Varieties Of Minimal Rational Tangents Research Articles

Birational Transformations on Irreducible Compact Hermitian Symmetric Spaces

Abstract We construct a sequence of explicit blow-ups and blow-downs on an irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover, this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties. The result was known in the special case where $X$ is of rank 2 and could be found in Zak’s monograph “Tangents and secants of algebraic varieties.”

Recognizing the G2-horospherical Manifold of Picard Number 1 by Varieties of Minimal Rational Tangents

Recognizing the G2-horospherical Manifold of Picard Number 1 by Varieties of Minimal Rational Tangents

Special lines on contact manifolds

In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and has a fixed standard splitting type of the restricted tangent bundle. In this paper we study singular contact lines and those with special splitting type. We provide restrictions on the families of such lines, and on contact Fano varieties which have reducible varieties of minimal rational tangents. We also show that the results about singular lines naturally generalise to complex contact manifolds, which are not necessarily Fano, for instance, quasiprojective contact manifolds or contact manifolds of Fujiki class C. In particular, in many cases the dimension of a family of singular lines is at most 2 less than the dimension of the contact manifold.

Complex differential geometry and its applications

<p/><p content-type="inline"/> “Complex differential geometry and its applications” originated from the author's prior work in Kähler geometry revolving around bounded symmetric domains $\Omega$, their finite-volume quotients $X_\Gamma~:=~\Omega/\Gamma$ and dual Hermitian symmetric spaces $S$ of the compact type. The author's solution of the generalized Frankel conjecture has revealed the importance of the collection of tangents to minimal rational curves in the curvature characterization of $S$. Together with Hwang, the author has developed the foundation of a differential-geometric theory of uniruled projective manifolds $(X,\mathcal~K)$ based on the variety of minimal rational tangents (VMRT), encapsulated in the VMRT structure $\pi:~\mathscr~C(X)~\to~X$ and the universal family $\rho:~\mathcal~U~\to~\mathcal~K$, enabling them to resolve classical problems in algebraic geometry such as the deformation rigidity of rational homogeneous spaces $G/P$ of Picard number 1, the Lazarsfeld problem and the characterization of uniruled projective manifolds equipped with reductive G-structures. Hwang-Mok established the Cartan-Fubini extension principle for uniruled projective manifolds $(X,\mathcal~K)$ of Picard number 1 under very mild conditions for the analytic continuation of VMRT-preserving local biholomorphisms, which was further developed by Hong-Mok resp. Mok-Zhang to a non-equidimensional version of Cartan-Fubini extension principle resp. analytic continuation of germs of complex submanifolds inheriting sufficiently non-degenerate sub-VMRT structures, resulting in particular in Schubert and Schur rigidity theorems.The geometric theory of VMRT has also motivated the author to examine algebraic subsets (defined via the Borel embedding) $Z~\subset~\Omega$, and their images $\pi_\Gamma(Z)$ under the quotient map $\pi_\Gamma:~\Omega~\to~X_\Gamma$, i.e., to examine an irreducible component $\mathcal~K$ of the Chow scheme ${\rm~Chow}(S)$, its universal family $\rho:~\mathcal~U~\to~\mathcal~K$ and evaluation map $\mu:~\mathcal~U~\to~X$, together with the quotient $\mathcal~U_\Gamma~:=~\mathcal~U_0/\Gamma$ of the restriction $\mathcal~U_0~:~=~\mathcal~U|_\Omega$. The fibered space $\mathcal~U_\Gamma$ may be regarded as the support of a meromorphic distribution $\mathscr~F$, such that the image under the natural projection of a certain $\mathscr~F$-saturated subvariety $\mathscr~Z\subset~\mathcal~U_\Gamma$ gives the Zariski closure of $q(Z)$. At the same time, the prior work of Mok-Zhong for the compactification under certain curvature assumptions of complete Kähler manifolds of finite volume yields differential-geometric proofs realizing $X_\Gamma$ and various holomorphic fiber bundles over $X_\Gamma$ as quasi-projective varieties, together with a proof of the algebraicity of natural meromorphic foliations defined on them, enabling him to prove the Ax-Lindemann Theorem for not necessary arithmetic lattices $\Gamma$ of rank 1. In the setting of Shimura varieties $X_\Gamma$, in which the arithmeticity of the lattices $\Gamma$ has enabled a flourishing of results on functional transcendence with the availability of o-minimal structures from model theory and the interpretation of $X_\Gamma$ as modular varieties of (mixed) Hodge structures, a combination of the differential-geometric and algebro-geometric perspective of the author together with the variety of techniques and results from o-minimality and Hodge theory, enabled Mok-Pila-Tsimerman to establish the long awaited Ax-Schanuel theorem on Shimura varieties, which, together with its many generalizations, has proven to be applicable to deep arithmetic problems such as the Pink-Zilber conjecture and the Shafarevich conjecture in number theory.

