Non-Projected Supermanifolds and Embeddings in Super Grassmannians

In this paper we give a brief account of the relations between non-projected supermanifolds and projectivity in supergeometry. Following the general results of arXiv:1706.01354, we study an explicit example of non-projected and non-projective supermanifold over the projective plane and we show how to embed it into a super Grassmannian. The geometry of super Grassmannians is also reviewed in details.

The problem of projectivity in supergeometry is a long-standing one. Indeed, large classes of complex supermanifolds whose reduced complex manifolds M red are projective -i.e. there exists an embedding M red ֒→ P n -are known to be non superprojective (henceforth, projective), that is they do not admit an embedding M ֒→ P n|m for some projective superspace P n|m . This is the case, for example, of a large class of complex super Grassmannians (see [M1] and section 4 of this paper). The problem of projectivity is related to another central problem characterizing the theory of complex supermanifold, that of the so-called non-projected supermanifolds: these are complex supermanifolds that do not possess a projection to their reduced manifold M → M red . Indeed, it has been shown that any projected supermanifold whose reduced manifold is projective, is also superprojective. In other words, if M red is a projective complex manifolds and M is projected, the embedding M red ֒→ P n can be lifted to an embedding of supermanifolds M ֒→ P n|m (see for example [BPW]). Notice that, for this to be true, the existence of the projection map M → M red is crucial: indeed if we let L red be a very-ample line bundle on M red , then π * L red will be very-ample on M , in the sense that π * L red will allow for the embedding at the level of the supermanifolds M ֒→ P n|m [BPW], [NPhD]. The story is different whenever a supermanifold is non-projected. The obstruction theory to find an embedding into projective superspace for a complex supermanifold has been studied for example in [BPW], back in the early days of supergeometry. There it is shown that the obstruction to extend the embedding map M red ֒→ P n at the level of the reduced complex manifolds, to an embedding M red ֒→ P n|m at the level of complex supermanifolds lies in in the cohomology groups H 2 (Sym 2k F M ) for k = 1, . . . , rank F M /2 and where the vector bundle F M = J M /J 2 M is constructed via a suitable quotient of the nilpotent bundle J M of the supermanifold, encoding the behavior of the anti-commutative nilpotent part of the geometry, see [M1], [NPhD]. This result has some obvious, yet remarkable, consequences: for example, by dimensional reasons, one sees that any supercurve, i.e. any supermanifold of dimension 1|m constructed over a projective curve, is actually projective, and the issues regarding projectivity start arising for dimension n|m, for n, m ≥ 2. Following these considerations, whist the literature fully acknowledged that in the realm of supergeometry projective superspaces P n|m are not as important as they are in ordinary complex algebraic geometry, nothing has been said, by the way, about which sort of space is to be considered when one looks for a universal embedding space for complex supermanifolds. In the recent [CNR] this problem has been taken on starting from dimension 2|2, working over the projective plane P 2 , and it has been shown that a large class of non-projected complex supermanifolds does not indeed admit projective embeddings, while all of these non-projected and non-projective supermanifolds admit embeddings in some complex super Grassmannians, thus hinting that the same might happen also in higher dimensions.
In the paper we consider again the problem of embedding a supermanifold into a super Grassmannians, enriching and clarifying the abstract results of [CNR] by very explicit constructions and examples. In particular, in the first section of the paper the key concepts of supergeometry are revised and the notation is fixed, also the main result of [CNR] are reported and put in context as to make the paper self-consistent. Next, following [M1], the supergeometry of complex super Grassmannians is explained. In the last section it is shown how to build maps to super Grassmannians and the example of the 2|2 dimensional supermanifold over P 2 characterized by a decomposable fermionic bundle F M = ΠO P 2 (−1) ⊕ ΠO P 2 (−2) is carried out in full details.

Basics of Supermanifolds
In this section we recall the basic definitions in the theory of (complex) supermanifolds. The interested reader might find more details in [M1] or [NPhD], which we will follow closely. The most important notion in supergeometry is the one of superspace, which is defined as follows.
where |M | is a topological space and O M is a sheaf of Z 2 -graded supercommutative rings (super rings for short) defined over |M | and such that the stalks O M ,x at every point of |M | are local rings. In other words, a superspace is a locally ringed space having structure sheaf given by a sheaf of super rings.
