We prove that the number of distinct homotopy types of limits of one-parameter semi-algebraic families of closed and bounded semi-algebraic sets is bounded singly exponentially in the additive complexity of any quantifier-free first order formula defining the family. As an important consequence, we derive that the number of distinct homotopy types of semi-algebraic subsets of $\mathbb{R}^k$ defined by a quantifier-free first order formula $Φ$, where the sum of the additive complexities of the polynomials appearing in $Φ$ is at most $a$, is bounded by $2^{(k+a)^{O(1)}}$. This proves a conjecture made by Basu and Vorobjov [On the number of homotopy types of fibres of a definable map, J. Lond. Math. Soc. (2) 2007, 757--776].
In this short note we show that the homotopy category of smooth compactifications of smooth algebraic varieties is equivalent to the homotopy category of smooth varieties over a field of characteristic zero. As an application we show that the functor sending a variety to the p-th step of the Hodge filtration of its complex cohomology is representable in the homotopy category of simplicial presheaves on smooth complex varieties. The main motivation are recent applications of homotopy theory to Deligne-Beilinson cohomology theories.
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 compute the set of naive pointed homotopy classes of endomorphisms of the projective line P^1 over the spectrum of a field. Our computation compares well with Fabien Morel's one of the motivic pointed homotopy classes of endomorphisms of P^1: there is an a priori monoid structure on the set of naive homotopy classes such that canonical map from this monoid to group of motivic homotopy classes is a group completion.
The purpose of this paper is to generalise Sullivan's rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toen's schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Maurer-Cartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the Baues-Lemaire conjecture. For compact Kaehler manifolds, the schematic homotopy groups can be described explicitly in terms of…
We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of "semi-algebraic differential forms" in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the DeRham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich's proof of the formality of the little cubes operad.
Using the machinery of etale homotopy theory a' la Artin-Mazur we determine the etale homotopy types of moduli stacks over $\bar{\Q}$ parametrizing families of algebraic curves of genus g greater than 1 endowed with an action of a finite group G of automorphisms, which comes with a fixed embedding in the mapping class group, such that in the associated complex analytic situation the action of G is precisely the differentiable action induced by this specified embedding of G in the mapping class group.
We study two different flavours of A^1-homotopy theory in the setting of spectral algebraic geometry, and compare them to classical A^1-homotopy theory. As an application we show that the spectral analogue of Weibel's homotopy invariant K-theory collapses to the classical theory. Along the way we give a new construction of nonconnective algebraic K-theory of stable infinity-categories via a generalization of the Bass-Thomason-Trobaugh construction.
In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.
We study the homotopy type of the simplicial set of continuous semi-algebraic simplexes of an algebraic variety defined over a real closed field, which we will call the real homotopy type. We prove an analogue of the theorem of Artin-Mazur comparing the real homotopy type with the étale homotopy type. This paper is part one of a sequence of papers on this topic.
We show the vanishing of the second homotopy group of the étale homotopy type of a smooth connected algebraic group over a separably closed field, completed away from the characteristic. This is an algebraic analogue of a classical theorem of Elie Cartan. Based on this result, we establish an explicit formula for the similarly completed second homotopy group of a homogeneous space.
We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. In a previous paper we have shown that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural "degree" filtration approximate closer and closer the homotopy type of the space of continuous maps and obtain bounds that describe the closeness of the approximation in terms of the degree. Moreover, we compute the homotopy groups of the spaces in low dimensions.
We construct a log algebraic version of the homotopy sequence for a quasi-projective normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic.
We prove that algebraic G-theory in is representable in unstable and stable motivic homotopy categories; in the stable category we identify it with the Borel-Moore theory associated to algebraic K-theory, and show that such an identification is compatible with the functorialities defined by Quillen and Thomason.
In this article, we show that the combination of the constructions done in SGA 6 and the A^1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.
We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic $K$-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.
Let $X$ be a homogeneous space of a connected linear algebraic group $G$ defined over the field of complex numbers $\mathbb C$. Let $x\in X({\mathbb C})$ be a point. We denote by $H$ the stabilizer of $x$ in $G$. When $H$ is connected, we compute the topological fundamental group $π_1^{\rm top}(X({\mathbb C}),x)$. Moreover, we compute the second homotopy group $π_2^{\rm top}(X({\mathbb C}),x)$.
C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way, we extend Bass-Quillen's fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-the…
Cazanave has identified the algebraic homotopy class of a rational function of $1$ variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question posed by Eisenbud for polynomial functi…
We show that the homotopy invariant algebraic K-theory of Weibel vanishes below the negative of the Krull dimension of a noetherian scheme. This gives evidence for a conjecture of Weibel about vanishing of negative algebraic K-groups.
