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Appendix Π-Grassmannians at A. Spaces Glimpse Appendix Π-Projective A and Obstructions Sheaf, Supermanifolds 5. Cotangent Non-Projected to Nod Geometry 4. A Π-Projective and Grassmannians Geometry 3. Super Π-Projective and Supergeometry 2. 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These are locally-free sheaves of rank 1|1 endowed with a specific odd symmetry, locally exchanging the even and odd components, called Π-symmetry. The spaces allowing for such sheaves to be defined were first constructed by Manin, see [M1]: Pn these are called Π-projective spaces Π and more in general Π-Grassmannians. The relevance of these geometric objects became apparent along with the generalisation to a supersymmetric context of the theory of elliptic curves and theta functions due to Levin. In particular, it was realised in [Lev1] and [Lev2] that the correct supergeometric generalisation of theta functions, called supertheta functions, should not be sections of a certain invertible sheaves, but instead sections of Π-invertible sheaves and every elliptic supersymmetric curves can be naturally embedded Pn into a certain product of Π-projective spaces Π by means of supertheta functions. Recently, following an observation due to Deligne, Kwok has provided in [Kw] a different description of Pn Π-projective spaces Π by constructing them as suitable quotients by the algebraic supergroup 1|1 ∗ Gm = D , the multiplicative version of the super skew field D, which is a non-commutative associative superalgebra, thus making apparent a connection between Π-projective geometry and the broader universe of non-commutative geometry. Pn In this paper we will provide a new construction of Π-projective spaces Π, showing how they arise naturally as non-projected supermanifolds over Pn, upon choosing the fermionic sheaf of 1 the supermanifold to be the cotangent sheaf ΩPn . More precisely, we will show that for n > 1 Π-projective spaces can be defined by three ordinary objects, a projective space Pn, the sheaf of 1 1-forms ΩPn defined on it and a certain cohomology class, called fundamental obstruction class, 1 n 2 1 n ω ∈ H (P , TPn ⊗ ΩPn ), where TPn is the tangent sheaf of P . In the case n = 1 one does not need any cohomologyV class and the data coming from the projective line P1 and the cotangent 1 P1 sheaf ΩP1 = OP1 (−2) are enough to describe the Π-projective line Π. Moreover we show that Π-projective spaces are all Calabi-Yau supermanifolds, that is, they have trivial Berezinian sheaf, a feature that makes them particularly interesting for physical applications. Finally, we provide some pieces of evidence that the relation with the cotangent sheaf of the underlying manifold is actually a characterising one in Π-geometry. Indeed, not only Π-projective spaces, but also more in general Π-Grassmannians can be constructed as certain non-projected supermanifolds starting from the cotangent sheaf of the underlying reduced Grassmannias. We show by means of an example that in this context the non-projected structure of the superman- ifold becomes in general more complicated and, in addition to the fundamental one, also higher obstruction classes enter the description. The paper is organised as follows: in the first section we briefly review some basic materials about (complex) supermanifolds, we give some elements of Π-projective geometry and, following Pn [M1], we introduce the Π-projective spaces Π as closed sub-supermanifolds of certain super Grass- mannians, whose construction will also be explained in the section. Having introduced the concept of non-projected supermanifold and the obstruction class in cohomology, detecting whether a super- manifold is non-projected or not, we finally construct the Π-projective spaces as the non-projected supermanifolds arising from the cotangent sheaf over Pn and a certain choice of a representative for the cohomology class obstructing the splitting of the supermanifold. We then show that Π- projective spaces have trivial Berezinian sheaf, thus proving they are examples of non-projected Calabi-Yau supermanifolds. In the last section we briefly hint at what happens in the more general context of Π-Grassmannians, by discussing the example of GΠ(2, 4).
Acknowledgments: I am in debt with my thesis supervisor Bert van Geemen for pointing out the details of a construction explained in the Appendix of this paper. I thank Sergio Cacciatori and Riccardo Re for their help, support and suggestions.
2. Super Grassmannians and Π-Projective Geometry By referring to [M1] as general reference for the theory of supermanifolds, we shall just give the most important definitions in order to establish some terminology and notation. Note that we will always be working in the (super)analytic category and so we let our ground field be the complex numbers C. SUPERGEOMETRY OF Π-PROJECTIVE SPACES 3
Definition 2.1 (Complex Supermanifold). A complex supermanifold is a locally ringed space (M , OM ), where M is a topological space and OM is a sheaf of supercommutative rings on M . We call OM the structure sheaf of the supermanifold.
If we let JM be the ideal of nilpotents contained in OM , then the following conditions are satisfied: M . • the pair ( , OMred ), where OMred = OM /JM defines a complex manifold, called the reduced space of the supermanifold (M , OM ). This corresponds to the existence of a morphism of supermanifolds ι : Mred → M for every supermanifold M , such that ι is actually a pair ι .= (ι, ι♯), with ι : M → M the identity on the underlying topological space and ♯ ι : OM → OMred is the quotient map by the ideal of nilpotents; 2 • the quotient JM /JM defines a locally-free sheaf of OMred -modules and it is called the fermionic sheaf. Notice that the fermionic sheaf has rank 0|q, if q is the odd dimension of the supermanifold M ; • M M • the structure sheaf O is locally isomorphic to the exterior algebra M F , seen as a O red Z2-graded algebra. A supermanifold whose structure sheaf is given bVy an exterior algebra is said to be split. If this is not the case, the supermanifold is called non-projected.
