MATHEMATICS

G. N. TYURINA

ON THE COHOMOLOGY OF COMPLEX HOMOGENEOUS MANIFOLDS

(Presented by Academician P. S. Aleksandrov, 30 XII 1959)

Let \(X\) be a homogeneous simply connected compact complex manifold. Denote by \(H^q(X,\Omega^p)\), or \(H^{p,q}(X)\), the cohomology group of the space \(X\) with coefficients in the sheaf \(\Omega^p\) of germs of holomorphic \(p\)-forms on \(X\), and set \(H''(X)=\sum_{p,q} H^{p,q}(X)\). In the case when \(X\) is a Kähler manifold, we have \((^1)\)*

\[ H^q(X,\Omega^p)=0 \quad \text{for } p\ne q; \tag{1} \]

\[ H^p(X,\Omega^p)=H^{2p}(X,C). \]

For the general case, Bott’s formula \((^2)\) is known, reducing the group \(H^{p,q}(X)\) to the cohomology groups of certain Lie algebras.

To each non-Kähler homogeneous manifold one can associate in a unique way a certain Kähler manifold \(Y\). In this case there exists a fibration \(X\to Y\), whose fiber is the torus \(T^{2n}\) of dimension \(2n\). In the present paper it is proved that \(H''(X)\) is isomorphic, as a bigraded space, to the cohomology group of the algebra \(H^*(Y,C)\otimes H^*(T^{2n},C)\), endowed with a certain differential and a certain bigrading. On a manifold \(X\) of the indicated type a semisimple complex group \(G\) acts transitively, and its maximal compact subgroup \(M\), so that \(X=G/U=M/V;\ V=M\cap U\). Denote by \(\mathfrak g,\mathfrak u,\mathfrak v\) the Lie algebras of the groups \(G,U,V\), and by \(\mathfrak v^c\) the complex envelope of the algebra \(\mathfrak v\). Let \(S'\) be some subset of the system \(S\) of simple roots of the algebra \(\mathfrak g\), and let \([S']\) be the set of positive roots generated by it. Denote by \(\mathfrak v(S')\) the subalgebra in \(\mathfrak g\) generated by all such root vectors \(e_\alpha\) that \(\alpha\) or \(-\alpha\in [S']\), and by \(\mathfrak h(S')\) the orthogonal complement to the subspace spanned by \([S']\) in the Cartan subalgebra, and by \(\mathfrak n(S')\) the subalgebra spanned by those \(e_\alpha\) for which \(\alpha>0\) and \(\alpha\notin [S']\). Wang showed \((^3)\) that for every manifold \(X\) of the type under consideration there exist an \(S'\subseteq S\), a decomposition \(\mathfrak h(S')=\mathfrak h_v+\mathfrak h_c\), where \(\mathfrak h_v\) and \(\mathfrak h_c\) are rational subspaces, and a complex subspace \(\mathfrak w\subset \mathfrak h_c\), \(\mathfrak w+\overline{\mathfrak w}=\mathfrak h_c\), such that

\[ \mathfrak u=\mathfrak v(S')+\mathfrak h_v+\mathfrak w+\mathfrak n(S'), \]

\[ \mathfrak v^c=\mathfrak v(S')+\mathfrak h_v. \]

Put

\[ [\mathfrak u]=\mathfrak u+\overline{\mathfrak w}. \]

* By \(H^k(X,C)\) we denote the cohomology groups of the space \(X\) with coefficients in the field of complex numbers \(C\).

and denote by \([U]\) the connected complex subgroup of the group \(G\) corresponding to this subalgebra. Then it follows from Wang’s results \((^{3})\) that \(Y=G/[U]\) is a simply connected compact Kähler manifold. Denote by \([\mathfrak v]\) the Lie algebra of the group \([V]=[U]\cap M\). We have

\[ [\mathfrak v]^c=\mathfrak v^c+\mathfrak h_c . \]

Let us note that, for different choices of \(\mathfrak w\subset \mathfrak h_c\), different complex structures may be induced on \(X\) by the fibration \((G,X,U)\).

We formulate the main result.

