BRAC University Journal, Vol. III, No. 1, 2006, pp. 59-63 STUDY ON CHOHOMOLOGY OF SUPERMANIFOLDS Khondokar M. Ahmed Department of Mathematics University of Dhaka Dhaka -1000, Bangladesh ABSTRACT In this present paper de Rham cohomology of graded manifolds, cohomology of graded differential forms and cohomology of DeWitt supermanifolds are studied. The structure of a G-supermanifold in general is not acyclic. So, the cohomology is defined via the complex of graded differential forms. This situation is investigated together with the graded Dolbeault cohomology of complex G- supermanifolds. A theorem is established that the structure sheaf of any DeWitt G-supermanifold is acyclic. Key words: Manifolds, chomology, supermanifolds. I. INTRODUCTION d 1 d 0 → Ñ → A → Ω A → 2 d The aim of this paper is to unfold a basic Ω A → L (2.1) cohomology theory for supermanifolds. This is exact. This 'graded Poincare′ Lemma' is most cohomology does not embody only trivial easily proved by working in local coordinates and extensions of results valid for differentiable proceeding on the analogy of the usual Poincare′ manifolds. For instance the natural analogue of the Lemma. By defining the de Rham cohomology of de Rham theorem does not holds in general and • (X,A), denoted by H DR (X , A), as the also in the case of complex super-manifolds there is generally no analogue of the Dolbeault theorem. cohomology of the complex of graded vector As a result the sturecture sheaf of a supermanifolds spaces A • (X ) , from (2.1) and using the ordinary does not need to be cohomology trivial [2]. The de Rham theorem, we obtain the following result fact that the cohomology of the complex of global (cf. [8] Theorem 4.7.). graded differential forms on a G-supermanifold ( M,A) depends on the G-supermanifold structure ( Proposition 2.1. There is a canonical isomorphism M,A) so that super de Rham cohomology is a fine k k DR A)' DR for all invariant of the supermanifold structure [3]. H (X , H (X ) k ≥ 0. k Here H DR (X ) denotes the usual de Rham II. de Rham COHOMOLOGY OF GRADED cohomology of X. MANIFOLDS III. COHOMOLOGY OF GRADED Graded manifolds are not very interesting as far as DIFFERENTIAL FORMS their cohomology is concerned. In the real case, the structure sheaf of a graded manifold (X,A) is fine Let (M,A) be a G-supermanifold. The sheaves k and therefore A and all sheaves Ω A of graded Ω k ⊗ B of smooth B -valued differential differential forms are acyclic. This implies that the Ñ L L • forms on M provide a resolution of the constant cohomology of the complex Ω A (X) coincides sheaf B on M, in the sense that the differential with the de Rham cohomology of X. In the L complex analytic case, a similar argument allows complex of sheaves of graded-commutative BL - one to prove a Dolbeault-type theorem. • 0 ∞ algebras Ω ⊗ Ñ B (with Ω ⊗ Ñ B ´C L ) is a The complex of sheaves A • is exact and moreover, L L it is a resolution of the constant sheaf Ñ on X; i.e., resolution of the constant sheaf BL , i.e. the the sequence of sheaves of Ñ-modules sequence Khondokar M. Ahmed Ak A1 k ∞ ω = dz ∧ ∧ dz ω ∈ Ω H (U) be ∞ d 1 d L A1LAk 0 → B → C L → Ω ⊗ Ñ B → L L an ∞ graded differential k-form on U (k>0); let 2 H Ω ⊗ Ñ BL → L (3.1) us set is exact. The cohomology associated with this ˆ k Ak −1 Kω(z) = (−1) dz ∧L∧ complex via the global section functor Γ(⋅, M ) , t • A1 B k −1 i.e. the cohomology of the complex Ω ⊗ Ñ BL , is dz z t ω (tz)dt. ∫ BA1LAk −1 • 0 denoted by H DR (M , B ) and is called the B - L L k We have an isomorphism [2] Ω G (U ) ' valued de Rham cohomology of M. Since BL is a k ∞ H ; it is therefore possible to introduce a finite dimensional real vector space, the universal Ω Ñ BL coefficient theorem [6] entails the isomorphism homotopy operator • • k k −1 H DR (M , BL ) ' H DR (M ) ⊗ Ñ BL . (3.2) K : Ω G (U ) → Ω G(U), defined by By virtue of the de Rham theorem, equation (3.2) K( a) Kˆ a. can be equivalently written as ω ⊗ = ω ⊗ • • One can indeed verify easily that H DR (M , B ) ' H (M , B ) . (3.3) L L dKλ + Kdλ = λ for any section λ ∈ Gk (U ) , By H • (M ,⋅) we designate interchangeable the so that, if dλ = 0 then λ = d(Kλ) . The case 1ech or sheaf cohomology functor, which coincide since the base space is