Characterizing subadjoint varieties among Legendrian varieties
Hwang, Jun-Muk1
1Center for Complex Geometry, Institute for Basic Science (IBS), Daejeon 34126 (Republic of Korea)
Annales de l'Institut Fourier, Online first, 23 p.

For a symplectic vector space $V$, a projective subvariety $Z \subset \mathbb{P} V$ is a Legendrian variety if its affine cone $\widehat{Z} \subset V$ is Lagrangian. In addition to the classical examples of subadjoint varieties associated to simple Lie algebras, many examples of nonsingular Legendrian varieties have been discovered which have positive-dimensional automorphism groups. We give a characterization of subadjoint varieties among such Legendrian varieties in terms of the isotropy representation. Our proof uses some special features of the projective third fundamental forms of Legendrian varieties and their relation to the lines on the Legendrian varieties.

Pour un espace vectoriel symplectique $V$, une sous-variété projective $Z \subset \mathbb{P} V$ est une variété legendrienne si son cône affine $\widehat{Z} \subset V$ est lagrangien. En plus des exemples classiques de variétés sous-adjointes associées à des algèbres de Lie simples, de nombreux exemples de variétés legendriennes non singulières ont été découverts qui ont des groupes d’automorphisme de dimension positive. Nous donnons une caractérisation des variétés sous-adjointes parmi ces variétés legendriennes en termes de représentation isotropique. Notre preuve utilise certaines caractéristiques particulières des troisièmes formes fondamentales projectives des variétés legendriennes et leur relation avec les lignes des variétés legendriennes.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3760
Classification: 14M15, 53A20, 53D10
Keywords: Legendrian variety, projective fundamental form, subadjoint varieties
Mots-clés : variété legendrienne, forme fondamentale projective, variétés sous-adjointes

Hwang, Jun-Muk  1

1 Center for Complex Geometry, Institute for Basic Science (IBS), Daejeon 34126 (Republic of Korea) 
@unpublished{AIF_0__0_0_A49_0,
     author = {Hwang, Jun-Muk},
     title = {Characterizing subadjoint varieties among {Legendrian} varieties},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3760},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Hwang, Jun-Muk
TI  - Characterizing subadjoint varieties among Legendrian varieties
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3760
LA  - en
ID  - AIF_0__0_0_A49_0
ER  - 
%0 Unpublished Work
%A Hwang, Jun-Muk
%T Characterizing subadjoint varieties among Legendrian varieties
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3760
%G en
%F AIF_0__0_0_A49_0
Hwang, Jun-Muk. Characterizing subadjoint varieties among Legendrian varieties. Annales de l'Institut Fourier, Online first, 23 p.

[1] Bryant, R. Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differ. Geom., Volume 17 (1982), pp. 455-473 | Zbl | DOI | MR

[2] Buczyński, J. Legendrian subvarieties of projective space, Geom. Dedicata, Volume 118 (2006), pp. 87-103 | Zbl | DOI | MR

[3] Buczyński, J. Toric Legendrian subvarieties, Transform. Groups, Volume 12 (2007), pp. 631-646 | DOI | Zbl | MR

[4] Buczyński, J. Hyperplane sections of Legendrian subvarieties, Math. Res. Lett., Volume 15 (2008), pp. 623-629 | DOI | Zbl | MR

[5] Buczyński, J. Some quasihomogeneous Legendrian varieties, Adv. Geom., Volume 9 (2009), pp. 257-277 | DOI | Zbl | MR

[6] Fischer, Gerd; Piontkowski, Jens Ruled varieties, Advanced Lectures in Mathematics, Vieweg & Sohn, 2001 | Zbl | DOI | MR

[7] Fu, B.; Hwang, Jun-Muk Classification of non-degenerate projective varieties with non-zero prolongation and applications to target rigidity, Invent. Math., Volume 189 (2012), pp. 457-513 | DOI | Zbl | MR

[8] Fu, B.; Hwang, Jun-Muk Special birational transformations of type (2,1), J. Algebr. Geom., Volume 27 (2018), pp. 55-89 | DOI | Zbl | MR

[9] Hwang, Jun-Muk Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry. Trieste, 2000 (Demailly, J. P.; Goettsche, L.; Lazarsfeld, R., eds.) (ICTP Lecture Notes), Abdus Salam International Centre for Theoretical Physics, 2001, p. 335-–393 | MR | Zbl

[10] Hwang, Jun-Muk Rigidity properties of holomorphic Legendrian singularities, Épijournal de Géom. Algébr., EPIGA, Volume 3 (2019), 18, 18 pages | Zbl | DOI | MR

[11] Hwang, Jun-Muk Prolongations of infinitesimal automorphisms of cubic hypersurfaces with nonzero Hessian, Acta Math. Vietnam., Volume 45 (2020), pp. 27-47 | DOI | Zbl | MR

[12] Hwang, Jun-Muk Legendrian cone structures and contact prolongations, Geometry, Lie theory and applications — The Abel Symposium 2019 (Hervik, Sigbjø rn; Kruglikov, Boris; Markina, Irina; The, Dennis, eds.) (Abel Symposia), Volume 16, Springer, 2022, pp. 131-145 | Zbl | DOI | MR

[13] Hwang, Jun-Muk; Mok, Ngaiming Prolongation of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kaehler deformation, Invent. Math., Volume 160 (2005), pp. 591-645 | Zbl | DOI | MR

[14] Hwang, Jun-Muk; Yamaguchi, K. Characterization of Hermitian symmetric spaces by fundamental forms, Duke Math. J., Volume 120 (2003), pp. 621-634 | Zbl | MR

[15] Kebekus, S. Lines on complex contact manifolds II, Compos. Math., Volume 141 (2005), pp. 227-252 | Zbl | MR | DOI

[16] Kobayashi, Shoshichi Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, 70, Springer, 1972 | Zbl | DOI | MR

[17] Landsberg, J. M.; Manivel, L. Legendrian varieties, Asian J. Math., Volume 11 (2007), pp. 341-359 | DOI | Zbl | MR

[18] Lanteri, A.; Palleschi, M. Projective manifolds containing many rational curves, Indiana Univ. Math. J., Volume 36 (1987), pp. 857-865 | DOI | Zbl | MR

[19] Li, Q. Deformation of the product of complex Fano manifolds, Comptes Rendus. Mathématique, Volume 356 (2018), pp. 538-541 | Numdam | DOI | Zbl

[20] Mok, Ngaiming Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents, Third International Congress of Chinese Mathematicians. Part 1, 2 (Lau, Ka-Sing; Xin, Zhou-Ping; Yau, Shing-Tung, eds.) (AMS/IP Studies in Advanced Mathematics), Volume 42, American Mathematical Society, 2008, pp. 41-61 | DOI | MR | Zbl

Cited by Sources: