相关论文: Examples of non-algebraic classes in the Brown-Pet…

Generalizing the classical work of Atiyah and Hirzebruch on non-algebraic classes, recently Quick proved the existence of torsion non-algebraic elements in the Brown-Peterson tower. We construct non-torsion non-algebraic elements in the…

Atiyah and Hirzebruch gave examples ofeven degree torsion classes in the singularcohomology of a smooth complex projective manifold, which arenot Poincar\'{e} dual to an algebraiccycle. We notice that the order ofthese classes are small…

For each odd prime p, we exhibit p-groups G of p-rank two such that (suitably defined) Chern classes of unitary representations of G fail to generate the following rings: 1. The even degree integral cohomology of G; 2. The final page of the…

In this note we discuss some examples of non torsion and non algebraic cohomology classes for varieties over finite fields. The approach follows the construction of Atiyah-Hirzebruch and Totaro.

We find new examples of complex surfaces with countably many non-isomorphic algebraic structures. Here is one such example: take an elliptic curve $E$ in $\mathbb P^2$ and blow up nine general points on $E$. Then the complement $M$ of the…

Let A=k+A_1+A_2.... be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A)=k(Y)[t,t^{-1},sigma], where sigma is an…

We use Brown-Peterson cohomology to obtain lower bounds for the higher topological complexity, TC_k(RP^n), of real projective spaces, which are often much stronger than those implied by ordinary mod-2 cohomology.

For an odd prime $p$, we study the image of the Thom map from Brown-Peterson cohomology of $BPU_n$ to the ordinary cohomology in dimensions $0\leq i\leq 2p+2$, where $BPU_n$ is the classifying space of the projective unitary group $PU_n$.…

We produce refinements of the known multiplicative structures on the Brown--Peterson spectrum $BP$, its truncated variants $BP\langle n \rangle$, Ravenel's spectra $X(n)$, and evenly graded polynomial rings over the sphere spectrum.…

We give new examples of algebraic integral cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of our examples have dimension 6, the lowest possible.…

It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be…

For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital algebra ${\mathcal B}$ over $\mathbb{C}$, its {\em projective spectrum} $P(A)$ or $p(A)$ is the collection of $z\in \mathbb{C}^n$, or respectively $z\in \mathbb{P}^{n-1}$ such…

The survey is devoted to associative $\Z_{\ge0}$-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We consider examples of such algebras…

We present some simple examples of smooth projective varieties in positive characteristic, arising from linear algebra, which do not admit a lifting neither to characteristic zero, nor to the ring of second Witt vectors. Our first…

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in…

We prove that for every reductive algebraic group $H$ with centre of positive dimension and every integer $K$ there is a smooth and projective variety $X$ and an algebraic $H$-torsor $P \to X$ such that the classifying map $X \to \Bclass H$…

A recent result of ours [GM] shows that all Hopf algebra liftings of a given diagram in the sense of Andruskiewitsch and Schneider are cocycle deformations of each other. Here we develop a "non-abelian" cohomology theory, which gives a…

We construct varieties B(r;An) such that a map X -> B(r;An) corresponds to a degree-n \'etale algebra on X equipped with r generating global sections. We then show that when n = 2, i.e., in the quadratic \'etale case, that the singular…

A particular case of Bergeron-Venkatesh's conjecture predicts that torsion classes in the cohomology of Shimura varieties are rather rare. According to this and for Kottwitz-Harris-Taylor type of Shimura varieties, we first associate to…

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a ``cohomology theory" for complex algebraic varieties. The theory is bigraded, functorial,…