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For k a field of positive characteristic and X a smooth variety over k , we compute the Hochschild cohomology of Grothendieck's differential operators on X. The answer involves the derived inverse limit of the Frobenius acting on the cohomology of the structure sheaf of X. [ABSTRACT FROM AUTHOR]

We introduce the wrapped Donaldson--Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi--Yau 3-fold.We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori--Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi--Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry. [ABSTRACT FROM AUTHOR]

To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively, like moduli of local systems on a Riemann surface and moduli of Higgs bundles. For instance, B is a moduli space of microlocal sheaves, which generalize local systems, and D carries the structure of a complex integrable system. We show the Euler characteristics of these varieties count spanning subtrees of the graph, and the point-count over a finite field for B is a generating polynomial for spanning subgraphs. This polynomial satisfies a deletion-contraction relation, which we lift to a deletion-contraction exact triangle for the cohomology of B. There is a corresponding triangle for D. Finally, we prove that B and D are diffeomorphic, the diffeomorphism carries the weight filtration on the cohomology of B to the perverse Leray filtration on the cohomology of D, and all these structures are compatible with the deletion-contraction triangles. [ABSTRACT FROM AUTHOR]

For G and H 1 , ... , H n finite groups, does there exist a 3-manifold group with G as a quotient but no H i as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments. [ABSTRACT FROM AUTHOR]

Let K be a finite extension of the p -adic numbers Q p with ring of integers O K and residue field κ. Let X a regular scheme, proper, flat, and geometrically irreducible over O K of dimension d , and X K its generic fiber. We show, under some assumptions on X , that there is a reciprocity isomorphism of locally compact groups H a r 2 d − 1 (X K , Z (d)) ≃ π 1 a b (X K) W from the cohomology theory defined in [10] to an integral model π 1 a b (X K) W of the abelianized fundamental group π 1 a b (X K). After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups. The key ingredient is the duality result in [10]. [ABSTRACT FROM AUTHOR]

Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincaré duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some one of its Poisson cochain complex if and only if the Poisson structure is pseudo-unimodular. This generalizes the geometric version due to P. Xu. The modular derivation and Batalin-Vilkovisky operator are also described by using the dual basis of the Kähler differential module. [ABSTRACT FROM AUTHOR]

We show that the pure mapping class group \mathcal {N}_{g}^{k} of a non-orientable closed surface of genus g\geqslant 2 with k\geqslant 1 marked points has p-periodic cohomology for each odd prime p for which \mathcal {N}_{g}^{k} has p-torsion. Using the Yagita invariant and cohomology classes obtained from some representations of subgroups of order p, we obtain that the p-period is less or equal than 4 when g\geqslant 3 and k\geqslant 1. Moreover, combining the Nielsen realization theorem and a characterization of the p-period given in terms of normalizers and centralizers of cyclic subgroups of order p, we show that the p-period of \mathcal {N}_{g}^{k} is bounded below by 4, whenever \mathcal {N}_{g}^{k} has p-periodic cohomology, g\geqslant 3 and k\geqslant 0. These results provide partial answers to questions proposed by G. Hope and U. Tillmann. [ABSTRACT FROM AUTHOR]

We study the procedure of regularization in the context of the Lipschitz version of de Rham calculus on metric simplicial complexes with bounded geometry. It provides us with the machinery to handle the de Rham homomorphism for Lπ-cohomologies. In this respect, we obtain the condition resolving the question of triviality of the kernel for de Rham homomorphism. In particular, we specify the nontrivial cohomology classes explicitly for a sequence of parameters π = missing nonincreasing monotonicity. [ABSTRACT FROM AUTHOR]

