scholarly journals Kodaira vanishing theorem and Chern classes for $\partial$-manifolds

1978 ◽  
Vol 54 (4) ◽  
pp. 107-108 ◽  
Author(s):  
Yoshiki Norimatsu
Keyword(s):  
Vanishing Theorem ◽  
Chern Classes ◽  
Partial Manifolds ◽  
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.

scholarly journals Tensors with eigenvectors in a given subspace

2021 ◽  
Author(s):  
Giorgio Ottaviani ◽  
Zahra Shahidi
Keyword(s):  
Algebraic Geometry ◽  
Linear Subspace ◽  
Chern Classes ◽  
Symmetric Tensors

AbstractThe first author with B. Sturmfels studied in [16] the variety of matrices with eigenvectors in a given linear subspace, called the Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore, we consider the Kalman variety of tensors having singular t-tuples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations.

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