Homogeneous hypersurfaces

Differential Geometry Seminar, ANU p. 1/14 Homogeneous hypersurfaces Michael Eastwood [based on joint work with Vladimir Ezhov] Australian National University

Differential Geometry Seminar, ANU p. 2/14 A classification problem Σ G/H where G/H = your favourite homogeneous space Examples of G/H R n =(SO(n) R n )/SO(n)= Euclidean space R n =(GL(n,R) R n )/GL(n,R)= affine space RP n = SL(n+1,R)/P= real projective space S n = SO(n+1,1)/P= conformal space C n =(GL(n,C) C n )/GL(n,C)= complex affine space Today: real or complex (equi-)affine space

Differential Geometry Seminar, ANU p. 3/14 Curves in the real affine plane straight line circle (or ellipse) parabola Y= X α r=e αθ (spira mirabilis (Jacob Bernoulli)) Y= e X Y= X logx Normal forms y= 0 y= x 2 y= x 2 ± x 4 + bx 5 + Y= X logx y= x 2 + x 4 + 8 5 x5 + Y= e X y= x 2 x 4 + 2 2 5 x5 + b 2 = invariant

Differential Geometry Seminar, ANU p. 4/14 Non-degenerate equi-affine surfaces in C 3 Z= XY Z 2 = XY+ 1 XY Z= 1 X 2 (Z+ Y 2 ) 3 = 1 Z= XY+ X 3 (Cayley surface) Z= XY+ logx Nomizu & Sasaki 1991 Guggenheimer 1963 Equi-affine symmetries of the Cayley surface X X Y Y+ t Z Z+ tx X X+ s Y Y 3sX 3 2 s2 Z Z+ sy 3 2 s2 X 1 2 s3 (Abelian)

Infinitesimal symmetries Cayley surface z= xy+ x 3 x x y y+ t z z+ tx x z + y 0 0 0 0 0 0 1 0 0 + 0 1 0 x x+s y y 3sx 3 2 s2 z z+ sy 3 2 s2 x 1 2 s3 3x y + y z + x 0 0 0 3 0 0 0 1 0 + 1 0 0 z= xy+ =F(x,y) [ F x, F y, 1] M x y F(x,y) + β = 0 symmetry algebra g Differential Geometry Seminar, ANU p. 5/14

Differential Geometry Seminar, ANU p. 6/14 Normal forms Non-degenerate z= x 2 + y 2 + or z= xy+ Blaschke normal (Leichtweiß1989) z= g ij x i x j + a ijk x i x j x k + x i x i + r i z a j jk a j jk+ 2(n+2) 3 r k a ijk trace-free Pick invariant a ijk a ijk e.g. z= xy+ ax 3 + by 3 + ab z= xy+ x 3 + y 3 + or z= xy+ x 3 +

Differential Geometry Seminar, ANU p. 7/14 Affine homogeneous surfaces in C 3 N1 z= xy+ x 3 + y 3 + ax 4 2bx 3 y 2axy 3 + by 4 + O(5) N2 z= xy+ x 3 + y 3 +bx 4 (4b+9)x 3 y 1 9 (2b+9)(8b+9)x2 y 2 (4b+9)xy 3 +by 4 +O(5) N3 z= xy+ x 3 + y 3 +bx 4 +(4b 9)x 3 y+(6b 9)x 2 y 2 +(4b 9)xy 3 + by 4 + O(5) N4 N5 z= xy+ x 3 + y 3 + 9 4 bx4 + 9 2 x2 y 2 + 9 4 (1/b)y4 + O(5) where b/= 0 z= xy+ x 3 + x 4 + bx 3 y+ O(5) where b/= 0 N6 z= xy+ x 3 + x 4 + bx 5 + O(6) N7 z= xy+ x 3 + x 3 y+ O(5) (the Archimedean screw) N8 z= xy+ x 3 + O(5)=xy+ x 3 (the Cayley surface) N9 z= xy+ x 2 y 2 + O(5)=(1 1 4xy)/2 (hyperboloid) N10 z= xy+ O(5)=xy D1 z= x 2 + x 2 y+ x 3 y+ x 2 y 2 + x 5 + 3x 3 y 2 + x 2 y 3 + O(6) ( twisted cubic) D2 z= x 2 + x 4 + ax 5 + O(6)=afunction of x alone (cylinder) D3 z= x 2 +x 2 y+x 2 y 2 +x 5 +x 2 y 3 +4x 5 y+x 2 y 4 +ax 7 +10x 5 y 2 +x 2 y 5 +O(8) (cone) D4 z= x 2 + x 2 y+ x 2 y 2 + x 2 y 3 + O(6)=x 2 /(1 y) (cone) D5 z= x 2 + O(5)=x 2 (cylinder) F z= O(3)=0