The requirement about the stalks being local rings is the same thing as asking that the even component of the stalk is a usual commutative local ring, for in superalgebra one has that if A = A 0 ⊕ A 1 a super ring, then A is local if and only if its even part A 0 is (see for example [Vara]). It is important to observe that one can always construct a superspace out of two classical data: a topological space, call it again |M |, and a vector bundle over |M |, call it E (analogously: a locallyfree sheaf of O |M | -modules). Now, we denote O |M | the sheaf of continuous functions (with respect to the given topology) on |M | and we put 0 E * = O |M | . The sheaf of sections of the bundle of exterior algebras • E * has an obvious Z 2 -grading (by taking its natural Z-grading mod 2) and therefore in order to realise a superspace it is enough to take the structure sheaf O M of the superspace to be the sheaf of sections valued in O |M | of the bundle of exterior algebras. This is what is called local model.
Definition 2.2 (Local Model S(|M |, E)). Given a pair (|M |, E), where |M | is a topological space and E is a vector bundle over |M |, we call S(|M |, E) the superspace modelled on the pair (|M |, E), where the structure sheaf is given by the O |M | -valued sections of the exterior algebra This is a minimal definition of local model: we have let |M | to be no more than a topological space and as such we are only allowed to take O |M | to be the sheaf of continuous functions on it. One can obviously work in a richer and more structured category, such as the differentiable, complex analytic or algebraic category: from now on, we will work in the complex analytic category and we consider local models based on the pair (M red , E), where M red is a complex manifold (its underlying topological space will be denoted with |M | and the sheaf of holomorphic functions on M red with O M red ) and where E is a holomorphic vector bundle on M red . We will call holomorphic local model a local model constructed on these kind of data .
The concept of local model enters in the definition of the main character of this paper.
where we have denoted with • E * the sheaf of sections of the exterior algebra of E considered with its Z 2 -gradation. In general, given two superspaces we can define a morphism relating these two.
Definition 2.4 (Morphisms of Superspaces). Given two superspaces M and N a morphism ϕ : (1) φ : |M | → |N | is a continuous map of topological spaces; (2) φ ♯ : O N → φ * O M is a morphism of sheaves of Z 2 -graded rings, having the property that it preserves the Z 2 -grading and that given any point x ∈ |M |, the homomorphism φ ♯ x : This definition applies in particular to the case of complex supermanifolds and enters the definition of sub-supermanifolds. Indeed, as in the ordinary theory, a sub-supermanifold is defined in general as a pair (N , ι), were N is a supermanifold and ι . .
is an injective morphism with some regularity property. In particular, depending on these regularity properties, we can distinguish between two kind of sub-supermanifolds. We start from the milder notion.
Making stronger requests, we can give instead the following definition. In what follows, we will always deal with closed embedded supermanifolds. Remarkably, it is possible to show that a morphism ι : N → M is an embedding if and only if the corresponding morphism ι ♯ : O M → O N is a surjective morphism of sheaves. Notice that, for example, given a supermanifold M , one always has a natural closed embedding: the map ι : M red → M , that embeds the reduced manifold underlying the supermanifold into the supermanifold itself.
We now introduce some further pieces of information carried by a supermanifold. M . It is crucial to note that modding out all of the nilpotent sections from the structure sheaf O M of the supermanifold M we recover the structure sheaf O M red of the underlying ordinary complex manifold M red , the local model was based on. We call the complex manifold M red the reduced manifold of the supermanifold M : loosely speaking, the reduced manifold arises by setting all of the nilpotents in O M to zero. In other words, more invariantly, attached to any complex supermanifold there is a short exact sequence that relates the supermanifold with its reduced manifold: We will refer to the short exact sequence (2.2) as the structural exact sequence of M . A very natural question arising when looking at the structural exact sequence (2.2) associated to a certain supermanifold is whether it is a split exact sequence or not, that is whether there exists a retraction -called projection in this context -π : Notice that, more precisely, this shall be recasted into the splitting of two exact sequences -the even and the odd part of (2.2) -, as we are only dealing with parity preserving morphisms. In particular, we shall give the following definition.
Definition 2.8 (Projected Supermanifold). We say that a supermanifold is projected if the even part of its structural exact sequence (2.2) splits: Notably, if also the odd part of the structural exact sequence attached to the supermanifold M is split, that is then the supermanifold M is called split : this expresses in a more invariant and meaningful form the isomorphism M ∼ = S(M red , ΠF * M ) : the supermanifold is globally isomorphic to the local model it is based onto. In other words, we might say that a supermanifold M is split if and only if is projected and the short exact sequence (2.5) is split. There indeed exists projected supermanifolds that are not split.
