Baohua Fu

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 $\mathc...

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 Bn via Cn. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symme...

Special nilpotent orbits in simple Lie algebras play an important role in representation theory. The nilpotent cone of a reductive 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 preci...

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...

We construct a new infinite family of four‐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 by the dihedral group of order , (2) as singular points of Calogero–Mos...

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...

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...

We propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalised tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Francesco Russo and Fyodor L. Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and, hence, the pseudoeff...

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 pr...

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...

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.

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...

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 smooth 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 inte...

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 this note, we classify smooth equivariant compactifications of G aⁿ that are Fano manifolds with index ≥n−2.

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...

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 inters...

In this note, we classify smooth equivariant compactifications of (G_a)^n which are Fano manifolds with index ≥ n − 2.

In this note, we classify smooth equivariant compactifications of $\mathbb{G}_a^n$ which are Fano manifolds with index $\geq n-2$.

Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that Euler-symmetric projective varieties can be classified by symbol systems, a class of algebraic objects modeled on the syst...

Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that Euler-symmetric projective varieties can be classified by symbol systems, a class of algebraic objects modeled on the syst...

The family of varieties of minimal rational tangents on a quasi-homogeneous projective manifold is isotrivial. Conversely, are projective manifolds with isotrivial varieties of minimal rational tangents quasi-homogenous? We will show that this is not true in general, even when the projective manifold has Picard number 1. In fact, an isotrivial fami...

The family of varieties of minimal rational tangents on a quasi-homogeneous projective manifold is isotrivial. Conversely, are projective manifolds with isotrivial varieties of minimal rational tangents quasi-homogenous? We will show that this is not true in general, even when the projective manifold has Picard number 1. In fact, an isotrivial fami...

Consider a simple algebraic group G of adjoint type, and its wonderful compactification X. We show that X admits a unique family of minimal rational curves, and we explicitly describe the subfamily consisting of curves through a general point. As an application, we show that X has the target rigidity property when G is not of type A_1 or C.

According to a well-known theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity,...

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...

We show that minimal rational components on a complete toric manifold X correspond bijectively to some special primitive collections in the fan defining X, and the associated varieties of minimal rational tangents are linear subspaces. Two applications are given: the first is a classification of n-dimensional toric Fano manifolds with a minimal rat...

We show that a compact Kähler manifold $X$ is a complex torus if both the continuous part and discrete part of some automorphism group $G$ of $X$ are infinite groups, unless $X$ is bimeromorphic to a non-trivial $G$ -equivariant fibration. Some applications to dynamics are given.

We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra.

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 i...

We show that a compact Kaehler manifold X is a complex torus if both the continuous part and discrete part of some automorphism group G of X are infinite groups, unless X is bimeromorphic to a non-trivial G-equivariant fibration. Some applications to dynamics are given.

The prolongation \(\mathfrak{g}^{(k)}\) of a linear Lie algebra \(\mathfrak{g}\subset \mathfrak{gl}(V)\) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras \(\mathfrak{g}\subset \mathfrak{gl}(V)\) with non-zero prolongations. If \(\m...

In a recent preprint, Y. Namikawa proposed a conjecture on Q-factorial terminalizations of nilpotent orbit closures and he proved his conjecture for classical simple Lie algebras. In this paper, we prove his conjecture for exceptional simple Lie algebras. For the birational geometry, contrary to the classical case, two new types of Mukai flops appe...

We shall show that the variety of minimal rational tangents on a complete toric manifold X is linear and minimal components in RatCurves^n(X) corresponds bijectively to some special primitive collections in the fan defining X.

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we shall show they are equivalent to well-known conjectures of Beauville and Murre.

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 o...

For stratified Mukai flops of type $A_{n,k}, D_{2k+1}$ and $E_{6,I}$, it is shown the fiber product induces isomorphisms on Chow motives. In contrast to (standard) Mukai flops, the cup product is generally not preserved. For $A_{n, 2}$, $D_5$ and $E_{6, I}$ flops, quantum corrections are found through degeneration/deformation to ordinary flops.

For stratified Mukai flops of type $A_{n,k}, D_{2k+1}$ and $E_{6,I}$, it is shown the fiber product induces isomorphisms on Chow motives. In contrast to (standard) Mukai flops, the cup product is generally not preserved. For $A_{n, 2}$, $D_5$ and $E_{6, I}$ flops, quantum corrections are found through degeneration/deformation to ordinary flops.

We prove that two Springer maps of the same degree over a nilpotent orbit closure are connected by stratified Mukai flops, and the latter is obtained by contractions of extremal rays of a natural resolution of the nilpotent orbit closure.

We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g^r_d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of P^1 whose ramification points ar...

We give two characterizations of hyperquadrics: one as non-degenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as $LQEL$-manifolds with large secant defects.

We construct a resolution of stratified Mukai flops of type A, D, E_{6, I} by successively blowing up smooth subvarieties. In the case of E_{6, I}, we construct a natural functor which induces an isomorphism between the Chow groups.