For homogeneous polynomials $G_1,\ldots ,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork's theory. In this article, we will construct an explicit cochain map from the Dwork complex of $G_1,\ldots ,G_k$ to the Monsky–Washnitzer complex associated with some affine bundle over the complement $\mathbb {P}^n\setminus X_G$ of the common zero $X_G$ of $G_1,\ldots ,G_k$ , which computes the rigid cohomology of $\mathbb {P}^n\setminus X_G$. We verify that this cochain map realizes the rigid cohomology of $\mathbb {P}^n\setminus X_G$ as a direct summand of the Dwork cohomology of $G_1,\ldots ,G_k$. We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz's comparison results in [19] for projective hypersurface complements to arbitrary projective complements. [ABSTRACT FROM AUTHOR]

We define a transverse Dolbeault cohomology associated to any almost complex structure j on a smooth manifold M. This we do by extending the notion of transverse complex structure and by introducing a natural j -stable involutive limit distribution with such a transverse complex structure. We relate this transverse Dolbeault cohomology to the generalized Dolbeault cohomology of .M; j / introduced by Cirici and Wilson in 2001, showing that the .p; 0/ cohomology spaces coincide. This study of transversality leads us to suggest a notion of minimally non-integrable almost complex structure. [ABSTRACT FROM AUTHOR]

In this paper, we study diffeological spaces as certain kinds of discrete simplicial presheaves on the site of cartesian spaces with the coverage of good open covers. The Čech model structure on simplicial presheaves provides us with a notion of ∞ -stack cohomology of a diffeological space with values in a diffeological abelian group A. We compare ∞ -stack cohomology of diffeological spaces with two existing notions of Čech cohomology for diffeological spaces in the literature Krepski et al. (Sheaves, principal bundles, and Čech cohomology for diffeological spaces. (2021). arxiv:2111 01032 [math.DG]), Iglesias-Zemmour (Čech-de-Rham Bicomplex in Diffeology (2020). http://math.huji.ac.il/piz/documents/CDRBCID.pdf). Finally, we prove that for a diffeological group G, that the nerve of the category of diffeological principal G-bundles is weak homotopy equivalent to the nerve of the category of G-principal ∞ -bundles on X, bridging the bundle theory of diffeology and higher topos theory. [ABSTRACT FROM AUTHOR]

We exhibit a group of type F whose second cohomology contains a weakly bounded, but not bounded, class. As an application, we disprove a long-standing conjecture of Gromov about bounded primitives of differential forms on universal covers of closed manifolds. [ABSTRACT FROM AUTHOR]

A d$d$‐dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal (∞,n)$(\infty,n)$‐category of d$d$‐bordisms (embedded into R∞$\mathbb {R}^\infty$ and equipped with a tangential (X,ξ)$(X,\xi)$‐structure) that lands in the Picard subcategory of the target symmetric monoidal (∞,n)$(\infty,n)$‐category. We classify these field theories in terms of the cohomology of the (n−d)$(n-d)$‐connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the (∞,n)$(\infty,n)$‐category of bordisms with Ω∞−nMTξ$\Omega ^{\infty -n}MT\xi$ as an E∞$E_\infty$‐space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math. 202 (2009), no. 2, 195–239) in the case n=1$n=1$, and of Bökstedt–Madsen (An alpine expedition through algebraic topology, vol. 617, Contemp. Math., Amer. Math. Soc., Providence, RI, 2014, pp. 39–80) in the n$n$‐uple case. We also obtain results for the (∞,n)$(\infty,n)$‐category of d$d$‐bordisms embedding into a fixed ambient manifold M$M$, generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN 2011 (2011), no. 3, 572–608) in the case n=1$n=1$. We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of n$n$‐vector spaces (for n⩽4$n \leqslant 4$), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum (Forum Math. 25 (2013), no. 5, 1067–1106. arXiv:0912.4706). [ABSTRACT FROM AUTHOR]

We develop a version of Seiberg–Witten Floer cohomology/homotopy type for a spinc${\rm spin}^c$ 4‐manifold with boundary and with an involution that reverses the spinc${\rm spin}^c$ structure, as well as a version of Floer cohomology/homotopy type for oriented links with nonzero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3‐manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov‐type inequalities that relate topological quantities of 4‐manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, the Nielsen realization problem for nonspin 4‐manifolds, and nonsmoothable unoriented surfaces in 4‐manifolds. [ABSTRACT FROM AUTHOR]