Notice that all of the complex supermanifolds having odd dimension 1 are projected and split because of dimensional reasons. When going up to odd dimension 2 a supermanifold can instead be non-projected -the short exact sequence (2.4) tells that O M ,0 an extension of O M red by the line bundle Sym 2 F M . If we call N = 2 supermanifold a complex supermanifolds having odd dimension equal to 2, we have the following important result.
The proof of the statement can be originally found in [M1] and it has been reproduced in full details in [NPhD].
3. Non-Projected N = 2 Supermanifolds over P 2 Using Theorem 2.9 of the previous section, in the recent [CNR] all the non-projected N = 2 supermanifolds over the projective plane P 2 were described through their characterizing cohomological invariants and their transition functions have been given. These non-projected supermanifolds reveal interesting features.
with M ij a 2 × 2 matrix with coefficients in O P 2 (U i ∩ U j ). Note that in the transformation (3.3) one can write M ij as a matrix with coefficients given by some even rational functions of z 1j , z 2j , because of the definitions (3.1) and the facts that θ hj ∈ J M and J 3 M = 0. Finally we note the transformation law for the products θ 1i θ 2i is given by Since det M is a transition function for the invertible sheaf Sym 2 F M ∼ = O P 2 (−3) over U i ∩ U j , this can be written, up to constant changes of bases in F ⌊ Ui and F ⌊ Uj , in the more precise form The meaning is that we can identify the base θ 1i θ 2i of Sym 2 F M ⌊ Ui with the standard base 1 Having set these conventions and notations we can give the following theorem, whose detailed proof can be found in [CNR].
Theorem 3.1 (Non-Projected N = 2 Supermanifolds over P 2 ). Every non-projected N = 2 supermanifold over P 2 is characterised up to isomorphism by a triple P 2 and ω is a non-zero cohomology class ω ∈ H 1 (T P 2 (−3)). The transition functions for an element of the family P 2 ω (F M ) from coordinates on U 0 to coordinates on U 1 are given by where λ ∈ C is a representative of the class ω ∈ H 1 (T P 2 (−3)) ∼ = C and M is a 2 × 2 matrix with coefficients in C[z 11 , z −1 11 , z 21 ] such that det M = 1 z 3 11 . Similar transformations hold between the other pairs of open sets.
We remark that the form of transition functions above is shared by all the supermanifolds P 2 ω (F M ), regardless the form of its fermionic sheaf F M , which is encoded in the matrix M . Some remarkable properties of this family of non-projected supermanifolds has been given by the authors in [CNR]. We condensate these results in the following theorem.
Theorem 3.2. Let M be a non-projected supermanifold in the family P 2 ω (F M ). Then: (1) M is non-projective, that is M cannot be embedded into any projective superspace of the kind P n|m ; (2) M can be embedded into a super Grassmannian.
In particular, let T M be the tangent sheaf of M , if we let V . (3.7) We observe that the theorem proves the existence of an embedding into some super Grassmannian, but it is not effective in that it does not give an esteem of the symmetric power of the tangent sheaf needed in order to set up the embedding. In the next sections we will first review the geometry of super Grassmannians and then, we will treat explicitly an interesting example of embedding into a super Grassmannian, by choosing a decomposable fermionic sheaf satisfying the hypotheses of theorem 3.1.

Elements of Super Grassmannians
This section is dedicated to the introduction of some elements of geometry of super Grassmannians. We remark that the this section contains no original result and it is fully expository: all of the results are originally due to Y. Manin and his school, see in particular [M1], [M2], [PenSko]. Nonetheless, we believe that since the cited literature is somewhat difficult and largely sketchy in the proofs of the various statements, it might be useful to have the constructions revised and readily at hand. In the present section our emphasis will be, anyway, on the non-projectedness and non-projectivity issues. Super Grassmannians are the supergeometric generalisation of the ordinary Grassmannians. This means that G(a|b; V n|m ) is a universal parameter space for a|b-dimensional linear subspaces of a given n|m-dimensional space V n|m . We will deal with the simplest possible situation, choosing the n|m-dimensional space V n|m to be a super vector space of the kind C n|m . We start reviewing how to construct a super Grassmannian by patching together the "charts" that cover it: this is a nothing but a generalization of the usual construction of ordinary Grassmannians making use of the so-called big cells.
