We show that a conjecture by Lawson holds, that is, the inclusion from the Chow variety $C_{p,d}(P^n)$ of all effective algebraic p-cycles of degree d in n-dimensional projective space to the space of effective algebraic p-cycles is 2d-connected. As a result, the homotopy and homology groups of $C_{p,d}(P^n)$ are calculated up to 2d. We also show an analogous statement for Chow variety $C_{p,d}(¶^n)$ over algebraically closed fields of arbitrary characteristic and compute their etale homotopy groups up to 2d.
We study the cohomology theory and the canonical Milnor-Witt cycle module associated to a motivic spectrum. We prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules, thus generalising Déglise's thesis. As a corollary, we recover a theorem of Ananyevskiy and Neshitov and we prove that the Milnor-Witt K-theory groups are birational invariants.
We compute the perverse delta-homotopy heart of the motivic stable homotopy category over a base scheme with a dimension function delta, rationally or after inverting the exponential characteristic in the equicharacteristic case. In order to do that, we define the notion of homological Milnor-Witt cycle modules and construct a homotopy-invariant Rost-Schmid cycle complex. Moreover, we define the category of cohomological Milnor-Witt cycle modules and show a duality result in the smooth case.
This short note regards 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 proper complex variety with good reduction mod $p$.
We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional cycles. Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme $X$ and a divisor $D \subset X$, we construct a canonical homomorphism from the Chow groups with modulus $\CH^i(X|D)$ to the relative motivic cohomology groups $H^{2i}(X|D, \Z(i…
We compute the quantized cycle class of a closed embedding of complex manifolds defined by Grivaux using homotopy perturbation theory. In the case of a diagonal embedding, our approach provides a novel perspective of the usual Todd class of a complex manifold.
We study the interplay of the homotopy coniveau tower, the Rost-Schmid complex of a strictly homotopy invariant sheaf, and homotopy modules. For a strictly homotopy invariant sheaf $M$, smooth $k$-scheme $X$ and $q \geqslant 0$ we construct a novel cycle complex $C^*(X, M, q)$ and we prove that in favorable cases, $C^*(X, M, q)$ is equivalent to the homotopy coniveau tower $M^{(q)}(X)$. To do so we establish moving lemmas for the Rost-Schmid complex. As an application we deduce a cycle complex model for Milnor-Witt motivic cohomology. Furthermore we prove that if $M$ is a strictly homotopy inv…
Making use of Gruson-Raynaud's technique of "platification par eclatement", Kerz and Strunk proved that the negative homotopy K-theory groups of a Noetherian scheme X of Krull dimension d vanish below -d. In this note, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy K-theory groups vanish below -1. Furthermore, in the case of cyclic quotient singularities, we provide an explicit "upper bound" for the first negative homotopy K-theory group.
In this note we study the functoriality of the coniveau filtration in motivic homotopy theory via a moving lemma over a base scheme, extending previous works of Levine and Bachmann-Yakerson. The main result is that the motivic stable homotopy category can be modeled on a smaller site, the "smooth-smooth site". The proof is based on a new approach to the purity theorem of Morel-Voevodsky using specialization maps, which turns out to hold even in absence of the $\mathbb{A}^1$-homotopy invariance property. Applications to the homotopy coniveau tower and to higher Chow-Witt groups are given.
In this note we study the local projective model structure on presheaves of complexes on a site, i.e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.
In this note we show that the protruncated shape of a spectral $\infty$-topos is a delocalization of its profinite stratified shape. This gives a way to reconstruct the extended étale homotopy groups (i.e., the non-profinitely complete étale homotopy groups) of a coherent scheme from its profinite Galois category.
We study the category $DM(S)$ of Beilinson motives (as described by Cisinski and Deglise) over a more or less general base scheme $S$, and establish several nice properties for a version $t_{hom}(S)$ of the perverse homotopy $t$-structure (essentially defined by Ayoub) for it. $t_{hom}(S)$ is characterized in terms of certain stalks of an $S$-motif $H$ and its Tate twists at fields over $S$; it is closely related to certain coniveau spectral sequences for the cohomology of (the Borel-Moore motives of) arbitrary finite type $S$-schemes. We conjecture that the heart of $t_{hom}(S)$ is given by c…
In this paper, we prove a form of purity property for the $(\mathbb{P}^1, \infty)$-invariant replacement $h_0^{\overline{\square}}(\mathfrak{X})$ of the Yoneda object $\mathbb{Z}_{\rm tr} (\mathfrak{X})$ for a modulus pair $\mathfrak{X}=(\overline{X}, X_\infty)$ over a field $k$, consisting of a smooth projective $k$-scheme and an effective Cartier divisor on it. As application, we prove the analogue in the modulus setting of Voevodsky's fundamental theorem on the homotopy invariance of the cohomology of homotopy invariant sheaves with transfers, based on a main result of "Purity of reciprocit…
The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, Rülling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of $0$-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of $0$-cycles with modulus and show their torsion and divisibility properties. Modulus is being brought to sheaf theory by Kah…
In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its $E_2$-page. Our description of the $E_2$-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work.
In this note we explain that homotopy coherent simplicial nerve has to used intead of the standard definition in the author's papers on formal deformation theory. A convenient version of the notion of fibered category is presented which is useful once one works with simplicial categories.
In this note we prove some structural properties of all the A1-homotopy invariants of corner skew Laurent polynomial algebras. As an application, we compute de mod-l algebraic K-theory of Leavitt path algebras using solely the kernel/cokernel of the incidence matrix. This leads naturally to some vanishing and divisibility properties of the algebraic K-theory of these algebras.
This short note is extracted from Section 3 and Appendix of the paper entitled with Intersection de Rham complexes in positive characteristic by the same named authors, where an explicit infinite homotopy from a Higgs complex to the Frobenius pushforward of the corresponding de Rham complex in positive characteristic has been provided. The verification details, which are omitted therein, are provided here.
Those are the notes of the two talks I gave in april 2013 in St-John (US Virgin Islands) during the Simons Symposium on non-Archimedean and tropical geometry. They essentially consist of a survey of Hrushovski and Loeser's work on the homotopy type of Berkovich spaces; the last section explains how the author has used their work for studying pre-image of skeleta.
We give a proof of the fact tha the subset of the rational curves form a closed analytic subset in the space of the 1-dimensional cycles of a complex space.
We provide some corrections and clarifications to the paper [Gr3] of the title. In particular, we clarify the "left/right" conventions on complex reflection groups and their braid groups. Most importantly, we fill in a gap related to the treatment of cuts in the Picard-Lefschetz theory part of the argument. The statements of the main results are not affected.
Motivated by the definition of homotopy $L_\infty$ spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a $2$-category with invertible $2$-morphisms, and that certain fiber product property holds in this $2$-category. In a subsequent paper, we construct the virtual fundamental cycle of a compact oriented Kuranishi manifold, and prove some of its basic properties. Manifest from this new formulation is the fact that $[0,1]$-type homotopy $L_\infty$ spaces are naturally Kuranishi manifolds. The former structured s…