Non-degenerate with non-vanishing Pick invariant z = xy + x 3 + y 3 + c 4,0 x 4 + c 3,1 x 3 y + c 2,2 x 2 y 2 + c 1,3 xy 3 + c 0,4 y 4 +O(5) c 4,0 c 3,1 = c 1,3 c 0,4 and c 2,2 =(2c 4,0 + c 1,3 )(4c 0,4 c 3,1 )/9 c 4,0 0 c 4,0 =0andc 0,4 =0 c 4,0 =0andc 1,3 =0 c 3,1 3 c 1,3 3 =0 c 3,1 = c 1,3 WLG 2c 0,4 + c 3,1 =0 (c 1,3 +9) 2 c 1,3 =0 c 1,3 = 9 c 1,3 =0 (2c 4,0 + c 1,3 )(c 4,0 3 c 0,4 3 )c 1,3 =0 c 1,3 = 2c 4,0 c 0,4 = c 4,0 WLG c 1,3 =0 16c 4,0 c 0,4 =81 N4 (2c 4,0 + c 1,3 )(4c 4,0 +9+c 1,3 )(4c 4,0 9 c 1,3 )=0 c 1,3 = 2c 4,0 c 1,3 = 4c 4,0 9 c 1,3 =4c 4,0 9 N1 N2 N3 47

Non-degenerate with vanishing Pick invariant z = xy + x 3 +O(4) z = xy +O(4) z = xy + x 3 + c 4,0 x 4 + c 3,1 x 3 y + c 2,2 x 2 y 2 + c 1,3 xy 3 + c 0,4 y 4 +O(5) c 0,4 = c 1,3 =0 q 3,3 c 2,2 =0 q 3,3 =0 c 2,2 =0 c 3,1 (3p 1,1 +4c 4,0 )c 2,2 =0 z = xy + c 4,0 x 4 + c 3,1 x 3 y + c 2,2 x 2 y 2 + c 1,3 xy 3 + c 0,4 y 4 +O(5) c 0,4 = c 1,3 = c 3,1 = c 4,0 =0 z = xy + x 2 y 2 +O(5) WLG p 3,3 = q 3,3 =0 no c 2,2 =0? yes c 3,1 =0 p 1,1 = 4c 4,0 /3 c 2,2 =0 c 2,2 =0 N9 no c z = xy + x 3 + x 4 + c 3,1 x 3 4,0 =0? y +O(5)WLG yes N10 no c 3,1 =0? no c 3,1 =0? yes yes q 3,3 = c 3,1 z = xy + x 3 + x 3 y +O(5)WLG p 1,1 = 2 p 1,1 = q 1,1 =0 z = xy + x 3 +O(5) N5 N7 N8 z = xy + x 3 + x 4 + c 5,0 x 5 + c 4,1 x 4 y + c 3,2 x 3 y 2 + c 2,3 x 2 y 3 + c 1,4 xy 4 + c 0,5 y 5 +O(6) N6 c 0,5 = c 1,4 = c 2,3 = c 3,2 = c 4,1 =0 48