(1) We let C n|m be such that n|m = c 0 |c 1 + d 0 |d 1 and look at C n|m as given by C c0+d0 ⊕ (ΠC) c1+d1 . This is obviously freely-generated, and we will write its elements as row vectors with respect to a certain basis, C n|m = Span{e 0 1 , . . . , e 0 n |e 1 1 , . . . , e 1 m }, where the upper indices refer to the Z 2 -parity.
(2) Consider a collection of indices I = I 0 ∪ I 1 such that I 0 is a collection of d 0 out of the n indices of C n and I 1 is a collection of d 1 indices out of m indices of ΠC m . If I is the set of such collections of indices I one gets This will give the number of super big cells covering the super Grassmannian.
(3) Choosing an element I ∈ I, we associate to it a set of even and odd (complex) variables, we call them {x αβ I | ξ αβ I }. These are arranged as to fill in the places of a d 0 |d 1 × n|m = a|b × (c 0 + d 0 )|(c 1 + d 1 ) matrix a way such that the columns having indices in I ∈ I I forms a (d 0 + d 1 ) × (d 0 + d 1 ) unit matrix if brought together. To makes this clear, for example, a certain choice of I ∈ I yields the following where we have chosen to pick that particular I ∈ I that underlines the presence of the the complex coordinates over the point. Whenever is represented as above, the superspace related to U I is called a super big cell of the Grassmannian, and denoted with Z I or, again, simply by U I (which encodes the topological information).
(5) We now show how to patch together two superspaces U I and U J for two different I, J ∈ I.
If Z I is the super big cell related to U I , we consider the super submatrix B IJ formed by the columns having indices in J. Let U IJ = U I ∩ U J be the (maximal) sub superspace of U I such that on U IJ the submatrix B IJ is invertible. As usual, the odd coordinates do not affect the invertibility, so that it is enough that the two determinants of the even parts of the matrix B IJ (that are respectively a d 0 × d 0 and a d 1 × d 1 matrix) are different from zero. When this is the case, on the superspace U IJ one has common coordinates {x αβ I | ξ αβ I } and {x αβ J | ξ αβ J }, and the rule to pass from one system of coordinates to the other one is provided by Z J = B −1 IJ Z I . For example, let us consider the following two super big cells: Looking at Z I , we see that the columns belonging to J are the first, the third and the fourth, so that Computing the determinant of the upper-right 2 × 2 matrix, we have invertibility of B IJ corresponds to x 2 = 0 (as seen from the point of view of U I . Likewise one would have foundx 2 = 0 by looking at Z J and U J ). The inverse of B −1 IJ is so that we can compute the coordinates of U J as functions of the ones of U I via the rule so that the change of coordinates can be read out of this. Observe that the denominator x 2 is indeed invertible on U IJ . (6) Patching together the superspaces U I one obtains the Grassmannian supermanifold G(d 0 |d 1 ; C n|m ) as the quotient supermanifold where we have written R for the equivalence relations generated by the change of coordinates that have been described above. Notice that, as a (complex) supermanifold a super Grassmannian has dimension We stress that the maps ψ UI : U I → G(d 0 |d 1 ; C n|m ) are isomorphisms onto (open) sub superspaces of the super Grassmannian, so that the various super big cells offer a local description of it, in the same way a usual (complex) supermanifold is locally isomorphic to a superspace of the kind C n|m . Clearly, the easiest possible example of super Grassmannians are projective superspaces, that are realised as P n|m = G(1|0; C n+1|m ), exactly as in the ordinary case: these are split supermanifolds, a feature that they do not in general share with a generic Grassmannian G(d 0 |d 1 ; C n|m ), as we shall see in a moment. For convenience, in what follows we call G a super Grassmannian of the kind G(d 0 |d 1 ; C n|m ) and we give the following, see [M1]. (4.9) Notice that this definition is well-posed, since one has that S G (U I )⌊ UIJ and S G (U J )⌊ UJI get identified by means of the transition functions B IJ .
One can have insights about the geometry of a super Grassmannian by looking at its reduced space -which, we recall, encloses all the topological information -, and also at the filtration of its trivial sheaf O G . We start observing that given a super Grassmannian G, one automatically has two ordinary even sub Grassmannians.
