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Register a new userUniqueness of equivariant compactifications of C^n by a Fano manifold of Picard number 1
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Let X be an $n$-dimensional Fano manifold of Picard number 1. We study how many different ways X can compactify the complex vector group C^n equivariantly. Hassett and Tschinkel showed that when X = P^n with n geq 2, there are many distinct ways that X can be realized as equivariant compactifications of C^n. Our result says that projective space is an exception: among Fano manifolds of Picard number 1 with smooth VMRT, projective space is the only one compactifying C^n equivariantly in more than one ways. This answers questions raised by Hassett-Tschinkel and Arzhantsev-Sharoyko.
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We introduce the notion of intrinsic Grassmannians which generalizes the well known weighted Grassmannians. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Plucker ideal $I_{d,n}$ of the Grassmannian $mathrm{Gr}(d,n)$. We give a complete classification of all smooth Fano intrinsic Grassmannians of type $(2,n)$ with Picard number two and prove an explicit formula to compute the total number of such varieties for an arbitrary $n$. We study their geometry and show that they satisfy Fujitas freeness conjecture.
Let $G$ be a simple complex algebraic group, $P$ a parabolic subgroup of $G$ and $N$ the unipotent radical of $P.$ The so-called equivariant compactification of $N$ by $G/P$ is given by an action of $N$ on $G/P$ with a dense open orbit isomorphic to $N$. In this article, we investigate how many such equivariant compactifications there exist. Our result says that there is a unique equivariant compactification of $N$ by $G/P$, up to isomorphism, except $P^n$.
Let $X$ be an $n$-dimensional smooth Fano complex variety of Picard number one. Assume that the VMRT at a general point of $X$ is smooth irreducible and non-degenerate (which holds if $X$ is covered by lines with index $ >(n+2)/2$). It is proven that $dim mathfrak{aut}(X) > n(n+1)/2$ if and only if $X$ is isomorphic to $mathbb{P}^n, mathbb{Q}^n$ or ${rm Gr}(2,5)$. Furthermore, the equality $dim mathfrak{aut}(X) = n(n+1)/2$ holds only when $X$ is isomorphic to the 6-dimensional Lagrangian Grassmannian ${rm Lag}(6)$ or a general hyperplane section of ${rm Gr}(2,5)$.
For a complex connected semisimple linear algebraic group G of adjoint type and of rank n, De Concini and Procesi constructed its wonderful compactification bar{G}, which is a smooth Fano G times G-variety of Picard number n enjoying many interesting properties. In this paper, it is shown that the wonderful compactification bar{G} is rigid under Fano deformations. Namely, for any family of smooth Fano varieties over a connected base, if one fiber is isomorphic to bar{G}, then so are all other fibers.
In this note, we classify smooth equivariant compactifications of $mathbb{G}_a^n$ which are Fano manifolds with index $geq n-2$.
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