Related papers: Birationally rigid complete intersections of high …

A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is…

We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.

Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the…

Any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles, of each dimension up to half its own dimension.

We study the birational geometry of hypersurfaces in projective varieties of the form $\mathbf{P}^1\times Z$, where $Z$ satisfies mild assumptions. Building on recent results of Herrera--Laface--Ugaglia, we study their Cox rings (when…

Let $\pi$ be a group satisfying the Farrell-Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincar\'e duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$…

We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10. We show that these threefolds are birationally isomorphic to Verra solids (hypersurfaces of bidegree $(2,2)$ in $ \P^2\times \P^2$).…

This paper is to look for bi-Frobenius algebra structures on quantum complete intersections. We find a class of comultiplications, such that if $\sqrt{-1}\in k$, then a quantum complete intersection becomes a bi-Frobenius algebra with…

Let $Z^\circ$ be a complete intersection inside $(\mathbb{C}^*)^n$ that compactifies to a smooth Calabi-Yau subvariety $Z$ inside a Fano toric variety $X$. We compute the skeleton of $Z^\circ$ and describe its decomposition into standard…

For a general cubic fourfold $X \subset \mathbb{P}^5$, we compute the Hodge numbers of the locus $S \subset F$ of lines of second type. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any…

In this paper we compute the cohomology of the Fano varieties of $k$-planes in the smooth complete intersection of two quadrics in $\mathbb{P}^{2g+1}$, using Springer theory for symmetric spaces.

We construct orbifolds with quasitoric boundary and show that they have stable almost complex structure. We show that a quasitoric orbifold is complex cobordant to finite disjoint copies of complex orbifold projective spaces. Finally some…

Mukai proved that most prime Fano fourfolds of degree 10 and index 2 are contained in a Grassmannian G(2,5). They are all unirational and some are rational, as remarked by Roth in 1949. We show that their middle cohomology is of K3 type and…

We prove that the basic intersection cohomology $ {I H}^{^{*}}_{_{\bar{p}}}{(M/\mathcal{F})}, $ where $\mathcal{F}$ is the singular foliation determined by an isometric action of a Lie group $G$ on the compact manifold $M$, is finite…

It is known that a necessary condition for the existence of K\"ahler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nystr\"om, it was generalized for (singular) Fano…

Following a combinatorial observation made by one of us recently in relation to a problem in quantum information [Nakata et al., Phys. Rev. X 7:021006 (2017)], we study what are the possible intersection cardinalities of a $k$-dimensional…

We study smooth projective complex varieties with ample cotangent bundle. Our main result is that in an abelian variety of dimension n, a complete intersection of at least n/2 general hypersurfaces of sufficiently high degrees has ample…

We give conditions for a uniruled variety of dimension at least 2 to be non-solid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit…

Let \phi: \mathbb{P}^{r}\dashrightarrow Z be a birational transformation with a smooth connected base locus scheme, where Z\subseteq\mathbb{P}^{r+c} is a nondegenerate prime Fano manifold. We call \phi a quadro-quadric special briational…

For a given topological type of a normal surface singularity, there are various types of complex structures which realize it. We are interested in the following problem: Find the maximum of the geometric genus and a condition for that the…