By using moving frames and directred digraphs, we study invariant (1,2)-symplectic structures on complex flag manifolds. Let $F$ be a flag manifold with height $k-1$. We show that there is a $k$-dimensional family of invariant (1,2)-symplectic metrics of any parabolic structure on $F$. We also prove any invariant almost complex structure $J$ on $F$ with height 4 admits an invariant (1,2)-symplectic metric if and only if $J$ is parabolic or integrable.
References
[1] A BESSE, Einstein Manifolds, Ergeb Math Grenzgeb (3) 10, Springer Verlag, Berlin, Heidelberg, New York, 1987. MR867684 0613.53001[1] A BESSE, Einstein Manifolds, Ergeb Math Grenzgeb (3) 10, Springer Verlag, Berlin, Heidelberg, New York, 1987. MR867684 0613.53001
[2] A BOREL, Kahlerian coset spaces of semi-simple Lie groups, Proc Nat Acad Sci U S A 40 (1954), 1147 1151 MR77878 0058.16002 10.1073/pnas.40.12.1147 0027-8424%2819541215%2940%3A12%3C1147%3AKCSOSL%3E2.0.CO%3B2-G[2] A BOREL, Kahlerian coset spaces of semi-simple Lie groups, Proc Nat Acad Sci U S A 40 (1954), 1147 1151 MR77878 0058.16002 10.1073/pnas.40.12.1147 0027-8424%2819541215%2940%3A12%3C1147%3AKCSOSL%3E2.0.CO%3B2-G
[3] A BOREL AND F HIRZEBRUCH, Characteristic classes and homogeneous spaces I, Amer J Math 80 (1958), 458-538 MR102800 0097.36401 10.2307/2372795[3] A BOREL AND F HIRZEBRUCH, Characteristic classes and homogeneous spaces I, Amer J Math 80 (1958), 458-538 MR102800 0097.36401 10.2307/2372795
[4] M BLACK, Harmonic maps into homogeneous spaces, Pitman Res Notes Math Ser 255, Longman, Harlow, 1991 MR1135793 0743.58002[4] M BLACK, Harmonic maps into homogeneous spaces, Pitman Res Notes Math Ser 255, Longman, Harlow, 1991 MR1135793 0743.58002
[5] F E BURSTALL, A twistor description of harmonic maps of a 2-sphere into a Grassmannian, Math Ann. 27 (1986), 61-74 MR834106 0575.58018 10.1007/BF01458017[5] F E BURSTALL, A twistor description of harmonic maps of a 2-sphere into a Grassmannian, Math Ann. 27 (1986), 61-74 MR834106 0575.58018 10.1007/BF01458017
[6] F E BURSTALL AND J H. RAWNSLEY, Twistor theory for Riemannian symmetric spaces, with application to harmonic maps of Riemann surfaces, Lecture Notes in Math 1424, Springer-Verlag, Berlin 1990 MR1059054 0699.53059[6] F E BURSTALL AND J H. RAWNSLEY, Twistor theory for Riemannian symmetric spaces, with application to harmonic maps of Riemann surfaces, Lecture Notes in Math 1424, Springer-Verlag, Berlin 1990 MR1059054 0699.53059
[7] F E BURSTALL AND S SALAMON, Tournaments, Flags and Harmonic maps, Math Ann 277 (1987), 249 265 MR886422 0597.58005 10.1007/BF01457363[7] F E BURSTALL AND S SALAMON, Tournaments, Flags and Harmonic maps, Math Ann 277 (1987), 249 265 MR886422 0597.58005 10.1007/BF01457363
[8] J EELLS AND S SALAMON, Twistorial constructions of harmonic maps of surfaces into four-manifolds, An Scuola Norm Sup Pisa Cl Sci (4) 12 (1985), 589-640 MR848842 0627.58019[8] J EELLS AND S SALAMON, Twistorial constructions of harmonic maps of surfaces into four-manifolds, An Scuola Norm Sup Pisa Cl Sci (4) 12 (1985), 589-640 MR848842 0627.58019
[9] J EELLS AND J H SAMPSON, Harmonic mappings of Riemannian manifolds, Amer J Math 86(1964), 109-160 MR164306 0122.40102 10.2307/2373037[9] J EELLS AND J H SAMPSON, Harmonic mappings of Riemannian manifolds, Amer J Math 86(1964), 109-160 MR164306 0122.40102 10.2307/2373037
[10] A LICHNEROWICZ, Applications harmoniques et varietes Kahleriennes, Sympos Math III, Bologna, (1970), 341-402 MR262993 0193.50101[10] A LICHNEROWICZ, Applications harmoniques et varietes Kahleriennes, Sympos Math III, Bologna, (1970), 341-402 MR262993 0193.50101
[11] C J C NEGREIROS, Harmonic maps from compact Riemann surfaces into flag manifolds, Thesis, Universit of Chicago (1987) or Indiana Univ. Math. J 37 (1988), 617-636 MR962926 0662.58016 10.1512/iumj.1988.37.37030[11] C J C NEGREIROS, Harmonic maps from compact Riemann surfaces into flag manifolds, Thesis, Universit of Chicago (1987) or Indiana Univ. Math. J 37 (1988), 617-636 MR962926 0662.58016 10.1512/iumj.1988.37.37030
[12] K. B REID AND L W BEINEKE, Tournaments, In: Select topics in graph theory, (Eds L W Beineke an R J Wilson), Academic Press, London-New York, 1978 MR543656[12] K. B REID AND L W BEINEKE, Tournaments, In: Select topics in graph theory, (Eds L W Beineke an R J Wilson), Academic Press, London-New York, 1978 MR543656
[13] J A WOLF AND A GRAY, Homogeneous spaces defined by Lie group automorphisms II, J Differentia Geom 2(1968), 115-159 MR236329 0182.24702 euclid.jdg/1214428252
[13] J A WOLF AND A GRAY, Homogeneous spaces defined by Lie group automorphisms II, J Differentia Geom 2(1968), 115-159 MR236329 0182.24702 euclid.jdg/1214428252