We define algebraic cobordism of classifying spaces, Ω^*(BG) and G-equivariant algebraic cobordism Ω^*_G(-) for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted F^j(Ω^*(-)), which are required for the definition to work. We show that G-equivariant cobordism satisfies the localization exact sequence. We calculate Ω^*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n+1). We also calculate Ω^*(BG) when G is a finite abelian group. A finite non-abelian group for which we calculate Ω^*(BG) is the quaternion group of order 8. In all the above cases, we check that Ω^*(BG) is isomorphic to MU^*(BG).
We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins-Morel isomorphism from the quotient of the algebraic cobordism spectrum by the generators of the Lazard ring to the motivic Eilenberg-MacLane spectrum, again after inverting the positive residue characteristics.
We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $\mathbb{P}^\infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $\mathbb{C}$ and passing to spaces of $\mathbb{C}$-valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we sho…
We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.
This is a sequel to our previous paper of oriented bivariant theory [14]. In 2001 M. Levine and F. Morel constructed algebraic cobordism $Ω_*(X)$ for schemes $X$ over a field $k$ in an abstract way and later M. Levine and R. Pandhairpande reconstructed it more geometrically. In this paper in a similar manner we construct an algebraic cobordism $Ω^*(X \xrightarrow {π_X} S)$ for a scheme $X$ over a fixed scheme $S$ in such a way that if the target scheme $S$ is the point $pt = \operatorname{Spec} k$, then $Ω^{-i}(X \xrightarrow {π_X} pt)$ is isomorphic to Levine-Morel's algebraic cobordism $Ω_i(…
Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological $K_0$-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordis…
We introduce an equivariant algebraic cobordism theory Ω^G for algebraic varieties with G-action, where G is a split diagonalizable group scheme over a field k. It is done by combining the construction of the algebraic cobordism theory Ωby F. Morel and M. Levine, with the notion of (G, F)-formal group law with respect to a complete G-universe and complete G-flag F as introduced by M. Cole, J. P. C. Greenlees and I. Kriz. In particular, we use their corresponding representing ring L_G(F) in place of the Lazard ring L. We show that localization property and homotopy invariance property hold in Ω…
In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott--Samelson classes to smaller flag varieties.
In this paper, we study Segre classes in algebraic cobordism. We also prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in the algebraic cobordism of the Grassmannian bundle.
Profinite etale cobordism is a cohomology theory for smooth schemes of finite type over a field. Using an idea of Friedlander, it is constructed as an etale topological analog of the algebraic cobordism theories of Voevodsky and Levine-Morel. With finite coefficients, profinite etale cobordism has a converging Atiyah-Hirzebruch spectral sequence starting from etale cohomology. There are interesting comparison maps from algebraic to profinite etale cobordism. We conjecture that algebraic and etale cobordism with finite coefficients are isomorphic after inverting a Bott element. For the developm…
In this paper we study the structure of the Algebraic Cobordism ring of a variety as a module over the Lazard ring, and show that it has relations in positive codimensions. We actually prove the stronger graded version. This extends the result of M.Levine-F.Morel claiming that this module has generators in non-negative codimensions. As an application we compute the Algebraic Cobordism ring of a curve. The main tool is Symmetric Operations in Algebraic Cobordism.
We describe additive (unstable) operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^*. We prove that there is 1-to-1 correspondence between the set of operations, and the set of transformations: A^n((P^{\infty})^{\times r}) ---> B^m((P^{\infty})^{\times r}) satisfying certain simple properties. This provides an effective tool of constructing such operations. As an application, we prove that (unstable) additive operations in Algebraic Cobordism are in 1-to-1 correspondence with the L\otimes_Z Q-linear c…
We give a more detailed construction of the operation "intersection with a pseudo-divisor" in algebraic cobordism. Using arguments from Levine-Morel, Algebraic Cobordism, sections 6.2, 6.3, this gives a new proof of the contravariant functoriality of algebraic cobordism for l.c.i. morphisms of schemes of finite type over a field of characteristic zero.