The space \(\overline{\mathfrak w}\) is contained in the center of the algebra \([\mathfrak v]^c\). Therefore the conjugate space \(\overline{\mathfrak w}^{*}\) is identified with a subspace of the space \(H^1([V],C)\). Transgression in the principal fiber space \((M,Y,[V])\) induces a linear map \(\tau:\overline{\mathfrak w}^{*}\to H^2(Y,C)\). Consider the algebra

\[ \theta=\bigwedge \overline{\mathfrak w}^{*}\otimes H(Y,C)\otimes \bigwedge \mathfrak w^{*} \]

with the double grading such that the elements of \(\bigwedge^l\overline{\mathfrak w}^{*}\otimes 1\otimes 1\) have bidegree \((l,0)\), the elements of \(1\otimes H^{2p}(Y,C)\otimes 1\) have bidegree \((p,p)\), and the elements of \(1\otimes 1\otimes \bigwedge^m\mathfrak w^{*}\) have bidegree \((0,m)\), and define in it a differential \(\delta\) for which

\[ \delta \mathfrak w^{*}=\delta H(Y)=0,\qquad \delta=\tau \text{ on } \overline{\mathfrak w}^{*}. \]

Theorem. The group \(H^{\prime\prime}(X)\), as a bigraded space, is isomorphic to \(H(\theta)=\bigwedge \mathfrak w\otimes H\bigl(H(Y,C)\otimes \bigwedge \overline{\mathfrak w}^{*}\bigr)\).

In proving the theorem we start from Borel’s formula \((^{2})\)

\[ H^q(X,\Omega^p)=H^q(\mathfrak u,\mathfrak v^c,\bigwedge^p(\mathfrak g/\mathfrak u)^{*}), \]

where on the right stands the cohomology group of the Lie algebra \(\mathfrak u\) relative to the subalgebra \(\mathfrak v^c\) with coefficients in the representation of the algebra \(\mathfrak u\) in the space \(\bigwedge(\mathfrak g/\mathfrak u)^{*}\), induced by the adjoint representation. Identify the spaces \(\mathfrak g/\mathfrak u\) and \(\mathfrak g/[\mathfrak u]\) with the complements \(\overline{\mathfrak n}_{*}=\mathfrak n(S')+\overline{\mathfrak w}\) and \(\overline{\mathfrak n}(S')\) to \(\mathfrak u\) and \([\mathfrak u]\) in the algebra \(\mathfrak g\).

Then we have

\[ \bigwedge(\overline{\mathfrak n}_{*})^{*} = \bigwedge(\mathfrak n(S'))^{*}\otimes \bigwedge \overline{\mathfrak w}^{*}, \]

and the operators from \([\mathfrak u]\) act on \(\bigwedge\overline{\mathfrak w}^{*}\) trivially, while on \(\bigwedge\overline{\mathfrak n}(S')^{*}\) their action is induced by the adjoint representation of \(\mathfrak u\) in \(\mathfrak g\).

We shall denote by \(C(L,L',P)\) the algebra of relative cochains of the Lie algebra \(L\) with respect to the subalgebra \(L'\), with coefficients in the representation \(P\), and by \(d\) the differential in this algebra.

Let

\[ A=C(\mathfrak u,\mathfrak v^c,\bigwedge(\overline{\mathfrak n}_{*})^{*}) \quad\text{and}\quad B=C(\mathfrak u,\mathfrak v^c,\bigwedge\overline{\mathfrak n}(S')^{*}). \]

We have

\[ A=B\otimes \bigwedge \overline{\mathfrak w}^{*}. \]

Let \(x\in \overline{\mathfrak w}^{*}\subset A\). Compute \(dx\). For this we choose in \(\mathfrak g\) a basis of root vectors and roots. Then, by the definition of the differential in \(A\) \((^{4})\), the element \([dx(e_\alpha)](e_{-\beta})\), where \(e_\alpha\in\mathfrak n(S')\), \(e_{-\beta}\in\overline{\mathfrak n}(S')\), is equal to \(0\) if \(\alpha\ne\beta\), and is equal to \(x(\alpha)\) if \(\alpha=\beta\). It is obvious that \([dx](\bar h)=0\), \(\bar h\in\overline{\mathfrak w}\), and \(dx(h)=0\), \(h\in\mathfrak w\). Compute \(H(B)\). As follows from \((^{2})\), the graded algebra associated with \(H(B)\) is isomorphic to the term \({}'E_\infty\) of a certain spectral sequence in which

\[ {}'E_2^q=\sum H^{p,q}(Y,C)\otimes \bigwedge \mathfrak w^{*}. \]

From dimension considerations and by virtue of (1), \({}'E_2={}'E_\infty\), and the algebra \(E_\infty\) is isomorphic to the algebra \(H(B)\). With the aid of the usual filtration of the algebra \(A\) by degree in the algebra \(B\), we construct the cohomological spectral sequence \((E_r,d_r)\).

