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Jun-Muk Hwang

Publications and source records attributed to Jun-Muk Hwang.

At least 37 records · Page 2Linked to original sources
arXiv2021

Extending Nirenberg-Spencer's question on holomorphic embeddings to families of holomorphic embeddings

Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As an application, we obtain a stronger version of the Cartan-Fubini type extension theorem for Fano manifolds of Picard number 1. We also propose a potential application on hyperplane sections of projective K3 surfaces.

math.CV↗
arXiv2021

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.

math.DG↗
arXiv2020

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 $\mathcal C \to M$. 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.

math.DG↗
arXiv2020

Legendrian cone structures and contact prolongations

We study a cone structure ${\mathcal C} \subset {\mathbb P} D$ on a holomorphic contact manifold $(M, D \subset T_M)$ such that each fiber ${\mathcal C}_x \subset {\mathbb P} D_x$ is isomorphic to a Legendrian submanifold of fixed isomorphism type. By characterizing subadjoint varieties among Legendrian submanifolds in terms of contact prolongations, we prove that the canonical distribution on the associated contact G-structure admits a holomorphic horizontal splitting.

math.DG↗
arXiv2019

Rigidity properties of holomorphic Legendrian singularities

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem on Fano contact manifolds. The second result is the deformation-rigidity of normal Legendrian singularities, meaning that any holomorphic family of normal Legendrian singularities is trivial, up to contactomorphic biholomorphisms of germs. Both results are proved by exploiting the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle on the contact manifold.

math.AG↗
arXiv2019

An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principle

A conjecture of Hirschowitz's predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan's equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if $A$ is a Fano manifold, or if the global sections of $W$ separate points of $A$. Our method shows more generally that for any unobstructed compact submanifold $A$ in a complex manifold, if the normal bundle is globally generated and its sections separate points of $A$, then a sufficiently general deformation of $A$ satisfies the formal principle. In particular, a sufficiently general smooth free rational curve on a complex manifold satisfies the formal principle.

math.AG↗
arXiv2019

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 odd-symplectic 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 pseudo-concavity type condition that certain vector bundles arising from Spencer complexes have no nonzero sections. The pseudo-concavity type condition is checked by exploiting geometry of minimal rational curves.

math.AG↗
arXiv2018

Special birational transformations of type (2,1)

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.

math.AG↗
arXiv2017

Rational curves and prolongations of G-structures

In a joint work with N. Mok in 1997, we proved that for an irreducible representation $G \subset {\bf GL}(V),$ if a holomorphic $G$-structure exists on a uniruled projective manifold, then the Lie algebra of $G$ has nonzero prolongation. We tried to generalize this to an arbitrary connected algebraic subgroup $G \subset {\bf GL}(V)$ and a complex manifold containing an immersed rational curve, but the proposed proof had a flaw.

math.AG↗
arXiv2017

Euler-symmetric projective varieties

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 systems of fundamental forms at general points of projective varieties. We study relations between the algebraic properties of symbol systems and the geometric properties of Euler-symmetric projective varieties. We describe also the relation between Euler-symmetric projective varieties of dimension n and equivariant compactifications of the vector group G_a^n.

math.AG↗
arXiv2017

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.

math.AG↗
arXiv2016

Prolongations of infinitesimal automorphisms of cubic hypersurfaces with nonzero Hessian

We study the connected component of the automorphism group of a cubic hypersurface over complex numbers. When the cubic hypersurface has nonzero Hessian, this group is usually small. But there are examples with unusually large automorphism groups: the secants of Severi varieties. Can we characterize them by the property of having unusually large automorphism groups? We study this question from the viewpoint of prolongations of the Lie algebras. Our result characterizes the secants of Severi varieties, among cubic hypersurfaces with nonzero Hessian and smooth singular locus, in terms of prolongations of certain type.

math.AG↗
arXiv2016

Geometry of webs of algebraic curves

A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$. Under two geometric assumptions on the web-structure, the pairwise non-integrability condition and the bracket-generating condition, we prove that the local differential geometry determines the global algebraic geometry of $X$, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X \subset {\bf P}^N$ is a Fano submanifold of Picard number 1, and the family of lines covering $X$ becomes a web. In this special case, we have a stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X, X' \subset {\bf P}^N, \dim X' \geq 3,$ are two such Fano manifolds of Picard number 1, then any surjective morphism $f: X \to X'$ is an isomorphism.

math.AG↗
arXiv2013

Varieties of minimal rational tangents on double covers of projective space

Let $ϕ: X \to \mathbb P^n$ be a double cover branched along a smooth hypersurface of degree $2m, 2 \leq m \leq n-1$. 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 $\mathbb C_x$ implies a certain rigidity property of the covering morphism $ϕ$. 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 $X$.

math.AG↗
arXiv2013

Uniqueness of equivariant compactifications of C^n by a Fano manifold of Picard number 1

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.

math.AG↗
arXiv2012

Webs of Lagrangian Tori in Projective Symplectic Manifolds

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkähler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville's. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt's theory of subnormal subgroups.

math.AG↗
arXiv2011

Slopes of smooth curves on Fano manifolds

Ross and Thomas introduced the concept of slope stability to study K-stability, which has conjectural relation with the existence of constant scalar curvature Kähler metric. This paper presents a study of slope stability of Fano manifolds of dimension $n\geq 3$ with respect to smooth curves. The question turns out to be easy for curves of genus $\geq 1$ and the interest lies in the case of smooth rational curves. Our main result classifies completely the cases when a polarized Fano manifold $(X, -K_X)$ is not slope stable with respect to a smooth curve. Our result also states that a Fano threefold $X$ with Picard number 1 is slope stable with respect to every smooth curve unless $X$ is the projective space.

math.AG↗