arXiv subjects
Publications and source records attributed to Marc Levine.
We define the algebraic elliptic cohomology theory coming from Krichever's elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic cobordism ring with rational coefficients, the ideal generated by differences of classical flops coincides with the kernel of Krichever's elliptic genus. This generalizes a theorem of B. Totaro in the complex analytic setting.
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.
We show that the spectral sequence converging to the stable homotopy groups of spheres, induced by the Betti realization of the slice tower for the motivic sphere spectrum, agrees with the Adams-Novikov spectral sequence, after a suitable re-indexing. The proof relies on an extension of Deligne's décalage construction to the Tot-tower of a cosimplicial spectrum.
Let X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic p. We show that for any coherent sheaf E on X, the index of X divides the Euler-Poincaré characteristic χ(X,E) if p=0 or p>dim(X)+1. If 0 dim(X)+1). When p=0, such statements also have implications for the possible multiplicities of singular fibers in degenerations of complex projective varieties.
Ananyevsky has recently computed the stable operations and cooperations of rational Witt theory. These computations enable us to show a motivic analog of Serre's finiteness result: Theorem: Let $k$ be a field. Then $π^{\mathbb{A}^1}_ n(\mathbb{S}^- _k )_*$ is torsion for $n > 0$. As an application we define a category of Witt motives over $k$ and show that rationally this category is equivalent to the minus part of $SH(k)_\mathbb{Q}$.
Let $x_1, x_2,\ldots$ be a system of homogeneous polynomial generators for the Lazard ring $\mathbb{L}^*=MU^{2*}$ and let $MGL_S$ denote Voevodsky's algebraic cobordism spectrum in the motivic stable homotopy category over a base-scheme $S$.Take $S$ essentially smooth over a field $k$. Relying on Hopkins-Morel-Hoyois isomorphism of the 0th slice $s_0MGL_S$ for Voevodsky's slice tower with $MGL_S/(x_1, x_2,\ldots)$ (after inverting the characteristic of $k$), Spitzweck computes the remaining slices of $MGL_S$ as $s_nMGL_S=Σ^n_TH\mathbb{Z}\otimes \mathbb{L}^{-n}$ (again, after inverting the characteristic of $k$). We apply Spitzweck's method to compute the slices of a quotient spectrum $MGL_S/(\{x_i:i\in I\})$ for $I$ an arbitrary subset of $\mathbb{N}$, as well as the mod $p$ version $MGL_S/(\{p, x_i:i\in I\})$ and localizations with respect to a system of homogeneous elements in $\mathbb{Z}[\{x_j:j\not\in I\}]$. In case $S=\text{Spec}\, k$, $k$ a field of characteristic zero, we apply this to show that for $\mathcal{E}$ a localization of a quotient of $MGL$ as above, there is a natural isomorphism for the theory with support \[ Ω_*(X)\otimes_{\mathbb{L}^{-*}}\mathcal{E}^{-2*,-*}(k)\to \mathcal{E}^{2m-2*, m-*}_X(M) \] for $X$ a closed subscheme of a smooth quasi-projective $k$-scheme $M$, $m=$dim${}_kM$.
Let $k$ be a field of characteristic zero and let $(F,R)$ be a Landweber exact formal group law. We consider a Landweber exact $T$-spectrum $\mathcal{E}:=R\otimes_{\mathbb{L}}\text{MGL}$ and its effective cover $f_0\mathcal{E}\to \mathcal{E}$ with respect to Voevodsky's slice tower. The coefficient ring $R_0$ of $f_0\mathcal{E}$ is the subring of $R$ consisting of elements of $R$ of non-positive degree; the power series $F\in R[[u,v]]$ has coefficients in $R_0$ although $(F,R_0)$ is not necessarily Landweber exact. We show that the geometric part $X\mapsto f_0\mathcal{E}^*(X):=(f_0\mathcal{E})^{2*,*}(X)$ of $f_0\mathcal{E}$ is canonically isomorphic to the oriented cohomology theory $X\mapsto R_0 \otimes_{\mathbb{L}} Ω^*(X)$, where $Ω^*$ is the theory of algebraic cobordism, as defined by Levine-Morel. This recovers results of Dai-Levine as the special case of algebraic $K$-theory and its effective cover, connective algebraic $K$-theory.
