Varieties Of Minimal Rational Tangents Research Articles

Holomorphic isometries of $${\mathbb {B}}^m$$ B m into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals

We study general properties of images of holomorphic isometric embeddings of complex unit balls \({\mathbb {B}}^m\) into irreducible bounded symmetric domains \({\varOmega }\) of rank at least 2. In particular, we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections \({\varLambda }\) of the compact dual \(X_c\) of \({\varOmega }\). The question naturally arises as to which linear sections \(Z = {\varLambda }\cap {\varOmega }\) are actually images of holomorphic isometries of complex unit balls. We study the latter question in the case of bounded symmetric domains \({\varOmega }\) of type IV, alias Lie balls, i.e., bounded symmetric domains dual to hyperquadrics. We completely classify images of all holomorphic isometric embeddings of complex unit balls into such bounded symmetric domains \({\varOmega }\). Especially we show that there exist holomorphic isometric embeddings of complex unit balls of codimension 1 incongruent to the examples constructed by Mok (Proc Am Math Soc 144:4515–4525, 2016) from varieties of minimal rational tangents, and that moreover any holomorphic isometric embedding \(f: {\mathbb {B}}^m \rightarrow {\varOmega }\) extends to a holomorphic isometric embedding \(f: \mathbb B^{n-1} \rightarrow {\varOmega }\), \(\dim {\varOmega }= n\). The case of Lie balls is particularly relevant because holomorphic isometric embeddings of complex unit balls of sufficiently large dimensions into an irreducible bounded symmetric domain other than a type-IV domain are expected to be more rigid.

Holomorphic isometries of the complex unit ball into irreducible bounded symmetric domains

LetΩ⊂S\Omega \subset Sbe an irreducible bounded symmetric domain of rank≥2\ge 2embedded as an open subset in its dual Hermitian symmetric manifoldSSof the compact type. Writec1(S)=(p+2)δc_1(S) = (p+2)\delta,δ∈H2(S,Z)≅Z\delta \in H^2(S,\mathbb {Z}) \cong \mathbb {Z}being the positive generator. We prove that there exists a nonstandard holomorphic embedding of the(p+1)(p+1)-dimensional complex unit ballBp+1B^{p+1}intoΩ\Omegawhich is isometric with respect to canonical Kähler-Einstein metricsggresp. hhnormalized so that minimal disks are of constant Gaussian curvature−2-2. We construct such holomorphic isometries using varieties of minimal rational tangents (VMRTs). We also prove thatn≤p+1n \le p+1for any holomorphic isometryf:(Bn,g)→(Ω,h)f: (B^n,g) \to (\Omega ,h). Our proofs rely on an extension theorem for holomorphic isometries of Mok (2012), the asymptotic behavior due to Klembeck (1978) of standard complete Kähler metrics on strictly pseudoconvex domains, and the fine structure of boundaries of bounded symmetric domains in their Harish-Chandra realizations due to Wolf (1972).

DEFORMATION RIGIDITY OF ODD LAGRANGIAN GRASSMANNIANS

In this paper, we study the rigidity under <TEX>$K{\ddot{a}}hler$</TEX> deformation of the complex structure of odd Lagrangian Grassmannians, i.e., the Lagrangian case <TEX>$Gr_{\omega}$</TEX>(n, 2n+1) of odd symplectic Grassmannians. To obtain the global deformation rigidity of the odd Lagrangian Grassmannian, we use results about the automorphism group of this manifold, the Lie algebra of infinitesimal automorphisms of the affine cone of the variety of minimal rational tangents and its prolongations.

On the extension theorem of Hwang and Mok

Abstract Jun-Muk Hwang and Ngaiming Mok developed a framework for studying complex Fano (more generally, uniruled) manifolds in terms of their intrinsic local differential-geometric structure: the varieties of minimal rational tangents (VMRT). In particular, their ‘Cartan–Fubini’ extension theorem shows that a Fano manifold of Picard number 1 (satisfying certain technical conditions) is determined, up to biholomorphism, by an analytic germ of its VMRT at a general point. We prove a characteristic-free analogue of this result, replacing the VMRT with families of formal arcs.

Varieties of minimal rational tangents on Veronese double cones

A rational curve C in a nonsingular variety X is standard if under the normalization f : P 1 ! C X, the vector bundle f T (X) decomposes asO(2)O (1) p O q for some nonnegative integers satisfying p +q = dimX 1. For a Fano manifold X of Picard number one and a general point x 2 X, a general rational curve of minimal degree through x is standard. It has been asked whether all rational curves of minimal degree through a general point x are standard. Our main result is a negative answer to this question.