NORMALISED TANGENT BUNDLE, VARIETIES WITH SMALL CODEGREE AND PSEUDOEFFECTIVE THRESHOLD

AbstractWe 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 pseudoeffective cones of the projectivised tangent bundles of rational homogeneous spaces of Picard number $1$ are explicitly determined by studying the total dual variety of minimal rational tangents (VMRTs) 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 $1$ .

Partial Compactification of Metabelian Lie Groups with Prescribed Varieties of Minimal Rational Tangents

Abstract We study minimal rational curves on a complex manifold that are tangent to a distribution. In this setting, the variety of minimal rational tangents (VMRTs) has to be isotropic with respect to the Levi tensor of the distribution. Our main result is a converse of this: any smooth projective variety isotropic with respect to a vector-valued anti-symmetric form can be realized as VMRT of minimal rational curves tangent to a distribution on a complex manifold. The complex manifold is constructed as a partial equivariant compactification of a metabelian group, which is a result of independent interest.

Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus q}$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C$, which is the germ of submanifolds $\mathcal{C}^C_{x} \subset \mathbb{P} T_{x} X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT $\mathcal{C}^{C}_{x} \subset \mathbb{P} D_{x}$. When $D \subset TX$ is a contact distribution, a well-known necessary condition is that $\mathcal{C}_{x}^{C}$ should be Legendrian with respect to the induced contact structure on $\mathbb{P} D_{x}$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D \subset TX$ and an unbendable rational curve $C \subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT $\mathcal{C}^C_{x} \subset \mathbb{P} D_{x}$ at some point $x \in C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.

Minimal Rational Curves and 1-Flat Irreducible G-Structures

1-Flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachhöfer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. The proof employs the method of Cartan connections. An interesting feature is that Cartan connections are considered not for the G-structures themselves, but for certain geometric structures on the spaces of minimal rational curves.

Characterizing symplectic Grassmannians by varieties of minimal rational tangents

We show that if the variety of minimal rational tangents (VMRT) of a uniruled projective manifold at a general point is projectively equivalent to that of a symplectic or an odd-symplectic Grassmannian, the germ of a general minimal rational curve is biholomorphic to the germ of a general line in a presymplectic Grassmannian. As an application, we characterize symplectic and oddsymplectic Grassmannians, among Fano manifolds of Picard number $1$, by their VMRT at a general point and prove their rigidity under global Kähler deformation. Analogous results for $G/P$ associated with a long root were obtained by Mok and Hong–Hwang a decade ago by using Tanaka theory for parabolic geometries. When $G/P$ is associated with a short root, for which symplectic Grassmannians are most prominent examples, the associated local differential geometric structure is no longer a parabolic geometry and standard machinery of Tanaka theory cannot be applied because of several degenerate features. To overcome the difficulty, we show that Tanaka’s method can be generalized to a setting much broader than parabolic geometries, by assuming a vanishing condition that certain vector bundles arising from Spencer complexes have no nonzero sections. In our application of this method to the characterization of symplectic (or odd-symplectic) Grassmannians, this vanishing condition is checked by exploiting geometry of minimal rational curves.

Cone structures and parabolic geometries

A cone structure on a complex manifold M is a closed submanifold \(\mathcal {C}\subset \mathbb {P}TM\) of the projectivized tangent bundle which is submersive over M. A conic connection on \(\mathcal {C}\) specifies a distinguished family of curves on M in the directions specified by \(\mathcal {C}\). There are two common sources of cone structures and conic connections, one in differential geometry and another in algebraic geometry. In differential geometry, we have cone structures induced by the geometric structures underlying holomorphic parabolic geometries, a classical example of which is the null cone bundle of a holomorphic conformal structure. In algebraic geometry, we have the cone structures consisting of varieties of minimal rational tangents (VMRT) given by minimal rational curves on uniruled projective manifolds. The local invariants of the cone structures in parabolic geometries are given by the curvature of the parabolic geometries, the nature of which depend on the type of the parabolic geometry, i.e., the type of the fibers of \(\mathcal {C}\rightarrow M\). For the VMRT-structures, more intrinsic invariants of the conic connections which do not depend on the type of the fiber play important roles. We study the relation between these two different aspects for the cone structures induced by parabolic geometries associated with a long simple root of a complex simple Lie algebra. As an application, we obtain a local differential-geometric version of the global algebraic-geometric recognition theorem due to Mok and Hong–Hwang. In our local version, the role of rational curves is completely replaced by appropriate torsion conditions on the conic connection.

Unbendable rational curves of Goursat type and Cartan type

We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension n with normal bundles isomorphic toOP1(1)⊕p⊕OP1⊕(n−1−p) for some nonnegative integer p. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of p=1 and n≤5, which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role.

Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X0, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X0, X) of the subdiagram type whenever X0 is nonlinear. It remains unsolved whether rigidity holds when (X0, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X0, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure $$\overline{\omega} : \mathscr{C}(S) \rightarrow S$$ on a uniruled projective manifold $$(X,\,{\cal K})$$ equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve l emanating from a general point x ∈ S, there exists an immersed neighborhood Nl of l which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ⊂ X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ⩾ 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X0 into X which induces an isomorphism on second homology groups. By studying ℂ*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X0 ⊂ X. Under the additional hypothesis that all holomorphic sections in Γ(X0, Tx∣x0) lift to global holomorphic vector fields on X, we prove that the admissible pair (X0, X) is rigid. As examples we check that (X0, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ≅ ℂ2n, n ⩾ 3, and when X0 ⊂ X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

Rigidity of pairs of rational homogeneous spaces of Picard number $1$ and analytic continuation of geometric substructures on uniruled projective manifolds

Building on the geometric theory of uniruled projective manifolds by Hwang–Mok, which relies on the study of varieties of minimal rational tangents (VMRTs) from both the algebro-geometric and the differential-geometric perspectives, Mok, Hong–Mok and Hong–Park have studied standard embeddings between rational homogeneous spaces $X = G/P$ of Picard number $1$. Denoting by $S \subset X$ an arbitrary germ of complex submanifold which inherits from $X$ a geometric structure defined by taking intersections of VMRTs with tangent subspaces and modeled on some rational homogeneous space $X_0 = G_0 / P_0$ of Picard number $1$ embedded in $X = G/P$ as a linear section through a standard embedding, we say that $(X_0, X)$ is rigid if there always exists some $\gamma \in \mathrm{Aut}(X)$ such that $S$ is an open subset of $\gamma (X_0)$. We prove that a pair $(X_0, X)$ of sub-diagram type is rigid whenever $X_0$ is nonlinear, which in the Hermitian symmetric case recovers Schubert rigidity for nonlinear smooth Schubert cycles, and which in the general rational homogeneous case goes beyond earlier works dealing with images of holomorphic maps. Our methods apply to uniruled projective manifolds $(X, \mathcal{K})$, for which we introduce a general notion of sub-VMRT structures $\varpi : \mathscr{C} (S) \to S$, proving that they are rationally saturated under an auxiliary condition on the intersection $\mathscr{C} (S) := \mathscr{C} (X) \cap \mathbb{P} T (S)$ and a nondegeneracy condition for substructures expressed in terms of second fundamental forms on VMRTs. Under the additional hypothesis that minimal rational curves are of degree $1$ and that distributions spanned by sub-VMRTs are bracket generating, we prove that $S$ extends to a subvariety $Z \subset X$. For its proof, starting with a “Thickening Lemma” which yields smooth collars around certain standard rational curves, we show that the germ of submanifold $(S; x_0)$ and, hence, the associated germ of sub-VMRT structure on $(S; x_0)$ can be propagated along chains of “thickening” curves issuing from $x_0$, and construct by analytic continuation a projective family of chains of rational curves compactifying the latter family, thereby constructing the projective completion $Z$ of $S$ as its image under the evaluation map.

Correction to: A characterization of symplectic Grassmannians

We provide a characterization of Symplectic Grassmannians in terms of their Varieties of Minimal Rational Tangents.

Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-K\"ahler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces.

Variation of the Variety of Minimal Rational Tangents of Cyclic Coverings

AbstractLet π:X→ ℙnbe thed-cyclic covering branched along a smooth hypersurfaceY⊂ ℙnof degreed, 3 ≤d≤n. We identify the minimal rational curves onXwithd-tangent lines ofYand describe the scheme structure of the variety of minimal rational tangents 𝒞x⊂ ℙTx(X) at a general pointx∈X. We also show that the projective isomorphism type of 𝒞xvaries in a maximal way asxmoves over general points ofX.

Characterizing the homogeneous variety

AbstractIn this paper we consider the 15‐dimensional rational homogeneous variety of Picard number one F4(4), and provide a characterization of it in terms of its variety of minimal rational tangents.

Standard Embeddings of Smooth Schubert Varieties in Rational Homogeneous Manifolds of Picard Number 1

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional $F_4$-homogeneous manifold associated to a short simple root.

Isotrivial VMRT-structures of complete intersection type

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 family of varieties of minimal rational tangents needs not be locally flat in differential geometric sense. This leads to the question for which projective variety Z, the Z-isotriviality of varieties of minimal rational tangents implies local flatness. Our main result verifies this for many cases of Z among complete intersections.

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.