In what follows we will simply denote a supermanifold by M and its reduced space with Mred. An important class of example of (split) complex supermanifolds is provided by the projective super- n|m . n • m spaces, P .= (P , (C ⊗C OPn (−1))). As in ordinary complex algebraic geometry, also in algebraic supergeometryV projective superspaces Pn|m are particular cases of super Grassmannians. Indeed, the supersymmetric generalisation of an ordinary Grassmannians is a space parametrising a|b-dimensional linear subspaces of a given n|m-dimensional space V n|m. In what follows, we will consider the case V n|m given by a vector superspace isomorphic to Cn|m and we briefly review the construction of Grassmannian supermanifolds by the patching technique, via their “big cells” description. For concreteness, we will follow closely [CNR], inviting the reader willing to have a more general and detailed treatment of relative super Grassmannians to refer in particular to [M1] and to [M2] for super flag varieties. n|m n|m c0+d0 c1+d1 We take C be such that n|m = c0|c1 +d0|d1. One can then view C as C ⊕(ΠC) , where Π is the parity changing functor that indicates the reversing of the parity, and since Cn|m is freely-generated one can write its elements as row vectors with respect to a certain basis, Cn|m 0 0 1 1 = Span{e1,...,en|e1,...,em}, (2.1) where the upper indices refer to the Z2-parity. Now, we take a collection of indices, call it I = I0∪I1, n such that I0 is a collection of d0 out of the n indices of C and likewise I1 is a collection of d1 indices out of m indices of ΠCm. Then, if I is the set of such collections of indices I one finds that card(I) = card(I × I )= n · m : this will be the number super big cells that cover the super 0 1 d0 d1 Grassmannian. αβ αβ We then associate a set of even and odd (complex) variables {xI | ξI } to each element I ∈ II : these even and odd variables can be arranged to fill in the places of a d0|d1 × n|m = a|b × (c0 + d0)|(c1 + d1) super matrix in a way such that the columns having indices in I ∈ II forms a (d0 + d1) × (d0 + d1) unit matrix, as follows 1 .. xI . 0 ξI 1 Z .= , (2.2) I 1 . .. ξI 0 xI 1 d0·c0+d1·c1|d0·c1+d1·c0 We define the superspace UI → Spec C =∼ {pt} as the analytic superspace {pt}×C =∼ Cd0·c0+d1·c1|d0·c1+d1·c0 αβ αβ , having {xI | ξI } as the complex coordinates over the point. The super- space UI is called a super big cell of the super Grassmannian when it is represented as explained above, via the super matrix ZI . Two superspaces UI and UJ for two different I,J ∈ I can be glued together as follows. Given ZI , the super big cell of UI , one considers the super submatrix BIJ formed by the columns having . indices in J and we let UIJ .= UI ∩UJ be the maximal sub-superspace of UI such that on UIJ we 4 SIMONE NOJA have that BIJ is invertible. Notice that for this to be true it is sufficient that the two determinants of the even parts of the matrix BIJ are different from zero, as the odd parts do not affect invertibility. αβ αβ αβ αβ On the superspace UIJ one has two sets of coordinates, {xI | ξI } and {xJ | ξJ }. One can pass −1 from one system of coordinates to the other by the transformation rule ZJ = BIJ ZI . A concrete example of this procedure is provided in [CNR]. Glueing together all the superspaces UI we obtain the super Grassmannian G(d0|d1; n|m). This is given as the quotient . G(d0|d1; n|m) .= I∈I R, (2.3) S where R are the equivalence relations generated by the change of coordinates described above. As described in [M1] (Chapter 4, § 3), the maps ψUI : UI → G(d0|d1; n|m) are isomorphisms onto open sub-superspace of G(d0|d1; n|m): so that the super big cells provide a local description of the super Grassmannian, in the same way as a usual (complex) supermanifold is locally isomorphic to a superspace of the kind Cn|m. As mentioned above, as in the ordinary complex algebraic geometric context, (complex) pro- jective superspaces are the most immediate examples of super Grassmannians. Indeed, as one has Pn .= G(1; n + 1) in ordinary algebraic geometry, similarly one finds Pn|m .= G(1|0; n +1|m) in algebraic supergeometry. We will now provide the reader with some basic notions of Π-projective geometry. As far as the author is concerned the most straightforward and immediate way to introduce Π-projective spaces and their geometry is via super Grassmannians. This approach has also the merit to make clear that Π-projective spaces are in general embedded in super Grassmannians, putting forward super Grassmannians as universal embedding spaces in a supergeometric context. Our starting point is the definition of Π-symmetry. We will give a general definition on sheaves.
Definition 2.2 (Π-Symmetry). Let G be a locally-free sheaf of OM -modules of rank n|n on a supermanifold M , a Π-symmetry is an isomorphism such that pΠ : G −→ ΠG and such that 2 pΠ = id. We now work locally and use simply the supercommutative free module Cn|n = Cn ⊕ ΠCn, instead of a generic sheaf: we can therefore choose a certain basis of even elements such that n n|n C = Span{e1,...,en} and we generate a basis for the whole C as follows n|n C = Span{e1,...,en | pΠe1,...,pΠen}. (2.4) n n Clearly, the action of pΠ exchanges the generators of C and ΠC . We observe that somehow the presence of a Π-symmetry should remind us of a “physical super- symmetry”, as it transform even elements in odd elements and viceversa. Also, as supersymmetry requires a Hilbert space allowing for the the same amount of bosonic and fermionic states, similarly Π-symmetry imposes an equal number of even and odd dimensions for a certain “ambient space”, as it might be the supercommutative free module Cn|n above. Along this line, one can give the following Definition 2.3 (Π-Symmetric Submodule). Let M be a supercommutative free