We look for Hirzebruch L-polynomials with the form
L1(p1)=ap1
L2(p1,p2)=b1p21+b2p2
L3(p1,p2,p3)=c1p31+c2p1p2+c3p3
where the coefficients a,b1,b2,c1,c2.c3 must be computed.
The Hirzebruch signature formula says that for any closed smooth oriented 4-manifold M4 we have
τ(M4)=∫M4L1(p1(M4))
The Hirzebruch signature formula says that for any closed smooth oriented 8-manifold M8 we have
τ(M8)=∫M8L2(p1(M8),p2(M8))
The Hirzebruch signature formula says that for any closed smooth oriented 12-manifold M12 we have
τ(M12)=∫M12L3(p1(M12),p2(M12),p3(M12))
The total Chern classes for the considered complex projective spaces are
c(CP2)=1+3f+3f2
c(CP4)=1+5g+10g2+10g3+5g4
c(CP6)=1+7h+21h2+35h3+35h4+21h5+7h6
c(CP2×CP2)=1+3f2+3f1+3f22+9f1f2+3f12+9f1f22+9f12f2+9f12f22
c(CP2×CP2×CP2)=1+(3f3+3f2+3f1)+(3f32+9f3f2+9f3f1+3f22+9f1f2+3f12)+(9f32f2+9f32f1+9f3f22+27f3f1f2+9f3f12+9f1f22+9f12f2)+(9f32f22+27f32f1f2+9f32f12+27f3f1f22+27f3f12f2+9f12f22)+(27f32f1f22+27f32f12f2+27f12f22f3)+27f12f22f32
c(CP2×CP4)=1+(3f+5g)+(3f2+15gf+10g2)+(15gf2+30g2f+10g3)+(30g2f2+30g3f+5g4)+(30g3f2+15g4f)+15g4f2
where the cohomological generators are normalized according to
∫CP2f2=1
∫CP4g4=1
∫CP6h6=1
∫CP2×CP2f21f22=∫CP2f21∫CP2f22=(1)(1)=1
∫CP2×CP2×CP2f21f22f23=∫CP2f21∫CP2f22∫CP2f23=(1)(1)(1)=1
∫CP2×CP4f2g4=∫CP2f2∫CP4g4=(1)(1)=1
Now using the following expressions for the Pontryagin classes in terms of the Chern classes
p1=−2c2+c12
p2=−2c1c3+c22+2c4
p3=−2c2c4+c32−2c6+2c1c5
we obtain that
p1(CP2)=3f2
p1(CP4)=5g2
p1(CP6)=7h2
p1(CP2×CP2)=3f22+3f12
p1(CP2×CP2×CP2)=3f32+3f22+3f12
p1(CP2×CP4)=3f2+5g2
p2(CP4)=10g4
p2(CP6)=21h4
p2(CP2×CP2)=9f12f22+9f24+9f14=9f12f22+9(0)+9(0)=9f12f22
p2(CP2×CP2×CP2)=9f12f22+9f24+9f14+9f32f22+9f32f12+9f34=9f12f22+9(0)+9(0)+9f32f22+9f32f12+9(0)=9f12f22+9f32f22+9f32f12
p2(CP2×CP4)=15g2f2+10g4+9f4=15g2f2+10g4
p3(CP6)=35h6
p3(CP2×CP2×CP2)=27f12f22f32+27f14f22+27f12f24+27f32f14+27f32f24+27f34f12+27f34f22=27f12f22f32
p3(CP2×CP4)=30g4f2+45g2f4=30g4f2+45g2(0)=30g4f2
Using these results we have that
L1(p1(CP2))=ap1(CP2)=a(3f2)=3af2
L2(p1(CP4),p2(CP4))=b1p1(CP4)2+b2p2(CP4)=b1(5g2)2+b2(10g4)=25b1g4+10b2g4=(25b1+10b2)g4
L2(p1(CP2×CP2),p2(CP2×CP2))=b1p1(CP2×CP2)2+b2p2(CP2×CP2)=b1(3f21+3f22)2+b2(9f21f22)=b1(9f41+18f21f22+9f42)+9b2f21f22=18b1f21f22+9b2f21f22=(18b1+9b2)f21f22
L3(p1(CP6),p2(CP6),p3(CP6))=c1p1(CP6)3+c2p1(CP6)p2(CP6)+c3p3(CP6)=c1(7h2)3+c2(7h2)(21h4)+c3(35h6)=343c1h6+147c2h6+35c3h6=(343c1+147c2+35c3)h6
L3(p1(CP2×CP2×CP2),p2(CP2×CP2×CP2),p3(CP2×CP2×CP2))=c1p1(CP2×CP2×CP2)3+c2p1(CP2×CP2×CP2)p2(CP2×CP2×CP2)+c3p3(CP2×CP2×CP2)=c1(3f12+3f22+3f32)3+c2(3f12+3f22+3f32)(9f12f22+9f32f22+9f32f12)+27c3f12f22f32=27f12f32f22(6c1+3c2+c3)
L3(p1(CP2×CP4),p2(CP2×CP4),p3(CP2×CP4))=c1p1(CP2×CP4)3+c2p1(CP2×CP4)p2(CP2×CP4)+c3p3(CP2×CP4)=c1(3f2+5g2)3+c2(3f2+5g2)(15g2f2+10g4)+c3(30g4f2)=15f2g4(15c1+7c2+2c3)
Now, the Hirzebruch signature formula says that for CP2 we have
τ(CP2)=∫CP2L1(p1(CP2))=1
∫CP23af2=1
3a∫CP2f2=1
3a=1
a=13
then we obtain
L1(p1)=13p1
The Hirzebruch signature formula says that for CP4 we have
τ(CP4)=∫CP4L2(p1(CP4),p2(CP4))=1
∫CP4(25b1+10b2)g4=1
(25b1+10b2)∫CP4g4=1
25b1+10b2=1
The Hirzebruch signature formula says that for CP2×CP2 we have
τ(CP2×CP2)=∫CP2×CP2L2(p1(CP2×CP2),p2(CP2×CP2))=1
∫CP2×CP2(18b1+9b2)f21f22=1
(18b1+9b2)∫CP2×CP2f21f22=1
18b1+9b2=1
Solving the equations for b1 and b2 we obtain
b1=−145
b2=745
then we have that
L2(p1,p2)=−145p12+745p2
The Hirzebruch signature formula says that for CP6 we have
τ(CP6)=∫CP6L3(p1(CP6),p2(CP6),p3(CP6))=1
∫CP6(343c1+147c2+35c3)h6=1
(343c1+147c2+35c3)∫CP6h6=1
343c1+147c2+35c3=1
The Hirzebruch signature formula says that for CP2×CP2×CP2 we have
τ(CP2×CP2×CP2)=∫CP2×CP2×CP2L3(p1(CP2×CP2×CP2),p2(CP2×CP2×CP2),p3(CP2×CP2×CP2))=1
∫CP2×CP2×CP227f12f32f22=1
27(6c1+3c2+c3)∫CP2×CP2×CP2f12f32f22=1
27(6c1+3c2+c3)=1
The Hirzebruch signature formula says that for CP2×CP4 we have
τ(CP2×CP4)=∫CP2×CP4L3(p1(CP2×CP4),p2(CP2×CP4),p3(CP2×CP4))=1
∫CP2×CP415f2g4(15c1+7c2+2c3)=1
15(15c1+7c2+2c3)∫CP2×CP4f2g4=1
15(15c1+7c2+2c3)=1
Solving the equations for c1, c2 and c3 we obtain
c1=2945
c2=−13945
c3=62945
then we have that
L3=2945p13−13945p1p2+62945p3