In this case
\[ E_2=\Lambda \overline{\mathfrak{w}}^{*}\otimes H(B) =\Lambda \overline{\mathfrak{w}}^{*}\otimes H(Y,C)\otimes \Lambda \mathfrak{w}^{*}, \]
and the element \(d_r x\), where \(x\in\mathfrak{w}^{*}\), belongs to \(H(B)\) and is equal to the cohomology class of the cycle \(dx\). It can be proved that in the given spectral sequence the differentials \(d_r\) \((r>2)\) are trivial. Thus,
\[ H(A)=H(\Lambda \overline{\mathfrak{w}}^{*}\otimes H(Y,C))\otimes \Lambda \mathfrak{w}^{*}, \]
where the differential \(\delta\), with respect to which the homology group is taken, is induced by the differential \(d\) computed above. It remains to compute \(\delta\).

Put
\[ D=C([\mathfrak{u}],[\mathfrak{v}]^{c},\Lambda \overline{\mathfrak{n}}(S'_{\gamma})^{*}). \]
As was shown, \(d\overline{\mathfrak{w}}^{*}\subset D^{1,1}\). Define a mapping
\[ P:D^{p,q}\to C^{p+q}(\mathfrak{g},[\mathfrak{v}]^{c},c) \]
as follows:
\[ Pf(x_1,\ldots,x_{p+q})= \begin{cases} 0, & \text{if } x_p\in\overline{[\mathfrak{u}]},\\ [f(x_1,\ldots,x_p)](\hat{x}_{p+1},\ldots,\hat{x}_{p+q}), & \text{if } x_p\in[\mathfrak{u}], \end{cases} \]
where the \(x_i\) are arranged so that if \(x_i\in\overline{[\mathfrak{u}]}\), then also \(x_j\in\overline{[\mathfrak{u}]}\) for \(j>i\), and \(\hat{x}_k\) is equal to the coset of the element \(x_k\) in the space \(\mathfrak{g}/[\mathfrak{u}]\).

Consider the diagram
\[ \begin{array}{ccc} D^{p,q}(Y) & \xrightarrow{\ p\ } & C^{p+q}(Y)\\ \sigma''\downarrow & & \downarrow\sigma''\\ I^{p,q}(Y) & \xrightarrow{\ i\ } & I^{p+q}(Y) \end{array} \]
where \(I^{p,q}(Y)\) are invariant forms on \(Y\) of bidegree \((p,q)\) (degree \(r\)), \(\sigma\) is the known identification \((^{4})\), and \(i\) is the obvious inclusion. One can check preservation of bidegree under the mapping \(\sigma\); consequently, one can define
\[ \sigma''=i^{-1}\sigma P. \]
It is easy to verify, by directly comparing the corresponding formulas, that
\[ \sigma''d=d''\sigma'', \]
where \(d\) is the differential in the algebra \(A\), and \(d''\) is differentiation with respect to \(\overline{z}\) in \(I(Y)\).*

Since the manifold \(Y\) is Kähler, it is known \((^{5})\) that, upon passing to cohomology algebras, the mapping \(i\) in \(I^{p,q}(Y)\) and in \(I^{p+q}(Y)\) induces an isomorphism of the algebras \(H''(Y)\) and \(H(Y,C)\). Therefore the mapping \(P\) also induces an isomorphism \(P^{*}\) of the algebras \(H''(Y)\) and \(H(Y,C)\), computed by means of the complexes \(D\) and \(C(\mathfrak{g},[\mathfrak{v}],C)\).