Cartan-Fubini type extension of holomorphic maps respecting varieties of minimal rational tangents

A fundamental result in the theory of minimal rational curves on projective manifolds is Cartan-Fubini extension theorem proved by Hwang and Mok, which describes the extensibility of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents (VMRT), under a mild geometric assumption on the second fundamental forms of VMRT’s. Hong and Mok have developed Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends VMRT’s onto linear sections of VMRT’s and it satisfies a mild geometric condition formulated in terms of second fundamental forms on VMRT’s. In the current paper, we give a generalization of Hong and Mok’s result, under the same condition on second fundamental forms, assuming only that the holomorphic immersions send VMRT’s to VMRT’s. Our argument is different from Hong and Mok’s and is based on the study of natural foliations on the total family of VMRT’s. This gives a substantially simpler proof than Hong and Mok’s argument.

A classification theorem on Fano bundles

In this paper we classify rank two Fano bundles ℰ on Fano manifolds satisfying H 2 (X,ℤ)≅H 4 (X,ℤ)≅ℤ. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization ℙ(ℰ), that allows us to obtain the cohomological invariants of X and ℰ. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

Minimal Rational Curves on Complete Toric Manifolds and Applications

AbstractWe 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 rational component of degree n, and the second shows that any complete toric manifold satisfying certain combinatorial conditions on the fan has the target rigidity property.

Varieties of minimal rational tangents of codimension 1

Varieties of minimal rational tangents of codimension 1

Varieties of minimal rational tangents on double covers of projective space

Let \(\phi : X \rightarrow \mathbb{P }^n\) be a double cover branched along a smooth hypersurface of degree \(2m, 2 \le m \le n-1\). We study the varieties of minimal rational tangents \(\mathcal{C }_x \subset \mathbb{P }T_x(X)\) at a general point \(x\) of \(X\). We describe the homogeneous ideal of \(\mathcal{C }_x\) and show that the projective isomorphism type of \(\mathcal{C }_x\) varies in a maximal way as \(x\) varies over general points of \(X\). Our description of the ideal of \(\mathcal{C }_x\) implies a certain rigidity property of the covering morphism \(\phi \). As an application of this rigidity, we show that any finite morphism between such double covers with \(m=n-1\) must be an isomorphism. We also prove that Liouville-type extension property holds with respect to minimal rational curves on \(X\).

Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1

In a series of works one of the authors has developed with J.-M. Hwang a geometric theory of uniruled projective manifolds basing on the study of varieties of minimal rational tangents, and the geometric theory has especially been applied to rational homogeneous manifolds of Picard number 1. In Mok [Astérisque 322, pp. 151–205] and Hong-Mok [J. Diff. Geom. 86 (2010), pp. 539–567] the authors have started the study of uniruled projective subvarieties, and a method was developed for characterizing certain subvarieties of rational homogeneous manifolds. The method relies on non-equidimensional Cartan-Fubini extension and a notion of parallel transport of varieties of minimal rational tangents. In the current article we apply the notion of parallel transport to a characterization of smooth Schubert varieties of rational homogeneous manifolds of Picard number 1. Given a pair ( S , S 0 ) (S, S_0) consisting of a rational homogeneous manifold S S of Picard number 1 and a smooth Schubert variety S 0 S_0 of S S , where no restrictions are placed on S 0 S_0 when S = G / P S = G/P is associated to a long root (while necessarily some cases have to be excluded when S S is associated to a short root), we prove that any subvariety of S S having the same homology class as S 0 S_0 must be g S 0 gS_0 for some g ∈ Aut ( S ) g \in \text {Aut}(S) . We reduce the problem first of all to a characterization of local deformations S t S_t of S 0 S_0 as a subvariety of S S . By Kodaira stability, S t S_t is uniruled by minimal rational curves of S S lying on S t S_t . We establish a biholomorphism between S t S_t and S 0 S_0 which extends to a global automorphism by reconstructing S t S_t by means of a repeated use of parallel transport of varieties of minimal rational tangents along minimal rational curves issuing from a general base point. Our method is applicable also to the case of singular Schubert varieties provided that there exists a minimal rational curve on the smooth locus of the variety.

Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity

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 $\mathfrak{g}$ is the Lie algebra $\mathfrak{aut}(\hat{S})$ of infinitesimal linear automorphisms of a projective variety S⊂ℙV, its prolongation $\mathfrak{g}^{(k)}$ is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation $\mathfrak{aut}(\hat{S})^{(k)}$ is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety S⊂ℙV with $\mathfrak{aut}(\hat {S})^{(k)}\neq0$ , which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when S is linearly normal and Sec (S)≠ℙV, the blow-up Bl S (ℙV) has the target rigidity property, i.e., any deformation of a surjective morphism f:Y→Bl S (ℙV) comes from the automorphisms of Bl S (ℙV).

Characterization of Standard Embeddings between Rational Homogeneous Manifolds of Picard Number 1

Let X be a rational homogeneous manifold of Picard number 1. A subdiagram of the marked Dynkin diagram of X induces naturally an embedding of a rational homogeneous manifold Z into X. By a standard embedding of Z into X we will mean the composite of it with the action of an element of Aut 0 (X). In this paper, we prove a characterization of standard embeddings of Z into X by means of varieties of minimal rational tangents in the case where Z is nonlinear. We also provide a characterization of maximal linear spaces in X.

Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds

In a series of works, one of the authors has developed with J.- M. Hwang a geometric theory of uniruled projective manifolds, especially those of Picard number 1, basing on the study of varieties of minimal rational tangents. A fundamental result in this theory is a principle of analytic continuation under very mild assumptions, called Cartan-Fubini extension, of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents. In this article we develop a generalization of Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends varieties of minimal rational tangents onto linear sections of varieties of minimal rational tangents and that it satisfies a mild geometric condition formulated in terms of second fundamental forms on varieties of minimal rational tangents. Formerly such a result was known only in the very special case of irreducible Hermitian symmetric manifolds of rank at least two, and the proof relied on the existence of flattening coordinates, viz., Harish-Chandra coordinates, with respect to which the varieties of minimal rational tangents form a constant family. The proof of the main result, which is based on the deformation theory of rational curves, is differential-geometric in nature and is applicable to the general situation of uniruled projective manifolds without any assumption on the existence of special coordinate systems. As an application, we give a characterization of standard embeddings for certain pairs of rational homogeneous manifolds in terms of embeddings of varieties of minimal rational tangents.

Unobstructedness of deformations of holomorphic maps onto Fano manifolds of Picard number 1

We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or finite. The condition on the variety of minimal rational tangents holds for practically all known examples of Fano manifolds of Picard number 1, except the projective space. When the variety of minimal rational tangents is non-linear, the proof is based on an earlier result of N. Mok and the author on the birationality of the tangent map. When the varieties of minimal rational tangents of the Fano manifold is finite, the key idea is to factorize the given surjective morphism, after some transformation, through a universal morphism associated to the minimal rational curves.

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold $M$ of dimension $2n$, a projective manifold $X$ and a surjective holomorphic map $f:M \to X$ with connected fibers of positive dimension, we prove that $X$ is biholomorphic to the projective space of dimension $n$. The proof is obtained by exploiting two geometric structures at general points of $X$: the affine structure arising from the action variables of the Lagrangian fibration $f$ and the structure defined by the variety of minimal rational tangents on the Fano manifold $X$.

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ⩾ 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:= rank(Ωg’) ⩽ rank(Ω):= r, proving a conjecture of the author’s motivated by Hermitian metric rigidity. As a first step in the proof, Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1. Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding, this means that the germ of f at a general point preserves the varieties of minimal rational tangents (VMRTs).

Geometry of chains of minimal rational curves

Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method to study chains of minimal rational curves via varieties of minimal rational tangents and their higher secants. For many examples of Fano manifolds this method can be used to compute the minimal length of chains needed to join two general points. One consequence of our computation is a bound on the multiplicities of divisors at a general point of the moduli of stable bundles of rank two on a curve.

On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Let X be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, X should be biholomorphic to a rational homogeneous manifold G/P, where G is a simple Lie group, and P ⊂ G is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents C x , and (b) recovering the structure of a rational homogeneous manifold from C x . The author proves that, when b 4 (X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane P 2 , the 3-dimensional hyperquadric Q 3 , or the 5-dimensional Fano homogeneous contact manifold of type G 2 , to be denoted by K(G 2 ). The principal difficulty is part (a) of the scheme. We prove that C x C PT x (X) is a rational curve of degrees < 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = P 2 resp. Q 3 resp. K(G 2 ). Let κ be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that κ is smooth. Furthermore, it implies that at any point x ∈ X, the normalization κ x of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that κ x is a rational curve, our principal object of study is the universal family u of κ, giving a double fibration p: u → κ, μ: u → X, which gives P 1 -bundles. There is a rank-2 holomorphic vector bundle V on κ whose projectivization is isomorphic to p: u → κ. We prove that V is stable, and deduce the inequality d < 4 from the inequality c 2 1 (V) < 4c 2 (V) resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of d = 4 is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on V in the special case where c 2 1 (V) = 4c 2 (V).

Deformation rigidity of the rational homogeneous space associated to a long simple root

As a continuation of our previous works we study the conjecture on the rigidity under Kähler deformation of the complex structure of rational homogeneous spaces G/ P of Picard number 1, confirming its validity whenever G/ P is associated to a long simple root. For these rational homogeneous spaces the minimal G-invariant holomorphic distribution D is spanned by varieties of minimal rational tangents, and, excepting the symmetric and the contact cases, the complex structure of G/ P is completely determined by the nilpotent symbol algebra of the weak derived differential system of D. The problem is reduced, in a sense, to the invariance of this nilpotent symbol algebra under Kähler deformation. In our earlier works in relation to the question of the integrability of distributions spanned by varieties of minimal rational tangents we have established identities on Lie brackets using integral surfaces arising from pencils of rational curves. In the case on hand, at a point o∈ G/ P we prove that the nilpotent symbol algebra at o is nothing other than the universal Lie algebra generated by D o subject to these identities on Lie brackets, by verifying that they correspond to finiteness condition in the Serre presentation of the simple Lie algebra G.