scholarly journals Kodaira vanishing theorem for log-canonical and semi-log-canonical pairs

2015 ◽  
Vol 91 (8) ◽  
pp. 112-117
Author(s):  
Osamu Fujino
Keyword(s):  
Vanishing Theorem ◽  
Log Canonical ◽  
Kodaira Vanishing

scholarly journals Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties

2009 ◽  
Vol 146 (1) ◽  
pp. 193-219 ◽  
Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Sándor J. Kovács
Keyword(s):  
Differential Forms ◽  
Natural Generalization ◽  
Resolution Of Singularities ◽  
Vanishing Theorem ◽  
Exceptional Set ◽  
Extension Theorems ◽  
Normal Variety ◽  
Log Canonical ◽  
Smooth Locus

AbstractGiven a normal variety Z, a p-form σ defined on the smooth locus of Z and a resolution of singularities $\pi : \widetilde {Z} \to Z$, we study the problem of extending the pull-back π*(σ) over the π-exceptional set $E \subset \widetilde {Z}$. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along E. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov–Sommese vanishing theorem to log canonical threefold pairs follows.

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.

A vanishing theorem for log canonical pairs

2010 ◽  
Vol 132 (5) ◽  
pp. 1205-1221 ◽  
Author(s):  
Tommaso de Fernex ◽  
Lawrence Ein
Keyword(s):  
Vanishing Theorem ◽  
Log Canonical

scholarly journals Non-vanishing theorem for log canonical pairs

2011 ◽  
Vol 20 (4) ◽  
pp. 771-783 ◽  
Author(s):  
Osamu Fujino
Keyword(s):  
Vanishing Theorem ◽  
Log Canonical

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.

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