An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P^1-spectra equipped with the symmetric monoidal structure described in arXiv:0709.3905v1 [math.AG]. The algebraic cobordism P^1-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation. For a commutative monoid E in the category SH(k) we identify the set of monoid homomorphisms from MGL to E in the motivic stable homotopy category with the set of all orientations of E. This result was stated originally in a s…
In this paper, we prove determinant formulas for the $K$-theory classes of the structure sheaves of degeneracy loci classes associated to vexillary permutations in type $A$. As a consequence we obtain determinant formulas for Lascoux-Schützenberger's double Grothendieck polynomials associated to vexillary permutations. Furthermore, we generalize the determinant formula to algebraic cobordism.
We associate a bivariant theory to any suitable oriented Borel-Moore homology theory on the category of algebraic schemes or the category of algebraic G-schemes. Applying this to the theory of algebraic cobordism yields operational cobordism rings and operational G-equivariant cobordism rings associated to all schemes in these categories. In the case of toric varieties, the operational T-equivariant cobordism ring may be described as the ring of piecewise graded power series on the fan with coefficients in the Lazard ring.
We construct a cohomology theory using quasi-smooth derived schemes as generators and an analogue of the bordism relation using derived fibre products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.
We prove the exactness of a descent sequence relating the algebraic cobordism groups of a scheme and its envelopes. Analogous sequences for Chow groups and K-theory were previously proved by Gillet.
We study the equivariant cobordism theory of schemes for action of linear algebraic groups. We compare the equivariant cobordism theory for the action of a linear algebraic groups with similar groups for the action of tori and deduce some consequences for the cycle class map of the classifying space of an algebraic groups.
In this article we construct Symmetric operations for all primes (previously known only for p=2). These unstable operations are more subtle than the Landweber-Novikov operations, and encode all p-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map L ---> Z[b_1,b_2,...], providing an important structure on Algebraic Cobordism. Applications include: questions of rationality of Chow group elements - see [11], and the structure of the Graded Algebraic Cobordism. We also construct Steenrod operations of T.tom Dieck-style…
In this paper we provide generalisations of Pfaffian formulas for the degeneracy loci classes in the algebraic cobordism of symplectic/odd orthogonal Grassmann bundles.
We define four distinct oriented bivariant theories associated with algebraic cobordism in its two versions (the axiomatic $Ω$ and the geometric $ω$), when applied to quasi-projective varieties over a field $k$. Specifically, we obtain contravariant analogues of the algebraic bordism group $Ω_*(X)$ and the double point bordism group $ω_*(X)$, for $X$ a quasi-projective variety, and covariant analogues of the algebraic cobordism ring $Ω^*(X)$ and the double point cobordism ring $ω^*(X)$, for $X$ a smooth variety. When the ground field has characteristic zero, we use the universal properties of …
For a reductive connected group or a finite group over a field of characteristic zero, we define an equivariant algebraic cobordism theory by a generalized version of the double point relation of Levine-Pandharipande. We prove basic properties and the well-definedness of a canonical fixed point map. We also find explicit generators of the algebraic cobordism ring of the point when the group is finite abelian.
We define a notion of Fourier-Mukai transform on algebraic cobordism cycles with $\mathbb{Q}$-coefficients on an abelian variety. We use this to produce a Beauville decomposition of algebraic cobordism and study its consequences, including a decomposition of the cobordism motive of an abelian variety.
In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration $Ω^*_{(r)}(X)$ are unions of finitely presented $\mathbb{L}$-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules $\mathbb{L}/I(p,n)x$ where $\mathrm{deg\ } x\ge \frac{p^n-1}{p-1}$. As a corollar…
The algebraic cobordism group of a scheme is generated by cycles that are proper morphisms from smooth quasiprojective varieties. We prove that over a field of characteristic zero the quasiprojectivity assumption can be omitted to get the same theory.
We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson-Thomas theory.…
Following an idea of Totaro, we prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of $\ell$-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.
The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.
We compute generators for the Chow ring of the classifying space of PGL_3 (over the complex numbers) as defined by Totaro. We also find enough relations after inverting 3. We show that this ring is not generated by Chern classes (this is the first example of this kind among classical groups) and prove that Totaro's refined cycle class map to a quotient of complex cobordism of BPGL_3 is surjective
We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with any Bott-Samelson class in algebraic cobordism of the grassmannian. We also establish some results for generalized Schubert polynomials for hyperbolic formal group laws.