We define and study a notion of minimal exponent for a local complete intersection subscheme 푍 of a smooth complex algebraic variety 푋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 푉-filtration associated to 푍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r ⁢ (O X) , where 푟 is the codimension of 푍 in 푋. We also study its relation to the Bernstein–Sato polynomial of 푍. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension 1 case. A key ingredient for our main result is a description of the Kashiwara–Malgrange 푉-filtration associated to any ideal (f 1 , ... , f r) in terms of the microlocal 푉-filtration associated to the hypersurface defined by ∑ i = 1 r f i ⁢ y i . [ABSTRACT FROM AUTHOR]

To a strongly 퐺-graded algebra 퐴 with 1-component 퐵, we associate the group Picent gr ⁢ (A) of isomorphism classes of invertible 퐺-graded (A , A) -bimodules over the centralizer of 퐵 in 퐴. Our main result is a Picent version of the Beattie–del Río exact sequence, involving Dade's group G ⁢ [ B ] , which relates Picent gr ⁢ (A) , Picent ⁢ (B) , and group cohomology. [ABSTRACT FROM AUTHOR]

A difference Lie group is a Lie group equipped with a difference operator, equivalently a crossed homomorphism with respect to the adjoint action. In this paper, first we introduce the notion of a representation of a difference Lie group, and establish the relation between representations of difference Lie groups and representations of difference Lie algebras via differentiation and integration. Then we introduce a cohomology theory for difference Lie groups and justify it via the van Est theorem. Finally, we classify abelian extensions of difference Lie groups using the second cohomology group as applications. [ABSTRACT FROM AUTHOR]

Let G${\bf G}$ be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers R${\mathbb {R}}$. We describe a method, implemented on computer, to find the first Galois cohomology set H1(R,G)${\rm H}^1({\mathbb {R}},{\bf G})$. The output is a list of 1‐cocycles in G${\bf G}$. Moreover, we describe an implemented algorithm that, given a 1‐cocycle z∈Z1(R,G)$z\in {\rm Z}^1({\mathbb {R}}, {\bf G})$, finds the cocycle in the computed list to which z$z$ is equivalent, together with an element of G(C)${\bf G}({\mathbb {C}})$ realizing the equivalence. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two types of U(1)-gauge field theories on G 2 -manifolds. One is the U(1)-Yang–Mills theory which admits the classical instanton solutions. We show that G 2 -manifolds emerge from the anti-self-dual U(1) instantons, which is an analogy of Yang's result for Calabi–Yau manifolds. The other one is the higher-order U(1)-Chern–Simons theory as a generalization of Kähler–Chern–Simons theory. We introduce the notion of higher-order U(1)-instanton, as the vacuum configurations of higher-order U(1)-Chern–Simons theory. By suitable choice of gauge and regularization technique, we calculate the partition function under semiclassical approximation. Finally, to make sure of the invariance at quantum level under the large gauge transformations, we use Deligne–Beilinson cohomology theory to give the higher-order U(1)-Chern–Simons actions (U(1)-BF-type actions) for nontrivial U(1)-principle bundles. [ABSTRACT FROM AUTHOR]

In this paper, we show the non-existence of finitely generated integral |$p$| -adic cohomology that satisfies finite étale descent and the associated rational cohomology coincides with rigid cohomology. [ABSTRACT FROM AUTHOR]

We give an explicit description of a diagonal map on the Bardzell resolution for any monomial algebra, and we use this diagonal map to describe the cup product on Hochschild cohomology. Then, we prove that the cup product is zero in positive degrees for triangular monomial algebras. Our proof uses the graded-commutativity of the cup product on Hochschild cohomology and does not rely on explicit computation of the Hochschild cohomology modules. [ABSTRACT FROM AUTHOR]

