De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Let [Formula: see text] be an abelian surface, and [Formula: see text] be the sum of [Formula: see text] distinct theta divisors having normal crossings. We set [Formula: see text]. We study the structure of the nonvanishing twisted cohomology group [Formula: see text], where [Formula: see text] denotes a locally constant sheaf over [Formula: see text] defined by a multiplicative meromorphic function on [Formula: see text] infinitely ramified just along the divisor [Formula: see text] (as will be seen below, we will take as such a function a product of complex powers of theta functions). The de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text], associated to the twisted cohomology groups [Formula: see text], is [Formula: see text]-valued, where [Formula: see text] denotes a topologically trivial (i.e. Chern class zero) line bundle over [Formula: see text] determined by the locally constant sheaf [Formula: see text]. Therefore the main results of this paper, which give us information on the order of poles of meromorphic 2-forms on [Formula: see text] generating the cohomology group [Formula: see text], are divided into Theorems 4.5 and 4.6, according as the de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text] takes the values in a holomorphically nontrivial line bundle [Formula: see text] or a holomorphically trivial one [Formula: see text] ([Formula: see text] denoting the holomorphically trivial line bundle [Formula: see text]). Such a phenomenon does not occur in the case of the twisted cohomology of complex projective space with hyperplane arrangement.

THE DE RHAM COHOMOLOGY OF DIFFERENTIAL SPACES

Demonstratio Mathematica ◽

10.2478/dema-1989-0124 ◽

1989 ◽

Vol 22 (1) ◽

pp. 249-272 ◽

Cited By ~ 2

Author(s):

Wiesław Sasin

Keyword(s):

De Rham Cohomology ◽

De Rham ◽

Differential Spaces

Torus Embeddings and de Rham Complexes

Commutative Algebra and Combinatorics ◽

10.2969/aspm/01110111 ◽

2018 ◽

Cited By ~ 3

Author(s):

Masa-Nori Ishida

Keyword(s):

De Rham

On the Voevodsky motive of the moduli space of Higgs bundles on a curve

Selecta Mathematica ◽

10.1007/s00029-020-00610-5 ◽

2021 ◽

Vol 27 (1) ◽

Author(s):

Victoria Hoskins ◽

Simon Pepin Lehalleur

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Characteristic Zero ◽

Rational Point ◽

Triangulated Category ◽

Higgs Bundles ◽

Projective Curve ◽

Smooth Projective Curve ◽

De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

Forum of Mathematics Sigma ◽

10.1017/fms.2021.20 ◽

2021 ◽

Vol 9 ◽

Author(s):

Benjamin Antieau ◽

Bhargav Bhatt ◽

Akhil Mathew

Keyword(s):

Spectral Sequence ◽

Spectral Sequences ◽

Characteristic P ◽

De Rham

Abstract We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

A de Rham model for complex analytic equivariant elliptic cohomology

Advances in Mathematics ◽

10.1016/j.aim.2021.107575 ◽

2021 ◽

Vol 380 ◽

pp. 107575

Author(s):

Daniel Berwick-Evans ◽

Arnav Tripathy

Keyword(s):

Elliptic Cohomology ◽

De Rham

Classification of Degenerate Verma Modules for E(5, 10)

Communications in Mathematical Physics ◽

10.1007/s00220-021-04031-z ◽

2021 ◽

Author(s):

Nicoletta Cantarini ◽

Fabrizio Caselli ◽

Victor Kac

Keyword(s):

Infinite Number ◽

Verma Module ◽

Lie Superalgebra ◽

Verma Modules ◽

Singular Vectors ◽

Finite Dimensional ◽

De Rham ◽

Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000177 ◽

2021 ◽

pp. 1-48

Author(s):

Federico Scavia

Keyword(s):

Finite Groups ◽

Reductive Groups ◽

Orthogonal Groups ◽

De Rham Cohomology ◽

Algebraic Stacks ◽

Steenrod Operations ◽

De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

The Global Sections of Chiral de Rham Complexes on Compact Ricci-Flat Kähler Manifolds

Communications in Mathematical Physics ◽

10.1007/s00220-021-03975-6 ◽

2021 ◽

Vol 382 (1) ◽

pp. 351-379

Author(s):

Bailin Song

Keyword(s):

Kähler Manifolds ◽

Kahler Manifolds ◽

Ricci Flat ◽

De Rham

On equivariant disconnected rational homotopy theory

Georgian Mathematical Journal ◽

10.1515/gmj.2010.008 ◽

2010 ◽

Vol 17 (2) ◽

pp. 229-240

Author(s):

Marek Golasiński

Keyword(s):

Homotopy Theory ◽

Weak Equivalence ◽

Rational Homotopy Theory ◽

Rational Homotopy ◽

Hamiltonian Group ◽

Simplicial Set ◽

De Rham

Abstract An equivariant disconnected Sullivan–de Rham equivalence is developed using Kan's result on diagram categories. Given a finite Hamiltonian group G, let X be a G-simplicial set. It is shown that the associated system of algebras indexed by the category 𝒪(G) of a canonical orbit can be “approximated” (up to a weak equivalence) by such a system ℳ X with the properties required by nonequivariant minimal algebras.