.. Techniques of computations of Dolbeault cohomology of solvmanifolds Hisashi Kasuya Graduate School of Mathematical Sciences, The University of Tokyo. Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 1 / 21
Dolbeault cohomology (M, J): a complex Manifold. (±i-eigen decomposition) T C M = T 1,0 T 0,1 A p,q (M) = Γ(T 1,0 ) Γ(T 0,1 ) d = }{{} +(1,0) + }{{} +(0,1) H p,q (M) = Ker A p,q (M)/ (A p,q 1 (M)) Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 2 / 21
. Problem If we have techniques of computations of de Rham cohomology, can we get techniques of computations of. Dolbeault cohomology? Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 3 / 21
Nilmanifolds G: a simply connected nilpotent Lie group (g: Lie algebra ) Γ a lattice(cocompact discrete subgroup of G) Consider nilmanifold G/Γ invariant differental forms g A (G/Γ) on G/Γ.. Theorem (Nomizu) this inclusion induces isomorphism. H (g) = H (G/Γ) Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 4 / 21
Invariant complex structure J invariant complex structure on G J End(g) such that J 2 = id, [X, Y ] [JX, JY ] + [JX, Y ] + [X, JY ] = 0. g C = g 1,0 g 0,1 p,q g = p g 1,0 q g 0,1 p,q ( g, ) (A p,q (G/Γ), ) Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 5 / 21
Dolbeault cohomology of nilamnifolds. Fact On a nilmanifold G/Γ with invariant complex structure with some conditions, the inclusion ( p,q g, ) (A p,q (G/Γ), ) induces isomorphism. H p,q (g) = H p,q (G/Γ) Ex. (G, J) is a complex Lie group(sakane) General case is open problem Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 6 / 21
Solvable case Solvbale case is not known. In this talk we consider solvmanifold G/Γ with invariant complex structure. Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 7 / 21
Main object G = C n ϕ N such that: (1) N is simply connected nilpotent Lie group with a left-invariant complex structure. (2) For any t C n, ϕ(t) is a holomorphic automorphism of (N, J). (3) ϕ induces a semi-simple action on the Lie algebra of N. Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 8 / 21
Lie algebra of G a, n Lie algebras of C n, N g = a n By (2) and (3), we have a basis Y 1,..., Y m of n 1,0 such that ϕ(g)y i = α i (g)y i g 1,0 = X 1,..., X n, α 1 Y 1,..., α m Y m p,q g = p x1,..., x n, α 1 1 y 1,..., α 1 m y m q x1,..., x n, ᾱ1 1 ȳ1,..., ᾱm 1 ȳ m Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 9 / 21
Solvmanifold G/Γ Assume G has a lattice Γ. Then Γ = Γ Γ, Γ and Γ :lattices of C n and N holomorphic fiber bundle. N/Γ G/Γ C n /Γ Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 10 / 21
Line bundle α : character of C n L α = G C α /Γ A (G/Γ, L α ) = A (G/Γ) α 1 v α Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 11 / 21
Direct product of Dolbeault complexes Consider A, (G/Γ, L β ) L β L L = {L β } is the set of isomorphism classes of line bundles given by characters of C n. L = Hom(Γ, U(1)). Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 12 / 21
. Lemma For a character α of C, we have a unique unitary character. β such that αβ 1 is holomorphic. Take β 1,..., β m such that α i βi 1 are holomorphic. α 1,..., α m eigencharacters of the action ϕ : C n Aut(n 1,0 ) Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 13 / 21
Subcomplex A p,q Consider subcomplex A p,q = p x1,..., x n, α 1 1 y 1 (β 1 v β 1 1 ),..., α 1 m y m (β m v β 1 m ) q x1,..., x n, ᾱ 1 1 ȳ1 (γ 1 v γ 1 1 ),..., ᾱ 1 m ȳ m (γ m v γ 1 m ). of L β L A, (G/Γ, L β ) Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 14 / 21
. Lemma.. Theorem (K.) The inclusion (A p,q, ) p,q = ( (a n), ) A p,q L β L A, (G/Γ, L β ). induces a cohomology isomorphism. Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 15 / 21
. Corollary Consider the subcomplex B p,q A p,q (G/Γ) given by B p,q = x I x J α 1 K β Ky K ᾱ 1 L γ Lȳ L. I + K =p, J + L =q, R Γ (β K γ L )=1 Then B p,q is a subcomplex of p,q (a n) and the inclusion B p,q A p,q (G/Γ) induces a cohomology isomorphism. H p,q (Bp,q ) = H p,q (G/Γ). Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 16 / 21
Examples Let G = C ϕ C 2 such that ϕ(x + ( ) e x 0 1y) = 0 e x. Then for some a R the ( ) e x 0 matrix 0 e x is conjugate to an element of SL(2, Z). Hence for any 0 b R we have a lattice Γ = (az + b 1Z) Γ such that Γ is a lattice of C 2. Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 17 / 21
p,q p,q g = dz1, e x dz 2, e x dz 3 dz 1, e x d z 2, e x d z 3. = A p,q p,q dz1, e x dz 2 e 1y v e 1y, e x dz 3 e 1y v e 1y d z 1, e x d z 2 e 1y v e 1y, e x d z 3 e 1y v e 1y. Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 18 / 21
Remark In the case b = 2nπ, H p,q (G/Γ) isomorphic to the Dolbeault cohomolgy of complex 3-torus but G/Γ is not diffeomorphic to a torus. In the case b nπ, we have Hodge decomposition and symmetry H r (G/Γ) = H p,q (G/Γ) p+q=r H p,q (G/Γ) = H q,p (G/Γ) But G/Γ admits no Kähler metric. G/Γ = G/aZ Γ S 1 Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 19 / 21
Proof of theorem (Idea) (1) Considering the left Hermittian metric on L β, using -harmonic forms, the induced map H(A p,q ) H( L β L A, (G/Γ, L β )) is injective. Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 20 / 21
(2) Considering the spectral seqeunce of the holomorphic fibration N/Γ G/Γ C n /Γ, we have p,q E s,t 2 = i 0 H i,i s (C n /Γ, L α H p i,q s+i (N/Γ )) = H i,i s (C n /Γ ) H p i,q s+i (N/Γ ) = H p,q (a n) Hence E 2 degenerates and H(A p,q ) = H( L β L A, (G/Γ, L β )) Hisashi Kasuya (Graduate School of Mathematical Techniques Sciences, of computations The University of of Dolbeault Tokyo.) cohomology of solvmanifolds 21 / 21