As was noted above, \(\overline{\mathfrak{w}}^{*}\) is identified with the subspaces \(j\overline{\mathfrak{w}}^{*}\) and \(j^{*}\mathfrak{w}^{*}\) of the spaces \(C^{1}([\mathfrak{v}]^{c})\) and \(H^{1}([\mathfrak{v}]^{c})\). Let \(x\in\overline{\mathfrak{w}}^{*}\); then \(jx\in C^{1}([\mathfrak{v}]^{c})\subset C(\mathfrak{g})\), and let \(\widetilde{d}\) be the differential in the latter group. Then \((\widetilde{d}x)(e_{\alpha},e_{\beta})\) is equal to zero if \(\alpha+\beta\ne0\), and is equal to \(x(\alpha)\) if \(\alpha+\beta=0\). Comparing this formula with the value computed above for \(dx\) in the algebra \(A\), we see that the diagram
\[ \begin{array}{ccc} \Lambda^{1}\overline{\mathfrak{w}}^{*} & \xrightarrow{\ j\ } & C^{1}([\mathfrak{v}]^{c})\\ d\downarrow & & \downarrow\widetilde{d}\\ D^{1,1} & \xrightarrow{\ p\ } & C^{2}(\mathfrak{g},[\mathfrak{v}]^{c},C) \end{array} \]
is commutative. Upon passing in the lower row to cohomology groups, we obtain the diagram
\[ \begin{array}{ccc} \Lambda^{1}\overline{\mathfrak{w}}^{*} & \xrightarrow{\ j^{*}\ } & H^{1}([\mathfrak{v}],C)\\ d_2\downarrow & & \downarrow\tau\\ H^{1,1}(Y) & \xrightarrow{\ P^{*}\ } & H^{2}(Y,C), \end{array} \]

\[ \text{* This construction is also applicable to the original manifold } X; \]
it follows directly from it that
\[ H(\mathfrak{u},\mathfrak{v}^{c},\Lambda(\mathfrak{g}/\mathfrak{u})^{*}) \]
is isomorphic to the algebra of \(d''\)-cohomologies of the algebra of invariant forms on the manifold \(X\). Therefore Bott’s formula is equivalent to the assertion that \(H''(X)\) can be computed by means of invariant forms on \(X\).

where \(P^*\) is an isomorphism, \(d_2\) is the second differential in the spectral sequence \((E_r, d_r)\) defined above, and \(\tau\) is the transgression in the fiber space \((M, Y, [V])\). We map the algebra \(E_2 \, \Lambda \mathfrak m^* \otimes H''(Y) \otimes \Lambda \mathfrak p^*\) into the algebra \(\theta = \Lambda \mathfrak m^* \otimes H(Y, C) \otimes \Lambda \mathfrak p^*\) by means of \(1 \otimes P^* \otimes 1\). This isomorphism carries \(d_2\) into the differential \(\tau\) of the algebra \(\theta\) and induces an isomorphism of the groups \(H(A)\) and \(H(\theta)\). The theorem is proved.

Corollary. \(H^q(X, \mathfrak D)\)—the cohomology group of the space \(X\) with coefficients in the sheaf of germs of analytic functions—is isomorphic to \(\Lambda^q \mathfrak m^*\).

As an example, consider the space \(SU(3)\), endowed with a left-invariant complex structure. Let
\[ P(X, s, t) = \sum \dim H^{p,q}(X)s^q t^p . \]
Depending on the choice of complex structure on \(SU(3)\), the polynomial \(P(SU(3), s, t)\) will be equal either to \((1 + st)(1 + s^2t^3)(1 + s)\) or to \((1 + st^2)(1 + st + s^2t^2)(1 + s)\). This example was analyzed by Bott \((^2)\), but the values he obtained for \(\dim H^{p,q}(SU(3))\) do not coincide with those computed above.

In conclusion, the author expresses his gratitude to A. L. Onishchik, who supervised this work.

Moscow State University
named after M. V. Lomonosov

Received
25 XII 1959

REFERENCES

\({}^1\) A. Borel, F. Hirzebruch, Am. J. Math., 80, 458 (1958).
\({}^2\) R. Bott, Ann. Math., 66, No. 2 (1957).
\({}^3\) H. C. Wang, Am. J. Math., 76, 1 (1954).
\({}^4\) C. Chevalley, S. Eilenberg, Trans. Am. Math. Soc., 63, No. 1 (1948).
\({}^5\) P. Dolbeault, C. R., 236, 175 (1953).