A-module such that M = An ⊕ ΠAn. Then we say that a super submodule S ⊂ M is Π-symmetric if it is stable under the action of pΠ. This has as a consequence the following obvious lemma: Lemma 2.4. Let M be a supercommutative free A-module such that M = An ⊕ ΠAn together with a basis given by {e1,...,en | pΠe1,...,pΠen}. Then a super submodule of M is Π-symmetric if and n i i n i i only if for every element v = i=1 x ei + ξ pΠei it also contains vΠ = i=1(−ξ ei + x pΠei) P P The proof is clear, as vΠ is nothing but the Π-transformed partner of v. Notice, though, the presence of a minus sign due to parity reasons. Pn Such Π-symmetric submodules allow us to define Π-projective superspaces, we call them Π, and, more in general, Π-symmetric super Grassmannians. The construction follows closely the one of super Grassmannians of the kind G(1|1; n +1|n + 1), but we only take into account Π-symmetric free submodules characterised as by the Lemma 2.4. These, in turn, allow us to write down the n+1 Pn affine super cells covering the supermanifold Π, each of these related to an affine supermanifold of SUPERGEOMETRY OF Π-PROJECTIVE SPACES 5
. n|n the kind U˜i .= (Ui, C[z1i,...,zni,θ1i,...,θni]) =∼ C and where Ui are the usual open sets covering Pn. We try to make these considerations clear by considering the case of the the Π-projective line, P1 we call it Π. P1 Example 2.5 (Π-Projective Line Π). This is the classifying space of the Π-symmetric 1|1 di- 2|2 mensional super subspaces of C , corresponding to the super Grassmannian GΠ(1|1;2|2), where subscript refers to the presence of further Π-symmetry with respect to the ordinary case treated previously. This is covered by two affine superspaces, each isomorphic to C1|1, having coordinates in the super big-cells notation given by
1 x0 0 ξ0 x1 1 ξ1 0 . . ZU0 = ZU1 = . (2.5) 0 −ξ0 1 x0 −ξ1 0 x1 1 It is then not hard to find the transition functions in the intersections of the charts either by means of allowed rows and column operation or by the method explained above. By rows and columns operations, for example, one finds:
1 x0 0 ξ0 R0/x0,R1/x0 1/x0 1 0 ξ0/x0 −→ 0 −ξ0 1 x0 0 −ξ0/x0 1/x0 1 2 1/x0 1 0 ξ0/x0 R0−ξ0/x0R1 1/x0 1 −ξ0/x 0 −→ 0 0 −ξ0/x0 1/x0 1 0 −ξ0/x0 1/x0 1 2 2 1/x0 1 −ξ0/x0 0 R1+ξ0/x0R0 1/x0 1 −ξ0/x0 0 −→ 2 . 0 −ξ0/x0 1/x0 1 ξ0/x0 0 1/x0 1 One can then read the transition functions in the intersection of the affine charts, characterising the structure sheaf OP1 of the Π-projective line: Π
1 ξ1 x1 = , ξ1 = − 2 . (2.6) x0 x0 P1|1 This leads to the conclusion that the Π-projective line Π is the 1|1-dimensional supermanifold 1 ∼ 1 that is completely characterised by the pair (P , OP1 (−2)). We underline that OP1 (−2) = ΩP1 over P1: we will see in what follows that this is not by accident.
Before we go on we remark two facts. First, one can immediately observe that the Π-projective line is substantially different compared with the projective superline P1|1. Indeed P1|1 is (completely) 1 1|1 characterised by the pair (P , OP1 (−1)). Also, without going into details, we note that while P P1 can be structured as super Riemann surface, the Π-projective line Π cannot. Indeed, the fermionic P1 bundle FP1|1 = (OP1 )1 is given by OP1 (−2) and this does not define a theta characteristic on . Π Π Indeed, there is just one such, and it is given by OP1 (−1), therefore the only genus zero super 1|1 1 Riemann surface is given by the ordinary P = (P , OP1 (−1)). This has a certain importance in the mathematical formulation of superstring perturbation theory. Secondly, we recall that when the odd dimension is 1, the pair (Mred, FM ), consisting into the reduced space and the fermionic sheaf completely characterises the supermanifold M , as clearly ∼ ∼ ∼ (OM )0 = OMred and JM = (OM )1 = FM and multiplication in OM = OMred ⊕FM is defined by the 2 action of OMred on FM in a unique way, as FM is an ideal such that FM = 0. This is observed, for example, in [M1] (Chapter 4, § 2, Proposition 8). In case the supermanifold has odd dimension greater than 1 it is no longer true in general that the supermanifold is completely determined by the pair (Mred, FM ) - if this is the case, then the supermanifold is split -. We will discuss these issues in the next section of the paper. Pn M For n > 1, Π-projective spaces Π are not indeed determined just by the pair ( red, FM ). In the following theorem we use the same method as above to write down the generic form of the Pn transition functions of Π: we will see that a certain nilpotent correction appears in the even transition functions.
n . n Theorem 2.6. Let P . P Pn be the -dimensional -projective space and let ˜ C ∼ Π = ( , O Π ) n Π Ui = (Ui, [zji,θji]) = Cn|n Pn for i =0,...,n, j 6= i be the affine supermanifolds covering Π. In the intersections Ui ∩Uj 6 SIMONE NOJA for the transition functions characterising Pn have the following form: 0 ≤ i < j ≤ n +1 O Π
zℓi θjiθℓi θℓi zℓi ℓ 6= i : zℓj = + 2 , θℓj = − 2 θji; (2.7) zji zji zji zji 1 θji ℓ = i : zij = , θij = − 2 . (2.8) zji zji Pn Proof. Π is covered by n + 1 affine charts, whose coordinates are given in the super big cell notation by
z1i ··· 1 ··· zni θ1i ··· 0 ··· θni ZUi = , (2.9) −θ1i ··· 0 ··· −θni z1i ··· 1 ··· zni where the 1’s and 0’s sit at the i-th positions. Considering the super big cell ZUj for j 6= i one can find the transition functions by bringing ZUi in the form of ZUj by means of allowed rows and column operations (and remembering that it is not possible to divide by a nilpotent element) as P1 done above in the case of Π. It is easily checked that this yields the claimed result. Pn Later on in the paper we will see that the same transition functions characterising Π arise naturally upon the choice of the cotangent sheaf as the fermionic sheaf for a supermanifold over Pn.
