Related papers: Rationality of secondary classes

We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as $\mathbb P^3$ or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are smooth…

In view of the Segal construction each category with a coherent operation gives rise to a cohomology theory. Similarly each open stable differential relation $R$ imposed on smooth maps of manifolds determines cohomology theories $k^*$ and…

We prove that stably isomorphic vector bundles of rank d-1 on a smooth affine d-fold X over an algebraically closed field k are indeed isomorphic, provided d! is invertible in k. This answers an old conjecture of Suslin.

Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a P^d-bundle Y over a smooth variety Z. This conjecture is known if d>1, if dim(X)<5, or if Z admits a finite morphism to an Abelian…

Let $C$ be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to $C$. We prove that the decorated double…

Supersymmetric heterotic string models, built from a Calabi-Yau threefold $X$ endowed with a stable vector bundle $V$, usually lead to an anomaly mismatch between $c_2(V)$ and $c_2(X)$; this leads to the question whether the difference can…

Let $X$ be a smooth projective variety of dimension $n$, and let $E$ be an ample vector bundle over $X$. We show that any non-zero Schur class of $E$, lying in the cohomology group of bidegree $(n-1, n-1)$, has a representative which is…

A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is…

We sketch a geometric proof of the classical theorem of Atiyah, Bott, and Shapiro \cite{ABS} which relates Clifford modules to vector bundles over spheres. Every module of the Clifford algebra $Cl_k$ defines a particular vector bundle over…

In this paper, complex vector bundles of rank $r$ over $8$-dimensional spin$^{c}$ manifolds are classified in terms of the Chern classes of the complex vector bundles and the cohomology ring of the manifolds, where $r = 3$ or $4$. As an…

We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large…

We prove Conjecture F from [VW12] which states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. Moreover, we generalize this conjecture…

We prove that the norm of the Euler class E for flat vector bundles is $2^{-n}$ (in even dimension $n$, since it vanishes in odd dimension). This shows that the Sullivan--Smillie bound considered by Gromov and Ivanov--Turaev is sharp. We…

In this paper, we give a combinatorial formula for the \v{C}ech cocycles representing the power sums of the Chern roots of a holomorphic vector bundle over a complex manifold. By an observation motivation by author's previous paper, we also…

We study the physics of multiple M5-branes compactified on a hyperbolic 3-manifold. On the one hand, it leads to the 3d-3d correspondence which maps an $\mathcal{N}=2$ superconformal field theory to a pure Chern-Simons theory on the…

We consider a compact complex manifold $M$, and introduce the notion of two holomorphic Banach bundles $E,F$ over $M$ being compact perturbations of one another. Given two such bundles we show that if the cohomology groups $H^q(M,E)$ are…

Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and the…

Kontsevich conjectured that $\text{BDiff}(M, \text{rel }\partial)$ has the homotopy type of a finite CW complex for all compact $3$-manifolds with non-empty boundary. Hatcher-McCullough proved this conjecture when $M$ is irreducible. We…

A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X,L), the cohomologies of L over the GIT quotient X // G equal the invariant part of…

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra…