In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.

Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kähler moduli which give rise to realistic compactifications of the strongly coupled heterotic string. When vector bundles are constructed using extensions, we provide simple rules to determine lower and upper bounds to the region of the Kähler moduli space where such compactifications can exist. We show how small these regions can be, working out in full detail the case of the recently proposed Heterotic Standard Model. More explicitely, we exhibit Kähler classes in these regions for which the visible vector bundle is stable. On the other hand, there is no polarization for which the hidden bundle is stable. December, 2005 Recently, phenomenologically interesting Calabi-Yau compactifications of the heterotic string have appeared in the literature [2], [5]. Using certain elliptically fibered threefold with fundamental group Z Z 3 × Z Z 3 , and an SU (4) × Z Z 3 × Z Z 3 instanton living on the visible E 8 -bundle, give rise to an effective field theory on IR 4 which has the particle spectrum of the Minimal Supersymmetric Standard Model (MSSM), with no exotic matter but an additional pair of Higgs-Higgs conjugate superfields. In these models, vector bundles are constructed using vector bundle extensions, which correspond to Hermitian Yang-Mills connections when they are slope-stable. We use this specific construction to exemplify how a systematic selection of realistic Kähler moduli can be done. 1 The organization of the paper is as follows: section 2 contains an outline of the natural criteria for selecting Kähler moduli in realistic Calabi-Yau compactifications of the heterotic string. In section 3, we analyze the case of the Heterotic Standard Model, describe the geometry of the elliptic Calabi-Yau and construct its Kähler cone. Section 4 provides lower and upper bounds to the region of the Kähler cone that makes stable the observable vector bundle of the HSM. In such construction we find a destabilizing sub-line bundle for the hidden vector bundle, and exhibit Kähler classes that make stable the visible one. 1 Recently, Donagi and Bouchard [8] have also proposed an independent CY compactification of the heterotic string with the spectrum of the MSSM and no exotic matter, using a different Calabi-Yau with an explicitly slope-stable vector bundle in the observable sector. It would be also interesting to study in detail these questions with the vector bundle which has just appeared in [4], on the same CY [5].3 Being rigorous, we should work with the dimensionfull measure α ′ ω 3 , although this will be irrelevant for our purposes because α ′ factorizes out in the formulae that we use. In the small volume limit this approximation can fail, and we should use conformal field theory to give a more accurate estimation.

In this paper we use Donaldson's approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the grassmannians Gr(r, N ). We assure that these embeddings are asymptotically holomorphic in a precise sense. We study first the particular case of CP N obtaining control on N and by a simple corollary we improve in a sense a classical result about symplectic embeddings [Ti77]. The main reason of our study is the construction of singular determinantal submanifolds as the intersection of the embedding with certain "generalized Schur cycles" defined on a product of grassmannians. It is shown that the symplectic type of these submanifolds is quite more general that the ones obtained by Auroux [Au97] as zeroes of "very ample" vector bundles.

Let G be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal G-bundle on a Riemann surface and constructed a projective moduli space of such objects. We generalize Ramanathan's notion and construction to higher dimension, allowing also objects which we call semistable principal G-sheaves, in order to obtain a projective moduli space: a principal G-sheaf on a projective variety X is a triple (P, E, ψ), where E is a torsion free sheaf on X, P is a principal G-bundle on the open set U where E is locally free and ψ is an isomorphism between E| U and the vector bundle associated to P by the adjoint representation.We say it is (semi)stable if all filtrations E • of E as sheaf of (Killing) orthogonal algebras, i.e. filtrations withwhere P Ei is the Hilbert polynomial of E i . After fixing the Chern classes of E and of the line bundles associated to the principal bundle P and characters of G, we obtain a projective moduli space of semistable principal G-sheaves. We prove that, in case dim X = 1, our notion of (semi)stability is equivalent to Ramanathan's notion.

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