A vanishing theorem for L2-cohomology in the nilpotent case

Abstract We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P(V ).

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Generalized vanishing theorems for Yukawa couplings in heterotic compactifications

Journal of High Energy Physics ◽

10.1007/jhep05(2021)085 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Lara B. Anderson ◽

James Gray ◽

Magdalena Larfors ◽

Matthew Magill ◽

Robin Schneider

Keyword(s):

Vector Bundles ◽

Geometric Approach ◽

Low Energy ◽

Geometric Features ◽

Vanishing Theorems ◽

Vanishing Theorem ◽

Alternative Derivation ◽

Yukawa Couplings ◽

Geometric Methods ◽

Higher Dimensional

Abstract Heterotic compactifications on Calabi-Yau threefolds frequently exhibit textures of vanishing Yukawa couplings in their low energy description. The vanishing of these couplings is often not enforced by any obvious symmetry and appears to be topological in nature. Recent results used differential geometric methods to explain the origin of some of this structure [1, 2]. A vanishing theorem was given which showed that the effect could be attributed, in part, to the embedding of the Calabi-Yau manifolds of interest inside higher dimensional ambient spaces, if the gauge bundles involved descended from vector bundles on those larger manifolds. In this paper, we utilize an algebro-geometric approach to provide an alternative derivation of some of these results, and are thus able to generalize them to a much wider arena than has been considered before. For example, we consider cases where the vector bundles of interest do not descend from bundles on the ambient space. In such a manner we are able to highlight the ubiquity with which textures of vanishing Yukawa couplings can be expected to arise in heterotic compactifications, with multiple different constraints arising from a plethora of different geometric features associated to the gauge bundle.

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ON THE STABILITY OF SYZYGY BUNDLES

International Journal of Mathematics ◽

10.1142/s0129167x1100688x ◽

2011 ◽

Vol 22 (04) ◽

pp. 515-534 ◽

Cited By ~ 1

Author(s):

IUSTIN COANDĂ

Keyword(s):

Projective Space ◽

Complete Intersection ◽

Vector Bundles ◽

Elementary Proof ◽

Base Point ◽

Vector Spaces ◽

Similar Argument ◽

Vanishing Theorem ◽

The Stability ◽

Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

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