Degenerate z = x 2 + c 2,1 x 2 y + c 1,2 xy 2 + c 0,3 y 3 +O(4) c 0,3 = c 1,2 =0 c 2,1 =1WLG no c Add Quartic Terms 2,1 =0? c 4,0 =0WLG Add Quartic Terms yes c 0,4 = c 1,3 =0 c 2,2 =1 c 0,4 = c 1,3 = c 2,2 = c 3,1 =0 c 3,1 =0? yes no c 3,1 =1WLG Add Quintic Terms Add Quintic Terms c 4,0 =1WLG Add Quintic Terms no c 4,0 =0? yes c 0,5 = c 1,4 = c 2,3 = c 3,2 = c 4,1 =0 c 0,6 = c 1,5 = c 3,3 = c 4,2 = c 6,0 =0 c 2,4 =1 c 5,1 =4 Add Quintic Terms c 0,5 = c 1,4 = c 2,3 = c 3,2 = c 4,1 = c 5,0 =0 D2 c 0,5 = c 1,4 = c 3,2 = c 4,1 =0 c 2,3 =1 D5 c 0,5 = c 1,4 =0 c 2,3 =1 c 3,2 =3 c 5,0 =1WLG no c 5,0 =0? yes Add Sextic Terms c 4,1 =0WLG D4 q 1,3 =3/2 c 5,0 =1 p 1,3 = 5 c 6,0 =0WLG D1 D3 Add Septic Terms c 0,7 = c 1,6 = c 3,4 = c 4,3 = c 6,1 =0 c 2,5 =1 c 5,2 =10 49

Differential Geometry Seminar, ANU p. 11/14 Non-degenerate equi-affine hypersurfaces in C 4 Equation Basepoint I1 W= XY+ Z 2 (0,0,0,0) I2 W 2 = XY+ Z 2 + 1 (1,0,0,0) I3 W= XY+ Z 2 + X 3 (0,0,0,0) I4 W= XY+ Z 2 + logx (0,1,0,0) I5 W(XY+ Z) 2 = 1 (1,0,0,1) I6 W 2 (XY+ Z 2 ) 3 = 1 (1,0,0,1) A1 WXY Z= 1 (1,1,1,1) A2 WZ+ WY 2 + X 2 Z+ X 2 Y 2 = 1 (1,0,0,1) A3 6WXY Z 4X 3 Z 3X 2 Y 2 + W 2 Z 2 + 4WY 3 = 1 (1,0,0,1) A4 W= XY+ Z 2 + X 2 Z+ αx 4 (0,0,0,0)

Differential Geometry Seminar, ANU p. 12/14 Non-degenerate equi-affine hypersurfaces in C 4 Equation Basepoint B1 WZ 2 = Z+ X 3 + XY Z (1,0,0,1) B2 (W+ XY+ X 3 ) 2 Z= 1 (1,0,0,1) B3 (WZ+ Y 2 + X 2 Z) 4 = Z (1,0,0,1) B4 (W 2 + X 2 ) 2 (Z+ Y 2 ) 3 = 1 (1,0,0,1) B5 (W+ X 2 Z+ XY) 2 Z= 1 (1,0,0,1) B6 (W+ XY) 2 (Z+ X 2 )=1 (1,0,0,1) B7 WZ 2 = Z+ X 2 + XY Z (1,0,0,1) B8 (W+ Y Z+ X 2 Z) 5 = Z (1,0,0,1) B9 (WZ+ X 2 + Y Z 2 ) 5 = Z 4 (1,0,0,1) B10 W= XY+ Z 2 + XZ 2 (0,0,0,0) B11 W 2 = XY+ X 2 Y+ X 2 Z (1,1,0,1)

Invariants Pick invariant= a ijk a ijk a ijk a ijl a mnl a mnk Differential Geometry Seminar, ANU p. 13/14

Differential Geometry Seminar, ANU p. 14/14 THE END THANK YOU