Definition 4.2 (G 0 and G 1 ). Let G = G(d 0 |d 1 ; C n|m ) be a super Grassmannian. Then we call G 0 and G 1 the two purely even sub Grassmannians defined as (4.10) Given a super big cell U I , then G 0 and G 1 can be visualized as the upper-left and the lower-right part respectively and they come endowed with their tautological sheaves, we call them S 0 and S 1 . Notice, though, that S 1 defines a sheaf of locally-free O G1 -modules and, as such, it has rank 0|d 1 . Let us now consider an ordinary even complex Grassmannian G of the kind G(d; C n ) together with its tautological sheaf S G . One can then also define the sheaf orthogonal to the tautological sheaf, we call it S, whose dual fits into the short exact sequence Notice that in the case the Grassmannian corresponds to a certain projective space G(1|0; C n+1 ) = P n , the sheaf orthogonal to the tautological sheaf can be red off the Euler exact sequence twisted by the tautological sheaf itself S P n = O P n (−1), and, indeed, we have that S * G ∼ = T P n (−1), so that S G ∼ = Ω 1 P n (+1). In the case of a super Grassmannian G(d 0 |d 1 ; n|m) the sequence (4.11) gets generalized to the canonical sequence we now have all the ingredients to state the following theorem, whose proof is contained in [M1].
Theorem 4.3. Let G = G(d 0 |d 1 ; C n|m ) be a super Grassmannian and let G 0 and G 1 their even sub Grassmannians together with the sheaves S 0 , S 1 and S 0 , S 1 . Then the following (canonical) isomorphisms hold true 1) G red ∼ = G 0 × G 1 ; 2) Gr O G ∼ = Sym (S 0 ⊗ S 1 ⊕ S 0 ⊗ S 1 ), where by Sym we mean the super-symmetric algebra over O G0×G1 .
The fundamental example, yet enclosing all the features characterizing the peculiar geometry of super Grassmannians, is given by G(1|1, C 2|2 ) -which is of dimension 2|2. We now study its geometry in some details.
The Geometry of G(1|1; C C C 2|2 ): we start studying the geometry of G(1|1; C 2|2 ), we call it G for short, from its reduced manifold which is easily identified using the previous Theorem 4.3.
It is fair to observe that we would have gotten to the same conclusion by looking at the big cells of this super Grassmannian, after having set the nilpotents to zero. We thus have the following situation that helps us to recover the geometric data of G red and G out of those of the two copies of projective lines. Along this line, we recall that , and since the tautological sheaf on P 1 is O P 1 (−1), we have that Similarly, observing that the sheaf dual to the tautological sheaf on P 1 is given again by the sheaf O P 1 (+1), as the (twisted) Euler sequence reads and therefore S P 1 ∼ = (T P 1 (−1)) * ∼ = Ω 1 P 1 (+1) ∼ = O P 1 (−1), one has the following: This is enough to identify the fermionic sheaf of G, since F G = Gr (1) O G and therefore by virtue of the second point of the previous Theorem 4.3, one has F G ∼ = S 0 ⊗ S 1 ⊕ S 0 ⊗ S 1 , so Which, in turns, shows that and one can prove the following.
Theorem 4.5 (G(1|1; C 2|2 ) is Non-Projected). The supermanifold G = G(1|1; C 2|2 ) is in general non-projected. In particular, H 1 (T P 1 0 ×P 1 1 ⊗ Sym 2 F G ) ∼ = C ⊕ C. Proof. In order to compute the cohomology group H 1 (T P 1 0 ×P 1 1 ⊗ Sym 2 F G ), we observe that in general, on the product of two varieties, we have T X×Y ∼ = p * 1 T X ⊕ p * 2 T Y , where the p i are the projections on the factors, so that, in particular, we find Taking the tensor product with Sym 2 F G , one has Now, via the Künneth formula one has so that which concludes the proof.