Any complex-analytic vector bundle E admits naturally defined homotheties ϕ α , α ∈ C ⁎ , i.e. ϕ α is the multiplication of a local section by a complex number α. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to E. Further, we compute the automorphism supergroup of a Π-symmetric super-Grassmannian Π Gr n , k , and, using Galois cohomology, we classify the real structures on Π Gr n , k and compute the corresponding supermanifolds of real points. [ABSTRACT FROM AUTHOR]

We study the weight 11 part of the compactly supported cohomology of the moduli space of curves |${\mathcal{M}}_{g,n}$|⁠ , using graph complex techniques, with particular attention to the case |$n = 0$|⁠. As applications, we prove new nonvanishing results for the cohomology of |${\mathcal{M}}_{g}$|⁠ , and exponential growth with |$g$|⁠ , in a wide range of degrees. [ABSTRACT FROM AUTHOR]

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi-Jacobi forms of index zero and their differential equations, which are given as vector fields. In the case of elliptic curves, we compute explicitly the Gauss–Manin connection and such vector fields. [ABSTRACT FROM AUTHOR]

We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with their bimodule category centres over the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology. [ABSTRACT FROM AUTHOR]

This paper considers the (negative) cyclic open–closed map O C - , which maps the cyclic homology of the Fukaya category of a symplectic manifold to its S 1 -equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that O C - intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara–Levelt–Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental–Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to O C - in the semisimple case; we also consider the non-semisimple case. [ABSTRACT FROM AUTHOR]

Davydov–Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category C are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center Z (C) relative to C . From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of Z (C) . Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras Λ C k ⋊ C [ Z 2 ] , the Taft algebras and the small quantum group of sl 2 at a root of unity. [ABSTRACT FROM AUTHOR]

For an algebraic torus S , Blinstein and Merkurjev have given an estimate of 3-rd unramified cohomology H ¯ n r 3 (F (S) , Q / Z (2)) obtained from a flasque resolution of S. Based on their work, for the norm one torus W = R K / F (1) G m with K / F abelian, we compute the 3-rd unramified cohomology H ¯ n r 3 (F (W) , Q / Z (2)). [ABSTRACT FROM AUTHOR]

We describe singular homology of a manifold X$X$ via simplices σ:Δd→X$\sigma :\Delta _d\rightarrow X$ that satisfy Stokes' formula with respect to all differential forms. The notion is geared to the case of the tame geometry (definable manifolds with respect to an o‐minimal structure), where it gives a description of the period pairing with de Rham cohomology via definable simplices. [ABSTRACT FROM AUTHOR]

The cosmological polytope and bootstrap programs have revealed interesting connections between positive geometries, modern on-shell methods and bootstrap principles studied in the amplitudes community with the wavefunction of the Universe in toy models of FRW cosmologies. To compute these FRW correlators, one often faces integrals that are too difficult to evaluate by direct integration. Borrowing from the Feynman integral community, the method of (canonical) differential equations provides an efficient alternative for evaluating these integrals. Moreover, we further develop our geometric understanding of these integrals by describing the associated relative twisted cohomology. Leveraging recent progress in our understanding of relative twisted cohomology in the Feynman integral community, we give an algorithm to predict the basis size and simplify the computation of the differential equations satisfied by FRW correlators. [ABSTRACT FROM AUTHOR]

The notion of O -operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O -operator is a pair consisting of two O -operators satisfying a compatibility relation. A compatible O -operator algebra is an algebra together with a bimodule and a compatible O -operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O -operators and compatible O -operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O -operators and compatible O -operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O -operators. [ABSTRACT FROM AUTHOR]

