Related papers: Cup-length of oriented Grassmann manifolds via Gr\…

We use the framework of Quot schemes to give a novel description of the moduli spaces of stable n-pairs, also interpreted as gauged vortices on a closed Riemann surface with target Mat(r x n, C), where n >= r. We then show that these moduli…

An elementary combinatorial technique for computing lower bounds for the Yang indices of real Stiefel manifolds and oriented real Grassmannians is described. As a demonstration, it shown that the Yang index of $St(n,k)$, and consequently…

Let $F$ be a finite extension of $\mathbb{Q}_p$, let $\Omega_F$ be Drinfeld's upper half-plane over $F$ and let $G^0$ the subgroup of $GL_2(F)$ consisting of elements whose determinant has norm $1$. By working locally on $\Omega_F$, we…

We give effective bounds on the generation of pushforwards of log-pluricanonical bundles twisted by ample line bundles. This gives a partial answer to a conjecture proposed by Popa and Schnell. We prove two types of statements: first, more…

In this paper we calculate the Hochschild cohomology of graded skew-gentle algebras, together with its structure as graded commutative algebra under the cup product and its Lie algebra structure given by the Gerstenhaber bracket. One of the…

Griffiths' conjecture asserts that a holomorphic vector bundle is ample if and only if it admits a Hermitian metric with positive curvature. In this paper, we present a new proof of this conjecture on compact Riemann surfaces using a system…

We develop strong lower bounds for the span of the projective Stiefel manifolds $X_{n,r}=O(n)/(O(n-r)\times \mathbb Z/2)$, which enable very accurate (in many cases exact) estimates of the span. The technique, for the most part, involves…

The goal of this article is twofold. On one hand, we study the subvarieties of projective varieties which possess partially ample normal bundle; we prove that they are G2 in the ambient space. This generalizes results of Hartshorne and…

We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier results of Cecchini-Zeidler to all…

The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry.

We determine a nice simple formula for the largest Euclidean space for which there is an orientable n-manifold with a nonimmersion detected by Stiefel-Whitney classes. For Spin manifolds, we prove the analogue of the upper bound and…

We introduce notions of {\it upper chernrank} and {\it even cup length} of a finite connected CW-complex and prove that {\it upper chernrank} is a homotopy invariant. It turns out that determination of {\it upper chernrank} of a space $X$…

We consider the orthonormal frame bundle F(M) of a Riemannian manifold M. A construction of Sasaki defines a canonical Riemannian metric on F(M). We prove that for two closed Riemannian n-manifolds M and N, the frame bundles F(M) and F(N)…

In a series of papers, Weil initiated the investigation of translation invariant curvature measures of convex bodies, which include as prime examples Federer's curvature measures. In this paper, we continue this line of research by…

The gonality conjecture predicts that the gonality of a curve can be read off Koszul cohomology of line bundles of sufficiently large degree. We verify this conjecture for generic curves of odd genus. The even-genus case was previously…

In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.

We generalize the Fenchel theorem for strong spacelike closed curves of index $1$ in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to $2\pi$. Here strong spacelike means that the tangent…

We present some results that complement our prequels [arXiv:1809.08425,arXiv:1907.05770] on holomorphic vector bundles. We apply the method of the Quot-scheme limit of Fubini-Study metrics developed therein to provide a generalisation to…

Since Schwarzenberger and his celebrated paper called "Vector bundles on the projective plane" we know that any rank two vector bundle on $\P^2$ is a direct image of a line bundle on a double covering of the plane. This theorem suggests to…

In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order…