3. A Nod to Non-Projected Supermanifolds We have anticipated in the previous section that not all of the supermanifolds can be looked at simply as exterior algebras over an ordinary manifold, that is, not all of the supermanifolds are split supermanifolds. There exist non-projected supermanifolds. We now recast these ideas in a more precise form. In particular, we have seen in the introduction that the structure sheaf OM of a supermanifold can be seen as an extension of OMred by JM , that is we have a short exact sequence / / / / 0 JM OM OMred 0. (3.1) A very natural question that arises when looking at this exact sequence is whether is it split or not. In other words, one might wonder whether there exists a morphism of supermanifolds, we call it π : M → Mred, splitting the sequence as follows
♯ π❴ ~ s ❑ / / / / 0 JM OM OMred 0. (3.2) ι♯ M M Supermanifolds that do posses a splitting morphism π : → red such that π ◦ ι = idMred are called projected. If there exists a projection π : M → Mred for M , then its structure sheaf is such that OM =∼
OMred ⊕ JM and it is a sheaf of OMred -modules. This translates as follows at the level of the even transition functions of the (projected) supermanifold: considering a supermanifold of dimension p|q with an atlas given by {Ui,zκi|θℓi}i∈I where the zκi|θℓi are the even and odd local coordinates, for κ = 1,...,p and ℓ = 1,...,q, in an intersection Ui ∩Uj , for the even transition functions one simply gets
zκi(zj ,θj )= zκi(zj ), (3.3) where the zκi are holomorphic as functions of zj . That is, the even transition functions of a projected supermanifold are those of an ordinary complex manifold and indeed they correspond to those of Mred. In particular, obviously, split supermanifolds are projected supermanifolds (while the converse is not in general true) and their even transition functions are of the kind above. In the case there is no such projection π : M → Mred we say that the supermanifold is non- projected and its structure sheaf is not, in general, a sheaf of OMred -modules. Keeping the same notation as above, in the intersection Ui ∩Uj the even transition functions of a non-projected supermanifold then get corrections by some (even) combination of nilpotent odd coordinates and they can be written as ⌊q/2⌋ (2n) zκi(zj ,θj )= zκi(zj )+ ωij (zj ,θj ) · zκi. (3.4) nX=1 SUPERGEOMETRY OF Π-PROJECTIVE SPACES 7
(2n) 1 2n M M In the expression above ωij ∈ Z (T red ⊗ Sym F )(Ui ∩Uj ) is a representative of n-th even (2n) 1 2n obstruction class ω ∈ H (TMred ⊗ Sym FM ). A sufficient condition for a supermanifold to be non-projected is given by the following M 1 M 2 M Theorem 3.1. Let be a complex supermanifold. If 0 6= ωM ∈ H ( , TMred ⊗ Sym FM ) , then M is non-projected. Proof. this is the main result of [Green]. For a more recent account see [DonWit]. The interested reader might find a detailed proof in case the odd dimension of the supermanifold is equal to 2 in [CNR].
. (2) 1 2 Relying on this theorem, we call ωM = ω ∈ H (TMred ⊗ Sym FM ) the fundamental obstruction class of the supermanifold M . At this point, it is important to notice that the transition functions of the Π-projective space Pn Π we have found in Theorem 2.6 are a typical example of transition functions of a non-projected supermanifolds: in particular, it is crucial to note the θθ-correction in the even transition functions,
zℓi θjiθℓi zℓj = + 2 . (3.5) zji zji The bit having form θθ/z2 is indeed the signal of certain cocycle representing the fundamental Pn Pn obstruction class ω Π . The identification of ω Π will play a crucial role as we shall see shortly. Incidentally, before we go on, we stress that also the odd transition functions might not be just linear in the nilpotent odd coordinates - that is, they might fail to be the transition functions of a locally- free sheaf of OMred -modules. Again, this corresponds to the presence of certain non-null cohomology classes, but we will not dwell further into this topic as these odd obstruction classes will not be used 1 2 in the paper: we will only need the fundamental obstruction class ωM ∈ H (TMred ⊗ Sym FM ).
4. Cotangent Sheaf, Obstructions and Π-Projective Spaces For future use we start fixing the notation we are to adopt when working on ordinary projective Pn n spaces . We consider the usual covering by n+1 open sets {Ui}i=0 characterised by the condition . n Ui .= {[X0 : ... : Xn] ∈ P : Xi 6=0}. Defining the affine coordinates to be
. Xj zji .= , (4.1) Xi n we have that P gets covered by n + 1 affine charts as follows (Ui, C[z1i,...,zni]). This allows to easily write down the transition functions for two sheaves of interest, the tangent and the cotangent sheaf.
• Tangent Sheaf TPn : on the intersection Ui ∩Uj one finds:
∂zji = −zij zkj ∂zkj (4.2) Xk6=j
∂zki = zij ∂zkj k 6= j (4.3) 1 • Cotangent Sheaf ΩPn : on the intersection Ui ∩Uj one finds:
dzij dzji = − 2 (4.4) zij zkj dzkj dzki = − 2 dzij + k 6= j (4.5) zij zij We now consider a supermanifold of dimension n|n having reduced space given by Pn and a 1 fermionic sheaf FM given by ΠΩPn . This is a sheaf of OPn -modules of rank 0|n, that is locally- generated on Ui by n odd elements {θ1i,...,θni} that transform on the intersections Ui ∩Uj as the local generators of the cotangent sheaf, {dz1i,...,dzni}: in other words, the correspondence is dzki ↔ θki for k 6= i. 1 The crucial observation is that the fermionic sheaf FM = ΠΩPn determined by the cotangent sheaf 1 Pn ΩPn reproduces exactly the odd transition functions of Π. It is then natural to ask whether the 1 Pn n whole O Π is determined someway by ΠΩP . We will see that this question has an affermative Pn answer, by realising Π as the non-projected supermanifold whose even part of the structure sheaf 8 SIMONE NOJA
Pn Pn M O Π is determined by O and the fundamental obstruction class ω . 1 First we need to prove that the choice FM = ΠΩPn can actually give rise to a non-projected 1 2 1 supermanifold. For this to be true it is enough that H (TPn ⊗ Sym ΠΩPn ) 6= 0: this is achieved in the following
1 2 1 ∼ Lemma 4.1. H (TPn ⊗ Sym ΠΩPn ) = C
2 1 ∼ 2 1 Proof. It is important to realise that due to parity reason, one has Sym ΠΩPn = ΩPn , therefore 1 2 1 it amounts to evaluate H (TPn ⊗ ΩPn ) : this can be done using the Euler exact sequenceV tensored 2 . 2 1 by ΩPn .= ΩPn , this reads V V / 2 / 2 ⊕n+1 / 2 / 0 / ΩPn / ΩPn (+1) / TPn ⊗ ΩPn / 0. (4.6)
2 2 Using Bott formulas (see for example [OkScSp]) to evaluate the cohomology of ΩPn and ΩPn (+1) 1 2 ∼ 2 2 ∼ one is left with the isomorphism H (TPn ⊗ ΩPn ) = H (ΩPn ) = C, again by Bott formulas.