We study the cohomology of families of |$(\varphi ,\Gamma)$| -modules with coefficients in pseudoaffinoid algebras. We prove that they have finite cohomology, and we deduce an Euler characteristic formula and Tate local duality. We classify rank- |$1$| |$(\varphi ,\Gamma)$| -modules and deduce that triangulations of pseudorigid families of |$(\varphi ,\Gamma)$| -modules can be interpolated, extending a result of [ 29 ]. We then apply this to study extended eigenvarieties at the boundary of weight space, proving in particular that the eigencurve is proper at the boundary and that Galois representations attached to certain characteristic |$p$| points are trianguline. [ABSTRACT FROM AUTHOR]

We provide a cohomology of n-Hom–Lie color algebras, in particular, a cohomology governing oneparameter formal deformations. Then we also study formal deformations of the n-Hom–Lie color algebras and introduce the notion of Nijenhuis operator on an n-Hom–Lie color algebra, which may give rise to infinitesimally trivial (n − 1)th-order deformations. Furthermore, in connection with Nijenhuis operators, we introduce and discuss the notion of product structure on n-Hom–Lie color algebras. [ABSTRACT FROM AUTHOR]

For n a power of two, we give a complete description of the cohomology algebra H*(G͠n,3; ℤ2) of the Grassmann manifold G͠n,3 of oriented 3-planes in ℝn. We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for H*(G͠n,3; ℤ2). [ABSTRACT FROM AUTHOR]

We prove under certain conditions (local-global compatibility and vanishing of modulo |$p$| cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of |$\operatorname{\textsf{GL}}(N)$| over a CM field; we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as the first Tate-Shafarevich group of this motive. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. By a result of Y. Cai generalizing previous results by Galatius-Venkatesh on the graded cohomology group of a locally symmetric space, our maps relate the (non-Eisenstein) localization of the graded cohomology group for a locally symmetric space to the exterior algebra of the Selmer group of the Tate dual of the adjoint representation. We generalize this to Hida families as well. [ABSTRACT FROM AUTHOR]

We describe the ring structure of the rational cohomology of the Torelli groups of the manifolds |$\#^g S^n \times S^n$| in a stable range, for |$2n \geq 6$|⁠. Some of our results are also valid for |$2n=2$|⁠ , where they are closely related to unpublished results of Kawazumi and Morita. [ABSTRACT FROM AUTHOR]

Associated to a certain additive polynomial, we introduce an Artin–Schreier variety admitting an action of a finite unitary group. We calculate the character of the cohomology as a representation of a finite unitary group. One of our main ingredients is explicit character formulae for Weil representations of unitary groups due to Gérardin. We give another trace formula for a projective hypersurface admitting an action of a finite unitary group. [ABSTRACT FROM AUTHOR]

We prove that the rational cohomology |$H^{i}(\mathcal {T}_{g};\mathbf {Q})$| of the moduli space of trigonal curves of genus |$g$| is independent of |$g$| in degree |$i<\lfloor g/4\rfloor.$| This makes possible to define the stable cohomology ring as |$H^{\bullet }(\mathcal {T}_{g};\mathbf {Q})$| for a sufficiently large |$g.$| We also compute the stable cohomology ring, which turns out to be isomorphic to the tautological ring. This is done by studying the embedding of trigonal curves in Hirzebruch surfaces and using Gorinov–Vassiliev's method. [ABSTRACT FROM AUTHOR]

For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X\subset \mathbb {P}V \times \mathbb {P}V^{\vee }, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is the full flag variety of V, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0, the cohomology groups are described for all V by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When n=3, we recover the recursive description of characters from recent work of Linyuan Liu, while for general n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters. [ABSTRACT FROM AUTHOR]

We show the smoothness over the affine line of the Hodge moduli space of logarithmic t$t$‐connections of coprime rank and degree on a smooth projective curve with geometrically integral fibers over an arbitrary Noetherian base. When the base is a field, we also prove that the Hodge moduli space is geometrically integral. Along the way, we prove the same results for the corresponding moduli spaces of logarithmic Higgs bundles and of logarithmic connections. We use smoothness to derive specialization isomorphisms on the étale cohomology rings of these moduli spaces; this includes the special case when the base is of mixed characteristic. In the special case where the base is a separably closed field of positive characteristic, we show that these isomorphisms are filtered isomorphisms for the perverse filtrations associated with the corresponding Hitchin‐type morphisms. [ABSTRACT FROM AUTHOR]

