Cohomological rigidity for toric Fano manifolds of small dimensions or large Picard numbers

光滑环面法诺四维簇的上同调刚性

Cohomological rigidity of smooth toric Fano fourfolds

该研究证明整上同调环分次同构的光滑环面法诺四维簇必微分同胚，确立其微分同胚类由上同调环决定的刚性性质。

Cohomological rigidity for Fano Bott manifolds

In the present paper, we characterize Fano Bott manifolds up to diffeomorphism in terms of three operations on matrix. More precisely, we prove that given two Fano Bott manifolds $X$ and $X'$, the following conditions are equivalent: (1) the upper triangular matrix associated to $X$ can be transformed into that of $X'$ by those three operations; (2) $X$ and $X'$ are diffeomorphic; (3) the integral cohomology rings of $X$ and $X'$ are isomorphic as graded rings. As a consequence, we affirmatively answer the cohomological rigidity problem for Fano Bott manifolds.

Stable cohomology of the universal Picard varieties and the extended mapping class group

We study the moduli spaces which classify smooth surfaces along with a complex line bundle. There are homological stability and Madsen--Weiss type results for these spaces (mostly due to Cohen and Madsen), and we discuss the cohomological calculations which may be deduced from them. We then relate these spaces to (a generalisation of) Kawazumi's extended mapping class groups, and hence deduce cohomological information about these. Finally, we relate these results to complex algebraic geometry. We construct a holomorphic stack classifying families of Riemann surfaces equipped with a fibrewise h…

Diffeomorphism classes of the doubling Calabi-Yau threefolds with Picard number two

Previously we constructed Calabi-Yau threefolds by a differential-geometric gluing method using Fano threefolds with their smooth anticanonical $K3$ divisors (New York J. Math. 20: 1-33, 2014). In this paper, we further consider the diffeomorphism classes of the resulting Calabi-Yau threefolds (which are called the doubling Calabi-Yau threefolds) starting from different pairs of Fano threefolds with Picard number one. Using the classifications of simply-connected $6$-manifolds in differential topology and the $λ$-invariant introduced by Lee (J. Math. Pures Appl. 141: 195-219, 2020), we prove t…

The Picard group of the category of $C_n$-equivariant stable homotopy theory

For a finite group $G$, there is a map $RO(G) \to {\rm Pic}(Sp^G)$ from the real representation ring of $G$ to the Picard group of $G$-spectra. This map is not known to be surjective in general, but we prove that when $G$ is cyclic this map is indeed surjective and in that case we describe ${\rm Pic}(Sp^G)$ explicitly. We also show that for an arbitrary finite group $G$ homology and cohomology with coefficients in a cohomological Mackey functor do not see the part of ${\rm Pic}(Sp^G)$ coming from the Picard group of the Burnside ring. Hence these homology and cohomology calculations can be gra…