We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A^1-homotopy type. Arithmetic considerations involving the sheaf of A^1-connected components lead us to introduce several different notions of connectedness in A^1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of al…
In this paper, the homotopy groups of Chow variety $C_{p,d}(P^n)$ of effective $p$-cycles of degree $d$ is proved to be stable in the sense that $p$ or $n$ increases. We also obtain a negative answer to a question by Lawson and Michelsohn on homotopy groups for the space of degree two cycles.
We study some basic properties of schematic homotopy types and the schematization functor. We describe two different algebraic models for schematic homotopy types: co-simplicial Hopf alegbras and equivariant co-simplicial algebras, and provide explicit constructions of the schematization functor for each of these models. We also investigate some standard properties of the schematization functor helpful for the description of the schematization of smooth projective complex varieties. In a companion paper these results are used in the construction of a non-abelian Hodge structure on the schemati…
We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\ell$. More precisely, we prove the following. Let $\R$ be a real closed field and let \[ {\mathcal P} = \{P_1,...,P_m\} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], \] with ${\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m$. Let $S \subset \R^{\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms …
Making use of the recent theory of noncommutative motives, we prove that every additive invariant satisfies Galois descent. Examples include mixed complexes, Hochschild homology, cyclic homology, periodic cyclic homology, negative cyclic homology, connective algebraic K-theory, mod-l algebraic K-theory, nonconnective algebraic K-theory, homotopy K-theory, topological Hochschild homology, and topological cyclic homology.
In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map, in terms of the format of its graph. In particular we show that if a semi-algebraic set $S \subset {\R}^{m+n}$, where $\R$ is a real closed field, is defined by a Boolean formula with $s$ polynomials of degrees less than $d$, and $π: {\R}^{m+n} \to {\R}^n$ is the projection on a subspace, then the number of different homotopy types of fibres of $π$ does not exceed $s^{2(m+1)n}(2^m nd)^{O(nm)}$. As applications of our main results we prove single exponential bounds o…
This dissertation comprises three collections of results, all united by a common theme. The theme is the study of categories via algebraic techniques, considering categories themselves as algebraic objects. This algebraic approach to category theory is central to noncommutative algebraic geometry, as realized by recent advances in the study of noncommutative motives. We have success proving algebraic results in the general setting of symmetric monoidal and semiring $\infty$-categories, which categorify abelian groups and rings, respectively. For example, we prove that modules over the semiring…
We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ` minimum degree\rq filtration b…
Let U be a unipotent group over the field of complex numbers C, acting on a complex algebraic variety X. Assume that there exists a surjective morphism of complex algebraic varieties f: X --> Y whose fibres are orbits of U. We show that if X and Y are smooth and all orbits of U in X have the same dimension, then the induced map on C-points X(C) --> Y(C) is a homotopy equivalence. Moreover, if U, X, Y, and f are defined over the field of real numbers R, then the induced map on R-points X(R) --> Y(R) is surjective and induces homotopy equivalences on connected components.
We explain how, given a plane algebraic curve $\mathcal{C}\colon f(x,y) = 0$, $x_1 \in \mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\varepsilon > 0$, we can compute $δ> 0$ such that $|y_j(x_1) - y_j(x_2)| < \varepsilon$ for all holomorphic functions $y_j(x)$ which satisfy $f(x, y_j(x)) = 0$ in a neighbourhood of $x_1$ and for all $x_2$ with $|x_1 - x_2| < δ$. Consequently, we obtain an algorithm for reliable homotopy continuation of plane algebraic curves. As an example application, we study continuous deformation of closed discrete Darboux transforms. Moreover, we discuss a scheme fo…
In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under $\ell$-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the s…
We prove an analogue of the Morel-Voevodsky localization theorem over spectral algebraic spaces. As a corollary we deduce a "derived nilpotent invariance" result which, informally speaking, says that A^1-homotopy invariance kills all higher homotopy groups of a connective commutative ring spectrum.
A homotopical treatment of intersection cohomology recently developed by Chataur-Saralegui-Tanré associates a "perverse algebraic model" to every topological pseudomanifold, extending Sullivan's presentation of rational homotopy to intersection cohomology. In this context, there is a notion of "intersection-formality", measuring the vanishing of Massey products in intersection cohomology. In the present paper, we study the perverse algebraic model of complex projective varieties with isolated singularities. We endow such invariant with natural mixed Hodge structures. This allows us to prove so…
This is an expended and revised version of the preprint "Schematization of homotopy types". The purpose of this work is to introduce a notion of \emph{affine stacks}, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and $p$-adic homotopy theory. This gives a first solution to A. Grothendieck's \emph{schematization problem} described in \cite{gr}. We also use affine…
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.