In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order n + 1 deformation if and only if the obstruction class in the second cohomology group is trivial. [ABSTRACT FROM AUTHOR]

In this paper, we will compute the non-abelian cohomology of the universal complete curve in positive characteristic. This extends Hain's result on the non-abelian cohomology of generic curves in characteristic zero to positive characteristics. Furthermore, we will prove that the exact sequence of etale fundamental groups of the universal n-punctured curve in positive characteristic does not split. [ABSTRACT FROM AUTHOR]

On negatively curved compact manifolds, it is possible to associate to every closed form a bounded cocycle – hence a bounded cohomology class – via integration over straight simplices. The kernel of this map is contained in the space of exact forms. We show that in degree 2 this kernel is trivial, in contrast with higher degree. In other words, exact non-zero 2-forms define non-trivial bounded cohomology classes. This result is the higher dimensional version of a classical theorem by Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] for surfaces. As a consequence, one gets that the second bounded cohomology of negatively curved manifolds contains an infinite dimensional space, whose classes are explicitly described by integration of forms. This also showcases that some recent results by Marasco [Proc. Amer. Math. Soc. 151 (2023), pp. 2707–2715] can be applied in higher dimension to obtain new non-trivial results on the vanishing of certain cup products and Massey products. Some other applications are discussed. [ABSTRACT FROM AUTHOR]

The notion of Rota-Baxter groups was recently introduced by Guo, Lang and Sheng [19] in the geometric study of Rota-Baxter Lie algebras. They are closely related to skew braces as observed by Bardakov and Gubarev. In this paper, we study extensions of Rota-Baxter groups by constructing suitable cohomology theories. Among others, we find relations with the extensions of skew braces. Given an extension of Rota-Baxter groups, we also construct a short exact sequence connecting various automorphism groups, which generalizes the Wells short exact sequence. [ABSTRACT FROM AUTHOR]

Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct L ∞ -algebras from (quasi-)twilled Lie conformal algebras. And we show that the result of the twisting by a C [ ∂ ] -module homomorphism on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie conformal algebra if and only if the C [ ∂ ] -module homomorphism is a Maurer-Cartan element of the L ∞ -algebra. In particular, we show that relative Rota-Baxter type operators on Lie conformal algebras are Maurer-Cartan elements. Besides, we propose a new algebraic structure, called NS-Lie conformal algebras, that is closely related to twisted relative Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an application of twisting theory, we give the cohomology of twisted relative Rota-Baxter operators and study their deformations. [ABSTRACT FROM AUTHOR]

The main goal of this paper was to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, and α k -derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from the one given by Guan and Chen. This definition is inspired by the process of the queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we developed a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we established a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology. [ABSTRACT FROM AUTHOR]

Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra A$A$ forms a Lie algebra, and a restricted Lie algebra if A$A$ contains a field of characteristic p$p$. We deduce that the space of integrable classes in HH1(A)${\operatorname{HH}}^1(A)$ forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group. [ABSTRACT FROM AUTHOR]

Following ideas of Lurie, we give a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain, in particular, equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori. [ABSTRACT FROM AUTHOR]

A complete embedding is a symplectic embedding $\iota :Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $\iota (K)$ inside $M$. Under the assumption that the torsion exponents of $K$ are bounded, we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions. [ABSTRACT FROM AUTHOR]

Historically, it was known by the work of Artin and Mazur that the ℓ -adic homotopy type of a smooth complex variety with good reduction mod p can be recovered from the reduction mod p, where ℓ is not p. This short note removes this last constraint, with an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor of Mandell, we see that the étale comparison theorem in the prismatic theory reproduces the p-adic homotopy type for a smooth complex variety with good reduction mod p. [ABSTRACT FROM AUTHOR]