There are different ways to find the representatives in the obstruction cohomology group for G. We will first use the super big cells of G(1|1; C 2|2 ) to identifies these representatives and to establish that in the isomorphisms H 1 (T P 1 0 ×P 1 1 ⊗ Sym 2 F G ) ∼ = C ⊕ C the cohomology class corresponds to the choice ω G = (1, 1). This is an explicit and immediate way to do this. First, we observe that, since the reduced manifold underlying G(1|1; C 2|2 ) has the topology of (1) ℓ } ℓ=0,1 the open sets covering P 1 1 , we then have a system of open sets covering their product P 1 0 × P 1 1 given by These correspond to the following matrices Z Ui , out of which we can read the coordinates on the big cells: (4.27) Following the procedure illustrated above or by rows and columns operations on the Z Ui one find the transition rules between the various charts, (4.28) By looking at these transformation rules, we therefore have that in the isomorphism above the class is represented by (1, 1) ∈ C⊕C and the cocycles representing ω are given by ω = (ω 12 , ω 13 , ω 14 , ω 23 , ω 24 , ω 34 ), where the ω ij are (in tensor notation) (4.29) One can get to the same result also by means of a different computation, as remarked above.
Observing that H 1 (O P 1 0 ×P 1 1 (−2, 0)) ⊕ H 1 (O P 1 0 ×P 1 1 (0, −2)) is generated by the two elements we can then look at these generators in the intersections, keeping in mind that F G ∼ = ΠO P 1 0 ×P 1 1 (−1, −1)⊕ ΠO P 1 0 ×P 1 1 (−1, −1), in order to identify the cocycles that enter in the transition functions. We examine the various intersections. U 1 ∩ U 2 : The following identifications can be made These gives the transition functions above between ξ 1 and ξ 2 and between η 1 and η 2 . Notice that in the intersection U 1 ∩ U 2 only the bit H 1 (O P 1 0 ×P 1 1 (−2, 0)) contributes and we have therefore where we have denoted by ⊙ the supersymmetric product of the two (local) sections on F G , as represented above.
U 1 ∩ U 3 : here we have a contribution from H 1 (O P 1 0 ×P 1 1 (0, −2)) only and, therefore, we have to deal with ω 13 = ℓ 2 (1 ⊠ 1/Y 0 Y 1 ). By a completely analogous treatment as above, one finds that (4.33) U 1 ∩ U 4 : In this case we have both the contributions, so (4.34) so that by analogous manipulations as the one above one finds All the other ω ij are identified in the same way and enter one of these three categories.
To conclude, one then impose the cocycle conditions as to fix the various signs of the ℓ 1 and ℓ 2 above, that agree with the one we found above by looking at the coordinates of the big cells: choosing (ℓ 1 = 1, ℓ 2 = 1) -this can always be done up to a change of coordinates -, one gets the same even transition functions as above. This is enough to use the theorem classifying the complex supermanifold of dimension n|2 (see [M1] or [CNR]) as to conclude that G(1|1; C 2|2 ) can be defined up to isomorphism as follows Definition 4.6 (G(1|1; C 2|2 ) as a Non-Projected Supermanifold). The super Grassmnannian G(1|1; C 2|2 ) can be defined up to isomorphism as the 2|2 dimensional supermanifold characterised by the triple (P 1 −1) and where ω G = (ℓ 1 , ℓ 2 ), with ℓ 1 = 0 and ℓ 2 = 0, in the isomorphism ω G ∈ H 1 (T P 1 On a very general ground, apart from projective superspaces, super Grassmannians are in general non-projected: the case of G(1|1; C 2|2 ) we treated is just the first non-trivial example of nonprojected super Grassmannian. Now, jump to the second issue we are interested into: we show that G(1|1; C 2|2 ) is not a nonprojective supermanifold.