1 2 1 A consequence of the lemma is that each choice of a class 0 6= ωM ∈ H (TPn ⊗ Sym ΠΩPn ) gives n 1 rise to a non-projected supermanifold having reduced space P and fermionic sheaf ΠΩPn . Making use of the Bott formulas, as in proof of previous Lemma, is certainly the briefest and easiest to show the non vanishing of the cohomology group. Anyway, there is another more instructive way to achieve the same result: this, has also the merit to allow the right setting to find the 1 2 1 2 2 ∼ 2 1 representative of H (TPn ⊗ Sym ΠΩPn ). Keeping in mind that Sym ΠΩPn = ΩPn , one starts from the dual of the the Euler exact sequence, that reads V
/ 1 / ⊕n+1 / / 0 / ΩPn / OPn (−1) / OPn / 0. (4.7)
Taking its second exterior power one gets
/ 2 1 / 2 ⊕n+1 / 1 / 0 / ΩPn / OPn (−1) / ΩPn / 0. (4.8) V V n+1 2 ⊕n+1 ⊕( ) Notice that clearly OPn (−1) =∼ OPn (−2) 2 , and, more important, that the existence of this short exact sequenceV depends on the fact that OPn is of rank 1. A more careful discussion of the general framework for second exterior powers of short exact sequences of locally-free sheaf of OMred -modules is deferred to the Appendix. This short exact sequence can be in turn tensored by TPn as to yield
2 n+1 / 1 / ⊕( 2 ) / 1 / 0 / TPn ⊗ ΩPn / TPn (−2) / TPn ⊗ ΩPn / 0. (4.9) V Upon using the Euler exact sequence for the tangent sheaf twisted by OPn (−2), one sees that for n+1 n+1 0 ⊕( ) 1 ⊕( ) n> 1 the cohomology groups H (TPn (−2) 2 ) and H (TPn (−2) 2 ) are zero, and therefore 0 1 ∼ 1 2 1 the long exact cohomology sequence gives the isomorphism H (TPn ⊗ ΩPn ) = H (TPn ⊗ ΩPn ). 0 1 0 The global section generating H (TPn ⊗ΩPn ) is easily identified as the diagonal element inVC (TPn ⊗ 1 ΩPn ). By representing it locally, in the chart Ui, one has
0 1 n ∼ H (TP ⊗ ΩPn ) = ∂zji ⊗ dzji C. (4.10) Xj6=i and it is easily proved that it actually defines a global section. 1 2 1 We aim to lift this element to the generator of H (TPn ⊗ ΩPn ), making the isomorphism explicit: this will be the key step of our construction of Π-projectiveV spaces as non-projected supermanifolds. To achieve this, we need to study carefully the homomorphisms of sheaves entering the exact 2 1 2 sequence 4.8. First we consider the injective map ∧ ι : ΩPn → OPn (−1). This is given by V V 2 2 1 / 2 ⊕n+1 ∧ ι : ΩPn / OPn (−1) (4.11) V V df ∧ dg ✤ / ι(df) ∧ ι(dg) SUPERGEOMETRY OF Π-PROJECTIVE SPACES 9
1 ⊕n+1 where ι :ΩPn → O(−1) is the map that enters the dual of the Euler exact sequence, that is, working for example in the chart Ui, 1 / ⊕n+1 ι :ΩPn / OPn (−1) (4.12)
/ f0i 1 fnj df = fjidzji ✤ / ,..., − 2 Xj fji,..., . j6=i Xi Xi j6=i Xi P P 2 ⊕n+1 1 Getting back to 4.8 and keep working in the chart Ui, the map Φ2 : OPn (−1) → ΩPn is defined as follows V 2 ⊕n+1 / 1 Φ2 : OPn (−1) / ΩPn V (f ,...,f ) ∧ (g , ∧,g ) ✤ / X n X (f ⊗ g − g ⊗ f ) d Xk , 0 n 0 n i j=0 k6=i j j k j k Xi P P where d(Xk/Xi) = dzki. The reader can check that these maps give rise to an exact sequence of locally-free OPn -modules. Clearly, the maps entering the exact sequence 4.9 are just the same tensored by identity on the tangent sheaf. Upon knowing these map, we can prove the following lemma.