Lee and Pandharipande studied a "double point" algebraic cobordism theory of varieties equipped with vector bundles, and speculated that some features of that story might extend to the case of varieties with principal G-bundles. This note shows that this expectation holds rationally, and more generally after inverting the torsion index of the group, for reductive G. We show that (after inverting the torsion index) the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism, that the theory for a point is dual to Omega^*(BG), and describe how the theory f…
We prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We then provide a geometric interpretation to the double beta-polynomials of Fomin and Kirillov by specializing our formula to the case of connected K-theory.
We extend the derived Algebraic bordism of Lowrey and Schürg to a bivariant theory in the sense of Fulton and MacPherson, and establish some of its basic properties. As a special case, we obtain a completely new theory of cobordism rings of singular quasi-projective schemes. The extended cobordism is shown to specialize to algebraic $K^0$ analogously to Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.
The purpose of this article is to show that the bivariant algebraic $A$-cobordism groups considered previously by the author are independent of the chosen base ring $A$. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define th…
We prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We specialise our formula to connective K-theory providing a geometric interpretation to the double beta-polynomials of Fomin and Kirillov by computing the fundamental classes of Schubert varieties. As a corollary we obtain a Thom-Porteous formula generalising those of the Chow ring and of the Grothendieck ring of vector bundles.
We define and study the notion of numerical equivalence on algebraic cobordism cycles. We prove that algebraic cobordism modulo numerical equivalence is a finitely generated module over the Lazard ring, and it reproduces the Chow group modulo numerical equivalence. We show this theory defines an oriented Borel-Moore homology theory on schemes and oriented cohomology theory on smooth varieties. We compare it with homological equivalence and smash-equivalence for cobordism cycles. For the former, we show that homological equivalence on algebraic cobordism is strictly finer than numerical equival…
This paper deals with the existence of algebraic structures on compact Nash sets. We introduce the algebraic-topological notion of asymmetric Nash cobordism between compact Nash sets, and we prove that a compact Nash set is semialgebraically homeomorphic to a real algebraic set if and only if it is asymmetric Nash cobordant to a point or, equivalently, if it is strongly asymmetric Nash cobordant to a real algebraic set. As a consequence, we obtain new large classes of compact Nash sets semialgebraically homeomorphic to real algebraic sets. To prove our results, we need to develop new algebraic…
We describe all operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^* (in the case of a field of characteristic zero). We prove that such an operation can be reconstructed out of it's action on the products of projective spaces. This reduces the construction of operations to algebra and extends the additive case done earlier, as well as the topological one obtained by T.Kashiwabara. The key new ingredients which permit us to treat the non-additive operations are: the use of "poly-operations" and the "D…
We construct and study a theory of bivariant cobordism of derived schemes. Our theory provides a vast generalization of the algebraic bordism theory of characteristic 0 algebraic schemes, constructed earlier by Levine and Morel, and a (partial) non-$\Ab^1$-invariant refinement of the motivic cohomology theory $MGL$ in Morel--Voevodsky's stable motivic homotopy theory. Our main result is that bivariant cobordism satisfies the projective bundle formula. As applications of this, we construct cobordism Chern classes of vector bundles, and establish a strong connection between the cobordism cohomol…
For a linear algebraic group $G$ over a field $k$, we define an equivariant version of the Voevodsky's motivic cobordism $MGL$. We show that this is an oriented cohomology theory with localization sequence on the category of smooth $G$-schemes and there is a natural transformation from this functor to the functor of equivariant motivic cohomology. We give several applications. In particular, we use this equivariant motivic cobordism to study the cobordism ring of the classifying spaces and the cycle class maps from the algebraic to the singular cohomology of such spaces. This theory of motivic…
Let S be an essentially smooth scheme over a field of characteristic exponent c. Let MGL and HZ denote the algebraic cobordism spectrum and the motivic cohomology spectrum over S, respectively. We show that the canonical map MGL/(a1, a2, ...) -> HZ induced by the additive orientation of motivic cohomology becomes an equivalence after inverting c. As an application, we prove the convergence of the Atiyah-Hirzebruch spectral sequence for all Z[1/c]-local Landweber exact motivic spectra.