Proof. In order to prove the non-projectivity of G . . = G(1|1; C 2|2 ) we consider the following short exact sequence that comes from the structural exact sequence of G: Ordinary results in algebraic geometry yield H 0 (O P 1 0 ×P 1 1 (−2, −2)) = 0 = H 1 (O P 1 0 ×P 1 1 (−2, −2)), whereas H 2 (O P 1 0 ×P 1 1 (−2, −2)) ∼ = C. Likewise, one has H 0 (O * P 1 0 ×P 1 1 ) ∼ = C * and Pic(P 1 0 × P 1 1 ) = H 1 (O * P 1 0 ×P 1 1 ) ∼ = Z ⊕ Z, by means of the ordinary exponential exact sequence. This is enough to realize that the cohomology sequence induced by the sequence above splits in two exact sequences. The first one gives an isomorphism H 0 (O G,0 ) ∼ = C * , while the second one instead reads Thus in order to establish the fate of the cohomology group H 1 (O * G,0 ) one has to look at the boundary map δ : Pic(P 1 0 × P 1 1 ) → H 2 (O P 1 0 ×P 1 1 (−2, −2)). Let then us consider the following diagram of cochain complexes: (1, 0) could be represented by (six) cocycles g ij ∈ Z 1 (U i ∩ U j , O * P 1 0 ×P 1 1 ). Explicitly, these cocycles are the transition functions of the line bundle where, with an abuse of notation, we dimissed the second bit of the external tensor product, which is just the identity. Since the map j : ) is surjective, these cocycles are images of elements in C 1 (O * G,0 ). Notice that j is induced by the inclusion of the reduced variety P 1 0 ×P 1 1 into G, so the cochains in C 1 (O * G,0 ) are exactly the {g ij } i,j∈I we have written above (notice also that these are no longer cocycles in O * G,0 ). Using theČech coboundary map δ(j * O P 1 0 ×P 1 1 (1, 0)) over G, one finds, for example: Indeed, by looking at the affine coordinates in the big cells, these reads x 2 x 3 = 1+ ξ2η2 x2y2 and setting, as we have done above above (4.40) and taking their supersymmetric product one has ξ2η2 x2y2 = 1 X0X1 ⊠ 1 Y0Y1 . Now, by exactness of the diagram, this element is in the kernel of the map j : and it is a cocycle. Then, considering that the map i is induced by the map O P 1 By symmetry, the same applies to the second generator of Pic(P 1 0 × P 1 1 ), which is given by O P 1 0 ×P 1 1 (0, 1), thus that the map δ : Pic(P 1 By exactness, it follows that the only invertible sheaves on P 1 0 × P 1 1 that lift to the whole G are those of the kind O P 1 0 ×P 1 1 (a, −a), as the composition of the maps yields (a, −a) → (a, −a) → a − a = 0 as it should. Since these invertible sheaves have no cohomology, they cannot give any embedding in projective superspaces and this completes the proof.
Notice the subtlety: the above theorem says that Pic(P 1 0 × P 1 1 ) = 0 (actually Pic(P 1 0 × P 1 1 ) ∼ = Z), but still there are no ample invertible sheaves that allow for an embedding of G(1|1; C 2|2 ) into some projective superspaces. The fundamental consequence is that non-projectivity is not confined to this particular super Grassmannian only.
Theorem 4.8 (Super Grassmannians are Non-Projective). The super Grassmannian space G(a|b; C m|n ) for 0 < a < n and 0 < b < m is non-projective.
The upshot of this result is that, working in the context of algebraic supergeometry, it is no longer true that projective superspaces are a privileged ambient: this is a substantial departure from usual context of complex algebraic geometry, that deserves to be stressed out.

Maps and Embeddings into a Super Grassmannian: An Explicit Example
Having reviewed the geometry of super Grassmannians in the previous section, we now consider the problem of setting up maps to super Grassmannians. First we recall the universal property characterizing the construction of maps into projective superspaces P n|m , which is nothing but a direct generalization of the usual criterium in algebraic geometry for projective spaces P n , using invertible sheaves, i.e. for any supermanifold or superscheme M , any locally-free sheaf L of rank 1|0 on M and any vector superspace V having a surjective sheaf-theoretical map V ⊗ O M → L, then there exists a unique (up to isomorphisms) map Φ L : M → P n|m such that the inclusion L * → V * ⊗ O M is the pull-back of the inclusion O P n|m (−1) → O ⊕n+1|m P n|m coming from the Euler exact sequence. More concretely, this is sometimes reported simply asking L to be globally-generated, that is there exists a surjective sheaf-theoretical map H 0 (L) ⊗ O M → E, with dim H 0 (L) = n + 1|m. Then there exists a unique map up to isomorphism Φ L : M → P n|m such that E = Φ * L (O P n|m (1)) and such that, if H 0 (L) = span C {s i |ξ j }, then s i = Φ * L (X i ) and ξ j = Φ * L (Θ j ) for i = 0, . . . , n and j = 1, . . . , m, where X i |Θ j are the generating sections of H 0 (O P n|m (1)), where we recall that O P n|m (1) . .= π * O P n (1) = π −1 O P n (1)⊗ π −1 O P n O P n|m , see [CN], where invertible sheaves on projective superspaces are studied.