We introduce notions of unramified and totally ramified maps in great generality - for commutative rings, schemes, ring spectra, or derived schemes. We prove that the definition is equivalent to the classical definition in the case of rings of integers in number fields. The new definition leads directly (without computational techniques) to a calculation of topological Hochschild homology for rings of integers. We show that THH(R) is the homotopy cofiber of a map $R[ΩS^3]\otimes_RΩ^1_{R/\mathbb{Z}}\to R[ΩS^3\langle 3\rangle]$, so there is a long exact sequence $\cdots\to H_\ast(ΩS^3;Ω^1_{R/\ma…
For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/τ\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/τ$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/τ\text{-}\mathbf{Mod}_{harm}^b$ is equivalent to $\mathcal{D}^b({{BP}_*{BP}\text{-}\mathbf{Comod}}^{ev})$ as stable $\infty$-categories equipped with $t$-structures. As an application, for…
Extending a result of Schröer on a Grothendieck question in the context of complex analytic spaces, we prove that the surjectivity of the Brauer map $δ: Br(X) \rightarrow H_{\rm ét}^2(X,\mathbb{G}_{m, X})_{\rm tor}$ for algebraic schemes depends on their étale homotopy type. We use properties of algebraic $K(π, 1)$ spaces to apply this to some classes of proper and smooth algebraic schemes. In particular we recover a result of Hoobler and Berkovich for abelian varieties. Further, we give an additional condition for the surjectivity of $δ$ which involves pro-universal covers. All proposed condi…
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 paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that explicitly constructs an optimal start system, circumventing the typical bottleneck associated with polyhedral homotopy algorithms. The correctness of our algorithm follows from intersection theoretic computations of the algebraic degree of polynomial optimization programs and relies on explicitly solving the tropicalization of a corresponding Lagrange system.…
We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A^1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.
Using Serre's adelic interpretation of cohomology, we develop a `differential and integral calculus' on an algebraic curve X over an algebraically closed filed k of constants of characteristic zero, define algebraic analogs of additive multi-valued functions on X and prove corresponding generalized residue theorem. Using the representation theory of the global Heisenberg and lattice Lie algebras, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of glo…
For an algebraically closed field $k$ of characteristic 0, we give a cycle-theoretic description of the additive 4-term motivic exact sequence associated to the additive dilogarithm of J.-L. Cathelineau, that is the derivative of the Bloch-Wigner function, via the cubical additive higher Chow groups under one assumption. The 4-term functional equation of Cathelineau, an additive analogue of Abel's 5-term functional equation, is also discussed cycle-theoretically.
Numerical Algebraic Geometry uses numerical data to describe algebraic varieties. It is based on the methods of numerical polynomial homotopy continuation, an alternative to the classical symbolic approaches of computational algebraic geometry. We present a package, the driving idea behind which is to interlink the existing symbolic methods of Macaulay2 and the powerful engine of numerical approximate computations. The core procedures of the package exhibit performance competitive with the other homotopy continuation software.
We prove a conjecture of Morel identifying Voevodsky's homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy t-structure and have a trivial action of the Hopf map. This is done by relating these two kind of objects to Rost's cycle modules. Applications to algebraic cobordism and construction of cycle classes are given.
Motivated by the study of hyperkahler structures in moduli problems and hyperkahler implosion, we initiate the study of non-reductive hyperkahler and algebraic symplectic quotients with an eye towards those naturally tied to projective geometry, like cotangent bundles of blow-ups of linear arrangements of projective space. In the absence of a Kempf-Ness theorem for non-reductive quotients, we focus on constructing algebraic symplectic analogues of additive quotients of affine spaces, and obtain hyperkahler structures on large open subsets of these analogues by comparison with reductive analogu…
We introduce a functor $\mathfrak{M}:\mathbf{Alg}\times\mathbf{Alg}^\mathrm{op}\rightarrow\mathrm{pro}\text{-}\mathbf{Alg}$ constructed from representations of $\mathrm{Hom}_\mathbf{Alg}(A,B\otimes ? )$. As applications, the following items are introduced and studied: (i) Analogue of the functor $π_0$ for algebras and affine schemes. (ii) Cotype of Weibel's concept of strict homotopization. (iii) A homotopy invariant intrinsic singular cohomology theory for affine schemes with cup product. (iv) Some extensions of $\mathbf{Alg}$ that are enriched over idempotent semigroups. (v) Classifying homo…
