For a reduced hyperplane arrangement, we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement’s complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: Every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite-dimensional linear algebra. Finally, we give some $\mathscr {D}_{X}$-module applications: For example, we give a sharp restriction on the codimension one components of the multivariate Bernstein–Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito’s result that the roots of the Bernstein–Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$

We introduce $F$-gauges over a prism, construct syntomic cycle classes, and prove the prismatic Poincaré duality for proper smooth schemes.

We prove a version of the Lefschetz hyperplane theorem for fppf cohomology with coefficients in any finite commutative group scheme over the ground field. As consequences, we establish new Lefschetz results for the Picard scheme.

We prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.

We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$-fold delooping of the filtered loop space $E_2$-groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$-cogroupoid object in the $\infty $-category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on $S^1$, as well as the Todd class of the Lie algebroid $\mathbb {T}_{X}$; this is an invariant of a smooth and proper scheme X that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontrivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally, we record some consequences of this bit of structure at the level of Hochschild cohomology.

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.

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable $C^{\infty }$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable $C^{\infty }$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.

Abstract prepared by Rodrigo Figueiredo.

E-mail: [email protected]

URL: https://doi.org/10.11606/T.45.2019.tde-28042019-181150

In this paper we give an interpretation, in terms of derived de Rham complexes, of Scholze's de Rham period sheaf and Tan and Tong's crystalline period sheaf.

We provide a new formalism of de Rham–Witt complexes in the logarithmic setting. This construction generalises a result of Bhatt–Lurie–Mathew and agrees with those of Hyodo–Kato and Matsuue for log-smooth schemes of log-Cartier type. We then use our construction to study the monodromy action and slopes of Frobenius on log crystalline cohomology.

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.

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 $\text{SL}\left( 2,\mathbb{C} \right)$-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauß-Manin connection on the natural family of the smooth $\text{SL}\left( 2,\mathbb{C} \right)$-representation varieties of the one-holed torus.

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.

Given a mixed Hodge module $\mathcal{N}$ and a meromorphic function $f$ on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module $\mathcal{N}\otimes \mathcal{E}^{f}$, which extends the construction of Esnault et al. ($E_{1}$-degeneration of the irregular Hodge filtration (with an appendix by Saito), J. reine angew. Math. (2015), doi:10.1515/crelle-2014-0118). We show the strictness of the push-forward filtered ${\mathcal{D}}$-module through any projective morphism ${\it\pi}:X\rightarrow Y$, by using the theory of mixed twistor ${\mathcal{D}}$-modules of Mochizuki. We consider the example of the rescaling of a regular function $f$, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss–Manin systems of $f$ in terms of the Harder–Narasimhan filtration of the Kontsevich bundles associated with $f$.

This work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k-algebra and k has characteristic 0, we show that two objects, S1⊗A and ϵ(A), determine one another, functorially in A. The object S1⊗A is the S1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S1 :=Bℤ, while the object ϵ(A) is the de Rham algebra of A, endowed with the de Rham differential, and viewed as a ϵ-dg-algebra (see the main text). We define an equivalence φ between the homotopy theory of simplicial commutative S1-equivariant k-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1 ⊗A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying the S1-equivariant functions on the derived loop space LX of a smooth k-scheme X with the algebraic de Rham cohomology of X/k. As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separated k-schemes. By construction, these decompositions are moreover compatible with the S1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.

We introduce a valuation-theoretic approach to the problem of semistable reduction (i.e. existence of logarithmic extensions on suitable covers) of overconvergent isocrystals with Frobenius structure. The key tool is the quasicompactness of the Riemann–Zariski space associated to the function field of a variety. We also make some initial reductions, which allow attention to be focused on valuations of height 1 and transcendence degree 0.

We show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case (m = 0), not all invariant forms are closed. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

This paper addresses several questions related to the Hodge conjecture. First of all we consider the question, asked by Maillot and Soulé, whether the Hodge conjecture can be reduced to the case of varieties defined over number fields. We show that this is the case for the Hodge classes whose corresponding Hodge locus is defined over a number field. We also give simple criteria for this last condition to be satisfied. Finally we discuss the relation between this condition and the notion of absolute Hodge class.

In this article, we prove that a free divisor in a three-dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. Calderón-Moreno et al. conjectured this implication in all dimensions and proved it in dimension two. We prove a theorem that describes in all dimensions a special minimal system of generators for the module of formal logarithmic vector fields. This formal structure theorem is closely related to the formal decomposition of a vector field by Kyoji Saito and is used in the proof of the above result. Another consequence of the formal structure theorem is that the truncated Lie algebras of logarithmic vector fields up to dimension three are solvable. We give an example that this may fail in higher dimensions.

We prove that holonomic arithmetical ${\mathcal{D}}$-modules over curves have finite fibers. We also define L-functions associated with arithmetical ${\mathcal{D}}$-modules and, when the scheme is a curve, we show a cohomological formula. Furthermore, we prove that F-isocrystals over curves are holonomic.