arXiv subjects
Publications and source records attributed to Baohua Fu.
Let $G/P$ be a complex projective rational homogeneous variety of dimension $2m+1$. We prove that $G/P$ is an equivariant compactification of the Heisenberg group of dimension $2m+1$ if and only if it is isomorphic to either an adjoint variety, or the 3-dimensional smooth quadric $Q^3$, or a product $\mathbb{P}^{2m+1-d} \times Y$ with $1 \leq d=\dim Y \leq m$, where $Y$ is a product of cominuscule varieties.
We introduce contact symbol systems, a noncommutative analogue of symbol systems for projective fundamental forms, by replacing the polynomial algebra on a vector space by the graded dual of the universal enveloping algebra of a Heisenberg algebra. For a complex projective submanifold equipped with a contact structure, we define contact fundamental forms and prove that, at a general point, they form a contact symbol system, which gives a contact version of the classical result due to E. Cartan. Conversely, we prove that every contact symbol system can be realized as the contact fundamental forms of a projective variety with a dense open Heisenberg orbit, called the Heisenberg-symmetric variety associated to the contact symbol system. We show that the closure of a projectivized nilpotent orbit in a simple Lie algebra is Heisenberg-symmetric if and only if it is the adjoint variety, namely, the projectivization of the minimal nilpotent orbit. For adjoint varieties of non-symplectic simple Lie algebras, we prove the contact analogue of the Landsberg--Manivel strict prolongation property by using Yamaguchi's prolongation theory.
We establish a novel connection between the minimal nilpotent orbit $\mathbb{O}_n$ in $\mathfrak{sl}_n$ and the minimal nilpotent orbit closure $\overline{\mathbf{O}}_n$ in $\mathfrak{so}_{2n+2}$, which differs from the shared-orbit paradigm of Brylinski and Kostant, where no direct type-A--type-D relation appears. More precisely, we show that the affine closure of the cotangent bundle $\overline{T^*\mathbb{O}_n}^{\mathrm{aff}}$ is isomorphic to a $\mathbb{C}^*$-Hamiltonian reduction of $\overline{\mathbf{O}}_n$. This provides a quasi-classical analogue of a quantum result of Levasseur and Stafford. A detailed study of the geometry of this Hamiltonian reduction reveals that $\overline{T^*\mathbb{O}_n}^{\mathrm{aff}}$ has no symplectic resolution.
Let $Y$ be an irreducible non-singular affine $G$-variety with a $2$-large action. We show that the Hamiltonian reduction $T^*Y/\!\!/\!\!/G$ is a symplectic variety with terminal singularities, isomorphic to the affine closure of $T^*Z_{\text{reg}}$ where $Z:=Y/\!/G$. Furthermore, we provide sufficient conditions for the non-existence of a symplectic resolution for such varieties. These results yield three main applications: (i) providing a short proof of G. Schwarz's theorem on the graded surjectivity of the push-forward map $\mathcal{D}(Y)^G \to \mathcal{D}(Z)$; (ii) establishing the surjectivity of the symbol map on $Z$; and (iii) confirming the non-linear analog of a conjecture of Kaledin--Lehn--Sorger for $2$-large actions.
For a smooth quasi-affine variety $X$, the affine closure $\overline{T^*X} := \text{Spec}(\mathbb{K}[T^*X])$ contains $T^*X$ as an open subset, and its smooth locus carries a symplectic structure. A natural question is whether $\overline{T^*X}$ itself is a symplectic variety. A notable example is the conjecture of Ginzburg and Kazhdan, which predicts that $\overline{T^*(G/U)}$ is symplectic for a maximal unipotent subgroup $U$ in a reductive linear algebraic group $G$. This conjecture was recently proved by Gannon using representation-theoretic methods. In this paper, we provide a new geometric approach to this conjecture. Our method allows us to prove a more general result: $\overline{T^*(G/H)}$ is symplectic for any horospherical subgroup $H$ in $G$ such that $G/H$ is quasi-affine. In particular, this implies that the affine closure $\overline{T^*(G/[P,P])}$ is a symplectic variety for any parabolic subgroup $P$ in $G$.
The algebra of symmetric tensors $S(X):= H^0(X, \sf{S}^{\bullet} T_X)$ of a projective manifold $X$ leads to a natural dominant affinization morphism $$ \varphi_X: T^*X \longrightarrow \mathcal{Z}_X:= \text{Spec} S(X). $$ It is shown that $\varphi_X$ is birational if and only if $T_X$ is big. We prove that if $\varphi_X$ is birational, then $\mathcal{Z}_X$ is a symplectic variety endowed with the Schouten--Nijenhuis bracket if and only if $\mathbb{P} T_X$ is of Fano type, which is the case for smooth projective toric varieties, smooth horospherical varieties with small boundary and the quintic del Pezzo threefold. These give examples of a distinguished class of conical symplectic varieties, which we call symplectic orbifold cones.