1 2 1 Lemma 4.2 (Lifting). The cohomology group H (TPn ⊗ ΩPn ) is represented by {ωij }i 0 1 1 2 1 Pn Pn Proof. We need to lift the element ( j6=i ∂zji ⊗ dzji)i=0,...,n ∈ Z (T ⊗ ΩPn ) to Z (T ⊗ ΩPn ) as in the following diagram: P V 1 2 1 / / 1 2 ⊕n+1 C (TPn ⊗ Ω n ) / / C (TPn ⊗ OPn (−1) ) P O V V δ 0 2 ⊕n+1 / / 0 1 C (TPn ⊗ OPn (−1) ) / / C (TPn ⊗ ΩPn ) where the maps are induced by those definingV the short exact sequence 4.9. The first step is to 0 2 ⊕n+1 Pn Pn find the pre-image of the element j6=i ∂zji ⊗ dzji in C (T ⊗ O (−1) ). We work, for simplicity, in the chart Ui, and weP look for elements f’s and g’s suchV that n ! (j) (j) (j) (j) ∂zji ⊗ dzji = ∂zji ⊗ Xi Xk fk ⊗ gℓ − gk ⊗ fℓ dzℓi . Xj6=i Xj6=i Xℓ6=i kX=0 h i The condition is satisfied by the choice (j) δki (j) δℓj fk = gℓ = , (4.14) Xi Xi so that one finds that the pre-image in the chart Ui reads −1 ei ∧ ek Φ ∂z ⊗ dzji = ∂z ⊗ (4.15) 2 ji ki X2 Xj6=i Xk6=i i 2 ⊕n+1 where we have denoted {ei ∧ ek}i6=k a basis for the second exterior power OPn (−1) . Now we lift this element to a Cechˇ 1-cochain by means of the Cechˇ coboundaryV map δ as to get on an intersection Ui ∩Uj 1 zkj 1 . sij = si − sj ⌊Ui∩Uj = 2 ∂zkj ⊗ ej ∧ ei + ei ∧ ek + ej ∧ ek (4.16) X zij zij j kX6=i,j 2 It is not hard to verity that (sij )i6=j is the image through the injective map ∧ ι of elements 2 dzij ∧ dzkj 1 1 . n ωij = ⊗ ∂zkj ∈ Z (U, TP ⊗ ΩPn ), (4.17) zij Xk6=j ^ 10 SIMONE NOJA 1 Pn which represents the lifting of j6=i ∂zji ⊗ dzji, and generates the cohomology group H (T ⊗ 2 1 ΩPn ). P V Notice that one finds the actual elements that enter the transition functions of the non-projected supermanifold in the identification dzij ∧ dzkj ↔ θij θkj , where the second is to be understood as 2 1 the symmetric product, giving an element in Sym ΠΩPn . The previous lemma gives all the elements we need in order to recognise the Π-projective space Pn Pn Π as the non-projected supermanifold is associated to the cotangent sheaf on . In particular, we have the following n . n Theorem 4.3 . The -projected space P . P Pn is the non-projected (Π-projective spaces) Π Π = ( , O Π ) n 1 supermanifold uniquely identified by the triple (P , ΠΩPn , λ), where λ 6=0 is the representative of 1 2 1 ∼ ωM ∈ H (TPn ⊗ Sym ΠΩPn ) = C. Proof. Pn Pn It is enough to proof that the sheaf O Π can be determined out of the structure sheaf O n 1 1 of P , the fermionic sheaf ΠΩPn and the non-zero representative λ ∈ C \{0} of ωM ∈ H (TPn ⊗ 2 1 ∼ C Pn Sym ΠΩ ) = . We have already observed that the transition functions of (O Π )1 do coincide 1 with those of ΠΩPn . Moreover, up to a change of coordinate or a scaling, λ can be chosen equal to Pn 1. Then, we see that the transition functions of (O Π )0 are determined by 3.4 as a non-projected 2 1 extension of OPn by Sym ΠΩPn , as follows zkj θij θkj zkj θij θkj zki = + ∂zkj zki = + 2 ∂zki zki (4.18) zij zij zij z Xk6=j Xk6=j ij zkj θij θkj = + 2 (4.19) zij zij and clearly, zji = 1/zij. Here, we have used the result of the previous lemma 4.2 to write the (2) representatives of the fundamental obstruction class ωij . This completes the proof. Now that we have constructed Π-projective spaces as non-projected supermanifolds, we inves- tigate a property that all of the Π-projective spaces share, regardless their dimensions: they have trivial Berezinian sheaf. Supermanifolds having this property are said Calabi-Yau supermanifolds, as the Berezianian sheaf is - in some sense - the only meaningful supersymmetric generalisation of the canonical sheaf. Before we go into the proof of the theorem, we recall that given a supermanifold M , the Berezinian 1 sheaf BerM is, by definition the sheaf Ber(ΩM ). That is, given an open covering {Ui}i∈I of the underlying topological space of M , the Berezinian sheaf BerM is the sheaf whose transition functions {gij}i6=j∈I are obtained by taking the Berezinian of the super Jacobian of a change of coordinates in Ui ∩Uj . Pn Pn Theorem 4.4 ( Π are Calabi-Yau supermanifolds). Π-projective spaces Π have trivial Berezia- nian sheaf. That is, Pn ∼ Pn Ber Π = O Π . Proof. Let us consider the generic case n ≥ 2. It is enough to prove the triviality of the Berezianian sheaf in a single intersection. One starts computing the super Jacobian of the transition functions Pn in 2.7 and 2.8: this actually gives the transition functions of the cotangent sheaf to Π in a certain intersection Ui ∩Uj , that can be represented in a super matrix of the form A B [J ac] = (4.20) ij C D for some A and D even and B and C odd sub-matrices depending on the intersection Ui ∩Uj . −1 −1 Then one computes Ber [J ac]ij by means of the formula Ber(X) = det(A) det(D−CA B) that reduces the computation of the Berezinian of a super matrix X to a computation of determinants SUPERGEOMETRY OF Π-PROJECTIVE SPACES 11 of ordinary matrices. It can be easily checked that for every non-empty intersection Ui ∩Uj one gets 1 −1 −1 n+1 det A = − n+1 , det(D − CA