paracompact. k = 0 has been left out. However, if f ∈ G (U ) , by writing f as f = f ⊗ a with f ∈ In order to gain information not on the topological ∑ i i i or smooth structure of M, but rather on its G- ∞ H (U ) and ai ∈ BL , the condition df = 0 implies supermanifold structure, we therefore need to define a new cohomology, obtained via a resolution directly that f is a constant in BL . of BL , different from the differential complex k Definition 3.2. Given a G-supermanifold (M,A), (3.1). We consider the sheaves Ω A of graded the cohomology of the complex differential forms. The following result is a d 1 d A(M) → Ω A generalization of the usual Poincare′ lemma (cf. (M ) → 2 [5]). Ω A (M ) → L (3.5) • Proposition 3.1. Given a G supermanifold (M,A), denoted by H SDR (M , A), is called the super de the differential complex of sheaves of graded B - Rham cohomology of (M,A). L The operation of taking the SDR cohomology of a algebras on M G-supermanifold is functorial. Indeed, given a G- d 1 d 0 → BL → A → Ω A → morphism ( f ,φ) : (M , A) → (N, B), it is easily 2 • Ω A → L (3.4) proved that the morphism Ω B (N) → is a resolution of B . • L f∗Ω A(M) induced byφ commutes with the Proof. Since the claim to be proved is a local exterior differential and therefore yields a m,n ¤ matter, we may assume that (M,A) = (BL ,G); morphism of graded BL -modules φ : moreover, it is enough to • • H SDR (N,B ) → H SDR (M ,A ) . It should be show that, if U is an open ball around the origin in • m,n noticed that the functor H SDR (⋅) does not fulfill BL then any closed graded differential k-form k the Eilenberg-Steenrod [10] axiomatics for λ ∈ Ω G(U) is exact; i.e., there exists a graded cohomology (if it did, it would coincide with the k−1 differential (k -1)-form η ∈ Ω G (U) such that BL -valued de Rham cohomology functor) since it λ = dη . Given coordinates (z1 , , z m+n ) in U, does not satisfy the excision axiom. Moreover, the L • let functor H SDR (⋅) does not give rise to 60 Study on Chohomology of Supermanifolds topological invariants. On the other hand, it is Proposition 3.3. provides a usefull tool for investigating the cohomological properties of the easily varified that the graded BL -modules k structure sheaf of a G- supermanifold. For instance, H SDR (M , A)are invariants associated with the it suffices to exhibit a G-supermanifold (M,A) such G-supermanifold structure of M. Indeed, if 1 1 that H SDR (M , A) ≠ H DR (M , BL ) to deduce ( f ,φ) : (M , A) → (N, B) is a G-isomorphism, it that, in general, the sheaf A can not be expected to ¤ • is easily proved that φ : H SDR (N, B) be acyclic. • → H SDR (M , A). is an isomorphism. IV. COHOMOLOGY OF DEWITT SUPERMANIFOLDS The most natural thing to do to gain insight into the geometric significance of the groups Considering in M the fine topology we study the k cohomology of a De Witt supermanifold (M,A); H SDR (M , A) - which, as a matter of fact, are this is advantageous because in this way M is para- graded BL -modules - is to compare them with the compact. Thus we continue to confuse the sheaf cohomology groups H k (M , B ) , which have a and 1ech cohomologies with coefficients in L sheaves on M. natural structure of graded BL -modules as well. k k We need the following Lemma, which is obtained The morphisms Ω A (M ) → Ω from a result given in [19] by strengthening certain induced by the morphism (M ) ⊗ Ñ BL hypotheses. ∞ δ : A → C L give rise to a morphism of Lemma 4.1. Let X and Y be topological spaces, differential complexes, which induces in with Y locally euclidean and F a sheaf of abelian cohomology a morphism of graded BL -modules groups on X; let we assume that all groups k k k · : H SDR (M , A) → H (X , F) are finitely generated. Then for all k n ≥ 0 there is an exact sequence of abelian H DR (M , BL ) ∀k ≥ 0. (3.6) groups In degree zero, · 0 is an isomorphism, in that one has manifestly j k 0 C 0 0 → ⊕ H (X , F) ⊗ Ù H (Y, Ù) → H SDR (M , A)' (BL ) ' H DR (M , BL ) , j+k=n where C is the number of connected components of H n (X ×Y, π −1 F) → M, which we assume to be finite. In degree higher → ⊕ Tor[H j (X , F), H k (X , Ù)] → 0 than zero, we have, as a straightforward application j+k =n+1 of the abstract de Rham theorem, the following result.