Let $G$ be a connected linear algebraic group over a field $k$ of characteristic zero. For a principal $G$-bundle $π: E \to X$ over a scheme $X$ of finite type over $k$ and a parabolic subgroup $P$ of $G$, we describe the rational algebraic cobordism and higher Chow groups of the flag bundle $E/P \to X$ in terms of the cobordism of $X$ and that of the classifying space of a maximal torus of $G$ contained in $P$. As a consequence, we also obtain the formula for the cobordism and higher Chow groups of the principal bundles over the scheme $X$. If $X$ is smooth, this describes the cobordism ring …
In this paper, we study the algebraic cobordism spectrum $MSL$ in the motivic stable homotopy category of Voevodsky over an arbitrary perfect field $k$. Using the motivic Adams spectral sequence, we compute the geometric part of the $η$-completion of $MSL$ (modulo the maximal subgroup that is $l$-divisble for all primes $l\neq2, char k$). As an application, we study the Krichever's elliptic genus with integral coefficients, restricted to $MSL$. We determine its image, and identify its kernel as the ideal generated by differences of $SL$-flops. This was proved by B. Totaro in the complex analyt…
We show that the cycle map on a variety X, from algebraic cycles modulo algebraic equivalence to integer cohomology, lifts canonically to a topologically defined quotient of the complex cobordism ring of X. This more refined cycle map gives a topological proof that the Griffiths group is nonzero for some varieties X, without any use of Hodge theory. We also use this more refined cycle map to give examples of torsion algebraic cycles which map to 0 in Deligne cohomology but are not algebraically equivalent to 0, thus answering some questions by Colliot-Thelene and Schoen.
We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S) there is a unique morphism g : MSp -> BO of commutative ring T- spectra which sends the Thom class th^{MSp} to the Thom class th^{BO}. We show that the induced morphism of bigraded cohomology theories MSp^{*,*} -> BO^{*,*} is isomorphic to the morphism of bigraded cohomology theories obtained by applying to MSp^{*,*} the "change of (simply graded) coefficients rings" MSp^{4*,2*} -> BO^{4*,2*}. This is an algebraic version of the theorem of Conner and Floyd reconstructing real K-t…
We reconstruct derived Witt groups via special linear algebraic cobordism. There is a morphism of ring cohomology theories which sends the canonical Thom class in special linear cobordism to the Thom class in the derived Witt groups. We show that for every smooth variety X this morphism induces an isomorphism between MSL^{*,*}(X)[h^{-1}] with the "extended" coefficient ring MSL^{4*,2*}(pt) -> W^{2*}(pt) and Laurent polynomial ring over the derived Witt groups W^*(X), where h is the stable Hopf map. This result is an analogue of the result by Panin and Walter reconstructing hermitian K-theory u…
In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant $ν^+$: we study its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.
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…
The construction of double point cobordism groups of vector bundles on varieties in the work [Lee-P] (arXiv:1002.1500 [math.AG]) of Yuan-Pin Lee and Rahul Pandharipande gives immediately double point cobordism groups of filtered vector bundles on varieties. We note also that among the four basic operations -- direct sum, tensor product, dual, and Hom -- on vector bundles on varieties, only taking dual is compatible with double point cobordisms of vector bundles on varieties in general, by a demonstration on an example of vector bundles on Calabi-Yau 3-folds. A question on refined and/or higher…
We examine various versions of oriented cohomology and Borel-Moore homology theories in algebraic geometry and put these two together in the setting of an "oriented duality theory", a generalization of Bloch-Ogus twisted duality theory. This combines and exends work of Panin and Mocanasu. We apply this to give a Borel-Moore homology version $MGL'_{*,*}$ of Voevodsky's $MGL^{*,*}$-theory, and a natural map $\vartheta:Ω_*\to MGL'_{2*,*}$, where $Ω_*$ is the algebraic cobordism theory defined by Levine-Morel. We conjecture that $\vartheta$ is an isomorphism and describe a program for proving this…