We consider Voevodsky's slice tower for a finite spectrum E in the motivic stable homotopy category over a perfect field k. In case k has finite cohomological dimension (in characteristic two, we also require that k is infinite), we show that the slice tower converges, in that the induced filtration on the bi-graded homotopy sheaves for each term in the tower for E is finite, exhaustive and separated at each stalk. This partially verifies a conjecture of Voevodsky.
Let k be an algebraically closed field of characteristic zero. Let SH(k) denote the motivic stable homotopy category of T-spectra over k and SH the classical stable homotopy category. Let c:SH -> SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism π_n(E)-> π_{n,0}c(E) for all spectra E. Fix an embedding of k into the complex numbers and let Re:SH(k) -> SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum has Betti realization which is strongly convergent. This gives a spectral sequence "of motivic origin" converging to the homotopy groups of the classical sphere spectrum; this spectral sequence at E_2 agrees with the E_2 terms in the Adams-Novikov spectral sequence.
We examine the theory of connective algebraic K-theory, CK, defined by taking the -1 connective cover of algebraic K-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend CK to a bi-graded oriented duality theory (CK', CK) in case the base scheme is the spectrum of a field k of characteristic zero. The homology theory CK' may be viewed as connective algebraic G-theory. We identify CK' theory in bi-degree (2n, n) on some finite type k-scheme X with the image of K_0(M(X,n)) in K_0(M(X, n+1)), where M(X,n) is the abelian category of coherent sheaves on X with support in dimension at most n; this agrees with the (2n,n) part of the theory defined by Cai. We also show that the classifying map from algebraic cobordism identifies CK' with the universal oriented Borel-Morel homology theory Ω_*^{CK}:=Ω_*\otimes_L\Z[β] having formal group law u+v-βuv with coefficient ring \Z[β]. As an application, we show that every pure dimension d finite type k scheme has a well-defined fundamental class [X] in Ω_d^{CK}(X), and this fundamental class is functorial with respect to pull-back for lci morphisms. Finally, the fundamental class maps to the fundamental class in G-theory after inverting β, and to the fundamental class in CH after moding out by β.
Let k be a perfect field of characteristic different from two. We show that the filtration on the Grothendieck-Witt group GW(k) induced by the slice filtration for the sphere spectrum in the motivic stable homotopy category is the I-adic filtration, where I is the augmentation ideal in GW(k).
We study the slice filtration for S^1-spectra over a field k, and raise a number of questions regardings its properties. We show that the slices, except for the 0th slice, admit a further filtration whose layers are in a natural way the Eilenberg-Maclane spectra associated to a homotopy invariant complex of Nisnevic sheaves with transfer, i.e., an effective motive. We give an example to show that in general the 0th slice does not admit transfers.
M. Amer and A. Brumer have shown that, for two homogeneous quadratic polynomials f and g in at least 3 variables over a field k of characteristic different from 2, the locus f=g=0 has non-trivial solution over k if and only if, for a variable t, the equation f+tg=0 has a non-trivial solution over k(t). We consider a modified version of this result, and show that the projective variety over k defined by f_0=...=f_r=0, where the f_i are homogeneous polynomials over k of the same degree d\ge2 in n+1 variables (with n+1\ge r+2) , has a 0-cycle of degree 1 over k if and only if the generic hypersurface f_0+t_1f_1+...+t_rf_r=0 has a 0-cycle of degree 1 over k(t_1,...,t_r).
Relying on results of Hopkins-Morel, we show that, for $X$ a quasi-projective variety over a field of characteristic zero, the canonical map $Ω_n(X)\to MGL_{2n,n}'(X)$ is an isomorphism. Here $Ω_*(X)$ is the theory of algebraic cobordism defined by Levine-Morel, and $MGL_{*,*}'$ is the Borel-Moore homology version of the theory of algebraic cobordism defined via the algebraic Thom complex in the Morel-Voevodsky motivic stable homotopy category.
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 conjecture.
Following ideas of Bondarko, we construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme $S$ generated by the motives of smooth projective $S$-schemes, assuming that $S$ is itself smooth over a perfect field.
We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK_2(A) for a central simple algebra A of square-free index.
Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.