null correlation bundles on $\mathbb{P}^{5}$. We prove that special generalized null correlation bundles on $\mathbb{P}^{5}$ are stable under some numerical conditions. Moreover, we prove that the closure of the subvariety parametrizing stable special generalized null correlation bundles is an irreducible component of the moduli space of rank $4$ stable vector bundles with corresponding Chern classes on $\mathbb{P}^{5}$. As an application, we prove that the number of irreducible components of moduli space of rank $4$ stable vector bundles on $\mathbb{P}^5$ with some fixed Chern classes can be arbitrarily high.
We study the null-cone of a semi-simple algebraic group acting on a number of copies of its Lie algebra via the diagonal adjoint action. We show that the null cone has rational singularities in the case of SL3. We observe by example that the null-cone is not normal in general and that the normalization of the null-cone does not have rational singularities in general. This is achieved by computing cohomology of certain vector bundles on flag varieties.
We introduce the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over a compact Riemann surface $Σ$ and consider a natural involution, studying its fixed point locus when $Σ$ is $\mathbb{C} \mathbb{P}^1$ and establishing an identification with a moduli space of null polygons in Minkowski $3$-space.
In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.
We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible values of the second Chern class $n$ on the projective 3-space. The generalized null correlation bundles constituting open dense subsets of these components are defined as cohomology bundles of monads whose members are direct sums of line bundles of degrees depending on nonnegative integers $a,b,c$, where $b\ge a$ and $c>a+b$. We show that, in the wide range w…
Harder-Narasimhan filtrations of decorated vector bundles
School of Mathematics, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom（爱丁堡大学数学学院）；Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany（海德堡大学数学研究所）
We give a new method for calculating the cohomology of the normal bundles over rational varieties which are smooth projections of Veronese embeddings. The method can be used also when the projections are not smooth, in this case it provides information about the critical locus of maps between projective spaces.
We prove that a double cover of $\mathbb{P}^2$ ramified along a general smooth curve B of degree $2s$, for $s \geq 3$, supports a rank 2 special Ulrich bundle.
Let $S$ be a regular surface endowed with a very ample line bundle $\mathcal O_S(h_S)$. Taking inspiration from a very recent result by D. Faenzi on $K3$ surfaces, we prove that if $\mathcal O_S(h_S)$ satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank $2$. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in $\mathbb P^N$. Finally, we also discuss about the size of the families of Ulrich bundles on $S$ and we inspect…
Let $S$ be a surface with $p_g(S)=0$, $q(S)=1$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$. Moreover, we show that $S$ supports stable Ulrich bundles of rank $2$ if the genus of the general element in $\vert h\vert$ is at least $2$.
In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, for Q-Gorenstein klt-trivial fibrations over smooth projective curves whose fibers are all klt log Calabi-Yau pairs of Fano type.
We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically imm…
We use certain special prehomogeneous representations of algebraic groups in order to construct aCM vector bundles, possibly Ulrich, on certain families of hypersurfaces. Among other results, we show that a general cubic hypersurface of dimension seven admitsan indecomposable Ulrich bundle of rank nine, and that a general cubic fourfold admits an unsplit aCM bundle of rank six.
We offer a new approach to proving the Chen-Donaldson-Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a families of curves and its relation to the one-dimensional cycles in the moduli space of stable bundles on curves.
