Related papers: Degeneration of the modified diagonal cycle

We prove that the vanishings of the modified diagonal cycles of Gross and Schoen govern the $\mathbb{Z}$-linear relations between small $m$-diagonals $\text{pt}^{\{1,\ldots,n\}\setminus A}\times\Delta_A$ in the rational Chow ring of $X^n$…

In this note we demonstrate some unexpected properties that simple gluings of the simplest derived categories may have. We consider two special cases: the first is an augmented curve, i.e., the gluing of the derived categories of a point…

Let $G$ be a graph on $n$ vertices and $t$ an integer. The reconfiguration graph of $G$, denoted by $R_t(G)$, consists of all $t$-colorings of $G$ and two $t$-colorings are adjacent if they differ on exactly one vertex. The $t$-recoloring…

The degeneration of the Quillen metric for a one-parameter family of Riemann surfaces has been studied by Bismut-Bost and Yoshikawa. In this article we propose a more geometric point of view using Deligne's Riemann-Roch theorem. We obtain…

A family of periodic perturbations of an attracting robust heteroclinic cycle defined on the two-sphere is studied by reducing the analysis to that of a one-parameter family of maps on a circle. The set of zeros of the family forms a…

Let f be a degeneration of Kahler manifolds. The local invariant cycle theorem states that for a smooth fiber of the degeneration, any cohomology class, invariant under the monodromy action, rises from a global cohomology class. Instead of…

We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi-Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.

Connections between the second Chow group of a smooth projective variety and its third unramified cohomology group, with coefficients the roots of unity twisted twice, feature in several recent works. In this note we revisit a 1996 paper by…

We prove that the infinitesimal invariant of a higher Chow cycle of type (2,3-g) on a generic abelian variety of dimension g<4 gives rise to a meromorphic Siegel modular form of (virtual) weight Sym^{4}det^{-1} with bounded singularity, and…

We introduce a geometric refinement of Gromov-Witten invariants for $\mathbb P^1$-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a…

We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show…

For $N \geq 4$ we classify the $(N-3)$-degenerate smooth CR maps of the three-dimensional unit sphere into the $(2N-1)$-dimensional unit sphere. Each of these maps has image being contained in a five-dimensional complex-linear space and is…

Weak degeneracy of a graph is a variation of degeneracy that has a close relationship to many graph coloring parameters. In this article, we prove that planar graphs with distance of $3$-cycles at least 2 and no cycles of lengths $5, 6, 7$…

I construct regulator indecomposable higher Chow cycles in elliptic surfaces satisfying certain conditions. As an application I give an alternative proof of a theorem of Gordon and Lewis, which asserts that there is a real regulator…

Let $M$ and $N$ be modules over an artin algebra such that $M$ degenerates to $N$. We show that any submodule of $M$ degenerates to a submodule of $N$. This suggests that a composition series of $M$ will in some sense degenerate to a…

Based on results on Hurwitz-Brill-Noether theory obtained by H. Larson we give a picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve of genus $g$. This picture starts from irreducible components of $W^r_d(C)$…

For any acyclic quiver, Keller-Scherotzke provided a stratifying functor from the category of finite-dimensional modules of the singular Nakajima category to the derived category of the quiver. Under this functor, a degeneration of strata…

In this paper we show that Bloch's higher cycle class map with finite coefficients for quasi-projective equi-dimensional schemes over a field fits naturally in a long exact sequence involving Schreieder's refined unramified cohomology. We…

We consider a family of quintic threefolds specializing to a certain reducible threefold. We describe the space of genus zero stable morphisms to the central fiber (as defined by J. Li). As an elementary application of an extension of the…

Let $\mathcal S\to\mathbb A^1$ be a smooth family of surfaces whose general fibre is a smooth surface of $\mathbb P^3$ and whose special fibre has two smooth components, intersecting transversally along a smooth curve $R$. We consider the…