The nilpotent cone of a simple Lie algebra is partitioned into locally closed subvarieties called special pieces, each containing exactly one special orbit. Lusztig conjectured that each special piece is the quotient of some smooth variety by a precise finite group $H$, a result proved for the classical types by Kraft and Procesi. The present work is about exceptional types. Our main result is a local version of Lusztig's conjecture: the intersection of a special piece with a Slodowy slice transverse to the minimal orbit in the piece is isomorphic to the quotient of a vector space by $H$. Along the way, we complete our previous work on the generic singularities of nilpotent orbit closures, by providing proofs for the last two `exotic' singularities. Four further, non-isolated, exotic singularities are studied: we show that quotients $\overline{{\mathcal 0}_{\text{mini}}(\mathfrak{so}_8)}/\mathfrak{S}_4$, $S^2({\mathbb C}^2/\mu_3)$, $S^3({\mathbb C}^2/\mu_2)$ and $\overline{{\mathcal 0}_{\text{mini}}(\mathfrak{sl}_3)}/\mathfrak{S}_4$ occur as Slodowy slice singularities between nilpotent orbits in types $F_4$, $E_6$, $E_7$ and $E_8$ respectively. We also extend, to fields other than ${\mathbb C}$, the results of Brylinski and Kostant on shared orbit pairs. In the course of our analysis, we discover a shared pair which is missing from Brylinski and Kostant's classification.
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 deformation. Namely, for any regular family of Fano manifolds over a connected base, if one fiber is isomorphic to $\bar{G}$, then so are all other fibers. This answers a question raised by Bien and Brion in their work on the local rigidity of wonderful varieties.
To each complex composition algebra $\mathbb{A}$, there associates a projective symmetric manifold $X(\mathbb{A})$ of Picard number one, which is just a smooth hyperplane section of the following varieties ${\rm Lag}(3,6), {\rm Gr}(3,6), \mathbb{S}_6, E_7/P_7.$ In this paper, it is proven that these varieties are rigid, namely for any smooth family of projective manifolds over a connected base, if one fiber is isomorphic to $X(\mathbb{A})$, then every fiber is isomorphic to $X(\mathbb{A})$.
Motivated by geometric Langlands, we initiate a program to study the mirror symmetry between nilpotent orbit closures of a semisimple Lie algebra and those of its Langlands dual. The most interesting case is $B_n$ via $C_n$. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symmetry we seek may coincide with Springer duality in the context of special orbits. Unfortunately, such a naive statement fails. To remedy the situation, we propose a conjecture which asserts the mirror symmetry for certain parabolic/induced covers of special orbits. Then, we prove the conjecture for Richardson orbits and obtain certain partial results in general. In the process, we reveal some very interesting and yet subtle structures of these finite covers, which are related to Lusztig's canonical quotients of special nilpotent orbits. For example, there is a mysterious asymmetry in the footprint or range of degrees of these finite covers. Finally, we provide two examples to show that the mirror symmetry fails outside the footprint.
We propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalized tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and hence the pseudoeffective cones of the projectivized tangent bundles of rational homogeneous spaces of Picard number $1$ are explicitly determined by studying the total dual VMRT and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number $1$.
We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of $\mathbb{C}^4$ by the dihedral group of order $2d$, (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order $2d$ at equal parameters, (3) as singularities of a certain Slodowy slice in the $d$-fold cover of the nilpotent cone in ${\mathfrak{sl}}_d$.
It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.
In a recent preprint, Y. Namikawa proposed a conjecture on Q-factorial terminalizations and their birational geometry of nilpotent orbits. He proved his conjecture for classical simple Lie algebras. In this note, we prove his conjecture for exceptional simple Lie algebras. For the birational geometry, contrary to the classical case, two new types of Mukai flops appear.
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 ${\rm Gr}(2,5)$.
In this note, we classify smooth equivariant compactifications of $\mathbb{G}_a^n$ which are Fano manifolds with index $\geq n-2$.
Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if $X = \cap_{i=1}^r D_i \subset G/P$ is a general complete intersection of $r$ ample divisors such that $K_{G/P}^* \otimes \mathcal{O}_{G/P}(-\sum_i D_i)$ is ample, then $X$ is Fano. We first classify these Fano complete intersections which are locally rigid. It turns out that most of them are hyperplane sections. We then classify general hyperplane sections which are quasi-homogeneous.
A birational transformation f: P^n --> Z, where Z is a nonsingular variety of Picard number 1, is called a special birational transformation of type (a, b) if f is given by a linear system of degree a, its inverse is given by a linear system of degree b and the base locus S \subset P^n of f is irreducible and nonsingular. In this paper, we classify special birational transformations of type (2,1). In addition to previous works Alzati-Sierra and Russo on this topic, our proof employs natural C^*-actions on Z in a crucial way. These C^*-actions also relate our result to the problem studied in our previous work on smooth projective varieties with nonzero prolongations.