A very similar situation happens in the case of super Grassmannians, but instead of invertible sheaves one has to deal with locally-free sheaves of higher rank / vector bundles, in order to appropriately set up maps. Indeed, let G = G(a|b, V ) be a super Grassmannian, then it is has the following universal property that characterizes the maps toward it [CNR]: considering the restriction of the tangent sheaf to the reduced manifold P 2 , that is It is a general result that T M ⌊ M red ∼ = T M red ⊕F * M , see for example [M1] or [NPhD], anyway this result can be readily red off once one has the explicit form of the transition functions of the tangent sheaf. Indeed, using the chain rule and starting from the above lemma, with obvious notation, one finds: (5.4) so that the related Jacobian has the following matrix representation The transition functions in the other intersections can be found by S 3 -symmetry. We now recall that, having at disposal the structure sheaf of O M of we can also form a subsuperscheme of M through the pair (P 2 , O . We stress that this is not a supermanifold: indeed it fails to be locally isomorphic to any local model of the kind C m|n : more generally, it is locally isomorphic to an affine superscheme for some super ring. We call M (2) the superscheme defined by the pair (P 2 , O (2) M ) and we characterize its geometry in the following lemma.
Lemma 5.2 (The Superscheme M (2) ). Let M (2) be the superscheme as above. Then M is a projected scheme and its structure sheaf O (2) M ,0 ∼ = O P 2 .The structure sheaf gets endowed with a structure of O P 2 -module given by O P 2 ⊕ F M , that actually coincides with the parity splitting. We observe that in the O P 2 -algebra O Pushing the characterization of the tangent sheaf a little bit further, we have to study the geometry of tangent bundle T M when restricted to the sub superscheme M (2) . Once again it can be proved that the following general isomorphism holds true where the first two summands are the even part and the second two summands are the odd part of the sheaf. In particular, in our case one gets: is a locally-free of O P 2 -module, moreover the following isomorphism holds Proof. the claim is proved by computing where we have used that, since F M is a locally-free sheaf of O P 2 -module we have that F M ∼ = F M ⊗ O P 2 O P 2 . The first isomorphism is a standard result in modules theory (note we have suppressed the subscript M in the sheaf of nilpotent element J M for a better notation).
that we write multiplicatively as θ 1i ∂ θ1i and θ 2i ∂ θ2i (both taken mod J 2 Then, the first one has been just shown to be surjective, while the second one is well-known to be surjective as T M ⌊ P 2 is a direct sum of globally-generated sheaves of O P 2 -modules. This concludes the proof. The universal property, thus leads to the following Theorem 5.8 (Map to G(2|2, T M )). There exists a unique map Φ T M : M −→ G(2|2, C 12|12 ) up to isomorphism.
More can be said about this map, which is actually an embedding of M into G(2|2, C 12|12 ): that is, it is an injective map and its differential dΦ T M is injective as well. We prove this in a completely explicit fashion by realizing the actual embedding in a certain chart.
We explain the strategy to do this in a general setting: once one have a map into a super Grassmannian and a local basis {e 1 , . . . , e a |f 1 , . . . , f b } is fixed for E over some open set U, then, over U, the evaluation map V ⊗ O M → E is defined by a (a|b) × (n|m) matrix M U with coefficients in O M (U), and any reduction of M U into a standard form of type (5.24) by means of elementary row operations, is a local representation of the map Φ : M → G(a|b, C n|m ). One can then easily verify injectivity and the injectivity of the differential of this map via this local representation, as to establish whether the map constitutes an embedding.
In order to do this, we need the explicit form of the global sections generating T M . Notice that to keep the discussion the most general possibile we will keep a parameter λ ∈ C representing the cohomology class ω M ∈ H 1 (T P 2 (−3)) ∼ = C, which we recall to be the same λ appearing in the transition functions provided by Theorem 3.1.
The embedding is explicitly realized through the following Now, following what explained above, the coefficients of the expansion are mapped into 12|12 columns, so that the resulting matrix is a super Grassmannian of the kind G(2|2, C 12|12 ), represented in a certain super big-cell. The full super Grassmannian is then reconstructed via its transition functions, as explained in the previous section. In our particular case, the global sections lead to an image into G(2|2, C 12|12 ) as follows: where we have highlighted the super big-cell singled out by the four global sections {V 1 = ∂ z1 , V 2 = ∂ z2 , Ξ 1 = ∂ θ1 , Ξ 2 = ∂ θ2 } in the chart U 0 and the A i×10 , B i×10 , C i×10 , D i×10 for i = 1, 2, make up four 2 × 10 matrices: