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Conference Paper: Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents
Title  Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents 

Authors  
Issue Date  2011 
Citation  Conference Journées Complexes Lorraines 2011, Nancy, France, 26 May 2011 How to Cite? 
Abstract  Given a uniruled projective manifold X equipped with a moduli space of minimal rational curves K, we associate to (X, K) its varieties of minimal rational tangents (VMRTs) Cx at general points x ∈ X. A fundamental question is the extent to which the manifold X is determined by its VMRTs. A special yet very interesting case of the problem is the case of characterization of a rational homogeneous manifold S = G/P of Picard number 1. Given a Fano manifold of Picard number 1 (X, K) equipped with a moduli space of minimal rational curves such that the VMRT Cx(X) at a general point x ∈ X is biholomorphic to the VMRT Co(S) of the model manifold S at a base point o ∈ S, the problem is determine whether X is
biholomorphic to S. We call this the Recognition Problem for S = G/P.
The problem was first motivated in 2002 in the Speaker’s work on a very special case of the CampanaPeternell Conjecture on compact complex manifolds with nef tangent bundle. There, assuming that X is Fano of Picard number 1 and b4(X) = 1 (a condition later on removed by J.M. Hwang in 2007), and imposing the condition that the VMRT Cx(X) is of dimension 1 at a general point, it was shown that X is biholomorphic to P2, Q3 or the 5dimensional contact Fano homogeneous manifold K(G2). The idea was to determine VMRTs and to deduce therefrom that X ∼= S. The Recognition Problem of S = Pn follows from the work ChoMiyaokaShepherdBarron. The same problem for S being Hermitian symmetric of rank ≥ 2 or of the contact type was solved by Mok in 2008, and the other cases of S = G/P associated to long simple roots were later solved by HongHwang. The case where S is associated to a short simple root, e.g., the case of symplectic Grassmannians (other than the Lagrangian case), remains open.
In place of just solving the Recognition Problem we propose in a certain sense to reconstruct the manifold S = G/P from its VMRTs. The idea is to realize S as the base space of the Stein factorization of an iterated fibered space obtained successively from forming P1 bundles and bundles of VMRTs based on the procedure of adjunction of minimal rational curves. We will explain the approach in the simple case of the hyperquadric, where one shows that X is a hyperquadric by means of the CartanFubini Principle on analytic continuation developed by HwangMok.

Description  Venue: Institut Élie Cartan 
Persistent Identifier  http://hdl.handle.net/10722/254142 
DC Field  Value  Language 

dc.contributor.author  Mok, N   
dc.date.accessioned  20180607T03:40:29Z   
dc.date.available  20180607T03:40:29Z   
dc.date.issued  2011   
dc.identifier.citation  Conference Journées Complexes Lorraines 2011, Nancy, France, 26 May 2011   
dc.identifier.uri  http://hdl.handle.net/10722/254142   
dc.description  Venue: Institut Élie Cartan   
dc.description.abstract  Given a uniruled projective manifold X equipped with a moduli space of minimal rational curves K, we associate to (X, K) its varieties of minimal rational tangents (VMRTs) Cx at general points x ∈ X. A fundamental question is the extent to which the manifold X is determined by its VMRTs. A special yet very interesting case of the problem is the case of characterization of a rational homogeneous manifold S = G/P of Picard number 1. Given a Fano manifold of Picard number 1 (X, K) equipped with a moduli space of minimal rational curves such that the VMRT Cx(X) at a general point x ∈ X is biholomorphic to the VMRT Co(S) of the model manifold S at a base point o ∈ S, the problem is determine whether X is biholomorphic to S. We call this the Recognition Problem for S = G/P. The problem was first motivated in 2002 in the Speaker’s work on a very special case of the CampanaPeternell Conjecture on compact complex manifolds with nef tangent bundle. There, assuming that X is Fano of Picard number 1 and b4(X) = 1 (a condition later on removed by J.M. Hwang in 2007), and imposing the condition that the VMRT Cx(X) is of dimension 1 at a general point, it was shown that X is biholomorphic to P2, Q3 or the 5dimensional contact Fano homogeneous manifold K(G2). The idea was to determine VMRTs and to deduce therefrom that X ∼= S. The Recognition Problem of S = Pn follows from the work ChoMiyaokaShepherdBarron. The same problem for S being Hermitian symmetric of rank ≥ 2 or of the contact type was solved by Mok in 2008, and the other cases of S = G/P associated to long simple roots were later solved by HongHwang. The case where S is associated to a short simple root, e.g., the case of symplectic Grassmannians (other than the Lagrangian case), remains open. In place of just solving the Recognition Problem we propose in a certain sense to reconstruct the manifold S = G/P from its VMRTs. The idea is to realize S as the base space of the Stein factorization of an iterated fibered space obtained successively from forming P1 bundles and bundles of VMRTs based on the procedure of adjunction of minimal rational curves. We will explain the approach in the simple case of the hyperquadric, where one shows that X is a hyperquadric by means of the CartanFubini Principle on analytic continuation developed by HwangMok.   
dc.language  eng   
dc.relation.ispartof  Conference Journées Complexes Lorraines   
dc.title  Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents   
dc.type  Conference_Paper   
dc.identifier.email  Mok, N: nmok@hku.hk   
dc.identifier.authority  Mok, N=rp00763   
dc.identifier.hkuros  185026   
dc.publisher.place  Nancy, France   