Generalizations of the Kodaira Vanishing Theorem

1985 ◽  
pp. 133-158
Author(s):  
Bernard Shiffman ◽  
Andrew John Sommese
Keyword(s):  
Vanishing Theorem ◽  
Kodaira Vanishing

scholarly journals Local cohomology of Du Bois singularities and applications to families

2017 ◽  
Vol 153 (10) ◽  
pp. 2147-2170 ◽  
Author(s):  
Linquan Ma ◽  
Karl Schwede ◽  
Kazuma Shimomoto
Keyword(s):  
Commutative Algebra ◽  
Local Cohomology ◽  
American Mathematical Society ◽  
Vanishing Theorem ◽  
Du Bois ◽  
Local Cohomology Modules ◽  
Du Bois Singularities ◽  
Cohomology Modules ◽  
Kodaira Vanishing ◽  
Koszul Cohomology

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,\mathfrak{m})$ be a local ring; we prove that if $R_{\text{red}}$ is Du Bois, then $H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$ is surjective for every $i$. We find many applications of this result. For example, we answer a question of Kovács and Schwede [Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), 969–1000; MR 3531359] on the Cohen–Macaulay property of Du Bois singularities. We obtain results on the injectivity of $\operatorname{Ext}$ that provide substantial partial answers to questions in Eisenbud et al. [Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583–600] in characteristic $0$. These results can also be viewed as generalizations of the Kodaira vanishing theorem for Cohen–Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen–Macaulayness of the defining ideal of Du Bois singularities, which are characteristic-$0$ analogs and generalizations of results of Singh–Walther and answer some of their questions in Singh and Walther [On the arithmetic rank of certain Segre products, in Commutative algebra and algebraic geometry, Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), 147–155]. We extend results on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities first shown in Hochster and Roberts [The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117–172; MR 0417172 (54 #5230)]. We also prove that singularities of dense $F$-injective type deform.

Vanishing theorems on Kähler Lie algebroids

2020 ◽  
Vol 17 (04) ◽  
pp. 2050059
Author(s):  
Zahra Pirbodaghi ◽  
Morteza Mirmohammad Rezaii
Keyword(s):  
Hodge Theory ◽  
Lie Algebroids ◽  
Vanishing Theorems ◽  
Vanishing Theorem ◽  
Laplacian Operators ◽  
Kodaira Vanishing

In this paper, we extend the Hodge theory to complex Lie algebroids, introduce the Laplacian operators and decompose the cohomological groups with respect to these operators, and generalize Kähler and Nakano identities to Kähler Lie algebroids as well. Our main purpose is to extend Kodaira vanishing theorem to Kähler Lie algebroids.

Branched Coverings and Minimal Free Resolution for Infinite-Dimensional Complex Spaces

2003 ◽  
Vol 10 (1) ◽  
pp. 37-43
Author(s):  
E. Ballico
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Free Resolution ◽  
Complete Intersections ◽  
Vanishing Theorem ◽  
Sheaf Cohomology ◽  
Minimal Free Resolution ◽  
Branched Coverings ◽  
Infinite Dimensional ◽  
Complex Spaces

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 ).

scholarly journals Generalized vanishing theorems for Yukawa couplings in heterotic compactifications

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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