We recover the wreath product X ≔ Sym²(ℂ²/± 1) as a transversal slice to a nilpotent orbit in 𝔰𝔭6. By using deformations of Springer resolutions, we construct a symplectic deformation of symplectic resolutions of X.

We prove that two Springer maps over a nilpotent orbit closure with the same degree are connected by stratified Mukai flops and the latter is obtained by extremal contractions of a natural resolution of the nilpotent orbit closure.

We prove that two symplectic resolutions of a nilpotent orbit closures in a simple complex Lie algebra of classical type are related by Mukai flops in codimension 2. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

The projectivised nilpotent orbit closure P(\bar{O}) carries a natural contact structure on its smooth part. A resolution X \to P(\bar{O}) is called contact if the contact structure on P(O) extends to a contact structure on X. It turns out that contact resolutions, crepant resolutions and minimal models of P(\bar{O}) are all the same. In this note,...

This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions, which could also serve as an introduction to this subject.

We prove that two projective symplectic resolutions of are connected by Mukai flops in codimension 2 for a finite sub-group G <Sp(2n). It is also shown that two projective symplectic resolutions of are deformation equivalent.

We prove the conjecture that two projective symplectic resolutions for a symplectic variety $W$ are related by Mukai's elementary transformations over $W$ in codimension 2 in the following cases: (i). nilpotent orbit closures in a classical simple complex Lie algebra; (ii). some quotient symplectic varieties.

A resolution $Z \to X$ of a Poisson variety $X$ is called {\em Poisson} if every Poisson structure on $X$ lifts to a Poisson structure on $Z$. For symplectic varieties, we prove that Poisson resolutions coincide with symplectic resolutions. It is shown that for a Poisson surface $S$, the natural resolution $S^{[n]} \to S^{(n)}$ is a Poisson resolut...

Let S→φP1 be an elliptic fibration on a K3 surface S. Then the composition S[n]→πS(n)→symnφPn gives an Abelian fibration on S[n]. Let E be the exceptional divisor of π, then symnφ∘π(E) is of dimension n−1. We prove the inverse in this Note. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

We prove that for any two projective symplectic resolutions $Z_1$ and $Z_2$ for a nilpotent orbit closure in a complex simple Lie algebra of classical type, then $Z_1$ is deformation equivalen to $Z_2$.

We prove the uniqueness of crepant resolutions for some quotient singularities and for some nilpotent orbits. The finiteness of non-isomorphic symplectic resolutions for 4-dimenensional symplectic singularities is proved. We also give an example of symplectic singularity which admits two non-equivalent symplectic resolutions.

En utilisant des résultats de A. Beauville (Acta Math. 164 (1990) 211–235), nous donnons une description explicite des champs de vecteurs invariants par translation sur les jacobiennes affines des courbes spectrales qui justifie mathématiquement les travaux de F.A. Smirnov and V. Zeitlin (preprint math-ph/0203037). Dans le cas hyperelliptique, cett...

Let $S$ be a smooth complex connected analytic surface which admits a holomorphic symplectic structure. Let $S^{(n)}$ be its $n$th symmetric product. We prove that every projective symplectic resolution of $S^{(n)}$ is isomorphic to the Douady-Barlet resolution $S^{[n]} \to S^{(n)}$.

Let $\0$ be a nilpotent orbit in a semisimple complex Lie algebra $\g$. Denote by $G$ the simply connected Lie group with Lie algebra $\g$. For a $G$-homogeneous covering $M \to \0$, let $X$ be the normalization of $\bar{\0}$ in the function field of $M$. In this note, we study the existence of symplectic resolutions for such coverings $X$.

In this paper, firstly we calculate Picard groups of a nilpotent orbit 풪 in a classical complex simple Lie algebra and discuss the properties of being ℚ-factorial and factorial for the normalization 풪tilde; of the closure of 풪. Then we consider the problem of symplectic resolutions for 풪tilde;. Our main theorem says that for any nilpotent orbit 풪 i...

We give some necessary conditions for the existence of a symplectic resolution for quotient singularities. The McKay correspondence is also worked out for these resolutions.

In this note, firstly we give an easy proof of the factorization of symmetric matrices (see [Mos]), and then use it to prove the wellknown fact that the automorphism group of a non-degenerate symmetric bilinear form Q acts transitively on the locus of isotropic subspaces Σk(Q). Let V be an m-dimensional complex vector space. Proposition 1. Every sy...

These notes are collected from talks given by the authors at the University of Nice (october-december 2002)

For stratified Mukai flops of type An,k, D2k+1 and E6,I, it is shown that the fiber product induces isomorphisms on Chow mo- tives. In contrast to (standard) Mukai flops, the cup product is gen- erally not preserved. For An,2, D5 and E6,I flops, quantum cor- rections are found through degeneration/deformation to ordinary flops.