本文建立了Barth-Van de Ven-Tyurin-Sato定理的一个类似定理。证明了线性ind-Grassmannian中有限余维完备交上的有限秩向量丛同构于线丛的直和.
英文摘要：
In this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem. We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.
The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an $E_6$-variety) and the Freudenthal variety (an $E_7$-variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such resu…
A noncommutative Grassmannian NGr(m, n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d = n-m of the Grassmannian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmannian. The proof is quite different …
We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $\hbox{Gr}(3,6)$ with a codimension 2 linear subspace (with respect to the Plücker embedding). We prove that the motive of $Y$ is Kimura finite-dimensional. We also prove the generalized Hodge conjecture for all powers of $Y$.
On the Picard group of low-codimension subvarieties
我们介绍了一种方法来确定是否n维光滑的子簇的环境空间的维数最多为2n 2继承皮卡德群从环境空间（因为它发生时，环境空间是一个射影空间，根据结果的巴思和拉森）。作为应用，我们给出了一个肯定的答案（直到一些温和的自然数值条件），当周围空间是一个Grassmannian的线（从而改进了Barth，Van de Ven和Sommese的结果）或产品的两个射影空间的相同的维度。
We prove that in the parameter space of $M$-dimensional Fano complete intersections of index one and codimension two the locus of varieties that are not birationally superrigid has codimension at least $\frac12 (M-9)(M-10)-1$.
In this paper we compute the generating rank of $k$-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of $k$-Grassmannians arising from Hermitian forms of Witt index $n$ defined over vector spaces of dimension $N > 2n$. We also study generating sets for the $2$-Grassmannians arising from quadratic forms of Witt index $n$ defined over $V(N,{\mathbb F}_q)$ for $q=4,8,9$ and $2n \leq N \leq 2n+2$. We prove that for $N >6$ they can be generated over the prime subfield, thus determining their generating rank.
We prove that the Witt vector affine Grassmannian, which parametrizes W(k)-lattices in W(k)[1/p]^n for a perfect field k of charactristic p, is representable by an ind-(perfect scheme) over k. This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably h-descent results for vector bundles.