B) = −zij , (4.21) zij Pn so that Ber [J ac]ij = 1, proving triviality of Ber Π . The interested reader might find in the Appendix some explicit computation performed in the intersection U0 ∩U1. P1 ∼ Finally, the case n = 1 is trivial, as Π is split: in general for a split supermanifold BerM = ∗ KMred ⊗ det FM where KMred is the canonical sheaf of the reduced manifold, so that in the case of P1 Π one gets 1 ∗ BerP1 ∼ KP1 ⊗ (ΩP1 ) ∼ OP1 (−2) ⊗ OP1 (+2) ∼ OP1 . (4.22) Π = = = thus concluding the proof. Before we move to the next section some remarks are in order. (1) We did already know examples of split Calabi-Yau supermanifolds in every bosonic/even dimension: these are the well-known projective superspaces of the kind Pn|n+1 for every n ≥ 1. The previous theorem characterises Π-projective spaces as relatively simple examples of non-projected Calabi-Yau supermanifold for every bosonic/even dimension. In this context, P1 the Π-projective line Π is actually the only Calabi-Yau supermanifold of dimension 1|1 which has P1 as reduced space. (2) The proof of the previous theorem might appear inelegant and somehow cumbersome as in the case n ≥ 2 it requires explicit knowledge of transition functions of the supermanifold in order to confirm the triviality of the Berezinian sheaf. Still, to the best knowledge of the author, this is actually unavoidable when dealing with non-projected supermanifolds, as the machinery of adjunction theory has not been developed yet and it is not available in this context. In order to answer more sophisticated questions and for classification issues, it would definitely be useful and necessary to fill this gap and provide a suitable adjunction theory for non-projected supermanifolds. 5. A Glimpse at Π-Grassmannians In the previous section we have shown that Π-projective spaces arise as certain non-projected supermanifolds whose fermionic sheaf is related to the cotangent sheaf of their reduced manifold Pn and, as such, they have a very simple structure. Remarkably, something very similar happens more in general to Π-Grassmannians (see [M1]), backing the idea of a close connection between Π-symmetry in supergeometry and the ordinary geometry of cotangent sheaves of the ordinary reduced variety. Indeed we claim “all of the Π-Grassmannians GΠ(n,m) can be constructed as higher-dimensional non-projected supermanifolds whose fermionic sheaf is given precisely by the cotangent sheaf of their reduced manifold, the ordinary Grassmannian G(n,m)”. The difference that makes things trickier compared to the case of the Π-projective spaces, is that also higher obstruction classes - not only the fundamental one - might appear, leading to non- projected and non-split supermanifolds. The construction of Π-Grassmannian as non-projected supermanifolds related to te cotangent sheaf 1 ΩG(n,m) of the underlying Grassmannian G(n,m), the relation between their dimension, structure and the presence of higher obstructions to splitting will be the subject of a forthcoming paper. For the time being, in support of the above claim and as an illustrative example, we analyse the structure of the transition functions in certain big-cells of the Π-Grassmannian GΠ(2, 4): notice that, as in the ordinary context, this is the first Π-Grassmannian that is not a Π-projective space. We start considering the reduced manifold, the ordinary Grassmannian G(2, 4) and look at the change of coordinates between the big-cells 1 0 x11 x21 1 x12 0 x22 ZU1 = ZU2 = . (5.1) 0 1 y11 y21 0 y12 1 y22 12 SIMONE NOJA By row-operations one easily finds that x11 x11y21 x12 = − , x22 = x21 − , (5.2) y11 y11 1 y21 y12 = , y22 = . (5.3) y11 y11 In the correspondence {dxij ↔ θij , dyij ↔ ξij } of the local frames of the cotangent sheaf with 1 those of its parity-reversed version ΠΩG(2,4) we are concerned with, one has the following transition functions θ11 x11 y21 x11 x11y21 θ12 = − + 2 ξ11, θ22 = θ21 − θ11 − ξ21 + 2 ξ11, (5.4) y11 y11 y11 y11 y11 ξ11 ξ21 y21 ξ12 = − 2 , ξ22 = − 2 ξ11. (5.5) y11 y11 y11 Now look at the corresponding change of coordinates in U1 ∩U2 for GΠ(2, 4): the super big-cells then look like 1 0 x11 x21 0 0 θ11 θ21 0 1 y11 y21 0 0 ξ11 ξ21 Z = (5.6) U1 0 0 −θ −θ 1 0 x x 11 21 11 21 0 0 −ξ −ξ 0 1 y y 11 21 11 21 1 x12 0 x22 0 θ12 0 θ22 0 y12 1 y22 0 ξ12 0 ξ22 Z = . (5.7) U2 0 −θ 0 −θ 1 x 0 x 12 22 12 22 0 −ξ 0 −ξ 0 y 1 y 12 22 12 22 Again, by acting with row-operations on U1 one finds the following change of coordinates for GΠ(2, 4) in U1 ∩U2 x11 θ11ξ11 x11y21 θ11ξ21 x11 y21 x12 = − − 2 , x22 = x21 − + − 2 ξ11ξ21 − 2 θ11ξ11, y11 y11 y11 y11 y11 y11 1 y21 ξ11ξ21 y12 = , y22 = + 2 , y11 y11 y11 θ11 x11 y21 x11 x11y21 θ11ξ11ξ21 θ12 = − + 2 ξ11, θ22 = θ21 − θ11 − ξ21 + 2 ξ11 − 2 y11 y11 y11 y11 y11 y11 ξ11 ξ21 y21 ξ12 = − 2 , ξ22 = − 2 ξ11. y11 y11 y11 We observe the following facts: as in the the case of Π-projective spaces, the bosonic transition 2 1 functions get nilpotent “corrections” taking values in Sym ΠΩG(2,4)(U1 ∩U2). More important, here is the difference: the fermionic transition functions are almost the same but 1 not actually the same as those of ΠΩG(2,4) above! Indeed, in the transition functions of θ22 appears 3 1 θ11ξ11ξ21 a term taking values in Sym ΠΩ (U1 ∩U2) - the term − 2 - that tells that GΠ(2, 4) will G(2,4) y11 also be characterised by the presence of an higher fermionic obstructions! The study of the geometry of Π-Grassmannians as non-projected supermanifolds, their character- ising algebraic-geometric invariants, and their special relationship with the cotangent sheaf of their underlying manifolds will be discussed in a dedicated follow-up paper. Appendix A. As the construction is not readily available in literature, we clarify in what follows the structure of the maps entering the second exterior power of a short exact sequence of locally-free sheaf of OX -modules / vector bundles, where X is an ordinary complex manifold. We will work in full generality, even if the for the purpose of the paper it is enough to consider the (easier) special case in which the quotient sheaf is invertible. We start looking at the following exact sequence of locally-free sheaf of OX -modules: 0 / F ι / G π / H / 0. (A.1) SUPERGEOMETRY OF Π-PROJECTIVE SPACES 13 Then, in general, there is an exact sequence 2 φ 0 / 2 F ∧ ι / 2 G / Q / 0. (A.2) where the map φ has yet to be definedV and theV quotient bundle fits into 0 / F ⊗ H / Q / 2 H / 0. (A.3) V Indeed, to get an idea, locally, the first exact sequence splits to give G =∼ F ⊕ H. Taking the second exterior power one gets 2 2 2 G = F ⊕ (F ⊕ H) ⊕ H, (A.4) ^ ^ ^ therefore, keep working locally, taking the quotient by 2 F it gives V 2 2 Q =∼ G 2 F =∼ (F ⊗ H) ⊕ H, (A.5) V ^ that suggests why the second exact sequenceV is true. Notice that if we consider the case rank H = 1, then one has 2 H = 0, and then sequence for Q tells that Q =∼ F ⊗ H, therefore one finds that V 2 φ 0 / 2 F ∧ ι / 2 G / F ⊗ H / 0. (A.6) Getting back to the general setting,V in orderV to define the map φ we consider the alternating map Φ: G ⊗ G / H ⊗ G (A.7) / g1 ⊗ g2 ✤ / π(g1) ⊗ g2 − π(g2) ⊗ g1. Notice that one has the commutative diagram Φ / G ⊗ G 6H ⊗ G (A.8) ♥♥ φ2 ♥♥ q ♥♥♥ ♥♥♥ ♥♥♥ 2 G V and, by the usual universal property, the map Φ factors over 2 G to induce a map / V Φ2 : G ⊗ G / H ⊗ G (A.9) / g1 ∧ g2 ✤ / π(g1) ⊗ g2 − π(g2) ⊗ g1. 2 We note that Φ2 ◦ ∧ ι = 0, indeed: 2 Φ2 ◦ ∧ ι (f1 ∧ f2)=Φ2(f1 ∧ f2)= π(f1) ⊗ f2 − π(f2) ⊗ f1 =0, (A.10) since ι : F → G is an inclusion and since ker π = im ι =∼ F. 2 2 Now, since Φ2 ◦ ∧ ι = 0, one has that Φ2( F) = 0, so that one has that, in turn Φ2 factors through Q, as to yield a well-defined map φ2V: Q→F⊗H, as follows Φ / G ⊗ G 6H> ⊗ G (A.11) ♥♥ ⑥ Φ2 ♥♥ ⑥ q ♥♥♥ ⑥⑥ ♥♥♥ ⑥⑥ ♥♥♥ ⑥⑥ 2 G ⑥⑥ ⑥⑥ φ2 V ⑥⑥ φ ⑥⑥ ⑥⑥ ⑥⑥ Q Now, let us examine the following exact sequence obtained by tensoring with H⊗− the first exact sequence: 1⊗ι 1⊗π 0 / H⊗F / H ⊗ G / H ⊗ H / 0, (A.12) where the maps are the obvious ones. We observe that 14 SIMONE NOJA 2 • im 1 ⊗ ι ⊂ Φ2( G), indeed for g ∈ G such that π(g)= h we have that V (1 ⊗ ι)(h ⊗ f)= h ⊗ ι(f)= π(g) ⊗ f − π(f) ⊗ g =Φ2(g ∧ f) (A.13) as π(f) = 0 and confusing ι(f) with f (remember that ι is an immersion). This implies that H⊗F =∼ im 1 ⊗ ι ⊂ φ2(Q). 2 • im ((1 ⊗ π) ◦ Φ2) ⊂ H, indeed V (1 ⊗ π)(Φ2(g1 ∧ g2)) = (1 ⊗ π) (π(g1) ⊗ g2 − π(g2) ⊗ g1) = π(g1) ⊗ π(g2) − π(g2) ⊗ π(g1)= π(g1) ∧ π(g2). (A.14) 2 Conversely, working on the fibers, one has that H ⊂ im ((1 ⊗ π) ◦ Φ2), so that one 2 concludes that im ((1 ⊗ π) ◦ Φ2)= H that in turnV implies that V 2 im ((1 ⊗ π) ◦ φ2)= H. (A.15) ^ 2 This says that φ2 maps onto H. 2 As we have shown that the map φ2Vis such that F⊗H ⊂ im φ2, and that φ2 is onto H, by counting the dimensions, φ2 is actually also injective, therefore F⊗H⊂ φ2(Q) impliesV that F⊗G⊂Q, which establishes the second exact sequence for Q. Appendix B. In this Appendix we support the proof of Theorem 4.4, by explicitly working out the Berezinian n of the super Jacobian in the intersection U0 ∩U1 of the usual covering of P . Pn Given the transition functions of Π, as in 2.7 and 2.8, in the intersection U0 ∩U1, the super Jacobian matrix reads A B [J ac] = (B.1) 10 C D where one has 1 − 2 0 ······ 0 z10 z20 θ10θ20 1 − 2 − 2 3 0 ··· 0 z10 z10 z10 . .. A = . . (B.2) . .. . . zn0 θ10θn0 1 − 2 − 2 3 0 ··· 0 z z10 z10 10 0 ········· 0 θ20 θ10 2 − 2 0 ··· 0 z10 z10 . . .. B = . . (B.3) . . .. . . θn0 θ10 2 0 ··· 0 − 2 z10 z10 θ10 +2 3 0 ······ 0 z10 θ20 z20 θ10 − 2 +2 3 θ10 − 2 0 ··· 0 z10 z10 z10 . .. C = . . (B.4) . .. . . θn0 zn0 θ10 − 2 +2 3 θ10 0 ··· 0 − 2 z10 z10 z10 SUPERGEOMETRY OF Π-PROJECTIVE SPACES 15 1 − 2 0 ······ 0 z10 z20 1 − 2 0 ··· 0 z10 z10 . .. D = . . . (B.5) . .. . . zn0 1 − 2 0 ··· 0 z10 z10 1 Now, clearly det(A)= − n+1 and one can compute that z10 1 − 2 0 ······ 0 z10 z20 θ10θ20 1 − 2 − 3 0 ··· 0 z10 z10 z10 −1 . .. 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