Local acyclic fibrations and the de Rham complex

On algebraic fiber spaces

We construct a cohomological mixed Hodge complex on a fiber of a weakly semistable fibration using logarithmic de Rham complex.

De Rham上同调与非Witt空间的分层解释

Stratified Interpretation for De Rham Cohomology and Non-Witt Spaces

本文建立层对应过滤空间与分层de Rham复形理论研究奇异空间，证明分层de Rham上同调与相交上同调的同构，并发展非Witt空间的分层结构理论，为奇异几何与拓扑不变量研究提供新基础。

Rational $p$-adic Hodge theory for $d$-de Rham-proper stacks

In this follow-up paper we show that smooth Hodge-proper stacks over $\mathcal O_K$ are $\mathbb Q_p$-locally acyclic: namely the natural map between étale $\mathbb Q_p$-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the $\mathbb Q_p$-case of general conjectures made in our previous work. As a corollary, we get that if a smooth Artin stack over $K$ has a smooth Hodge-proper model over $\mathcal O_K$, its $\mathbb Q_p$-étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more gener…

Gauss-Manin connection and t-adic geometry

We show that the de Rham cohomology of any separated and smooth rigid variety over a field of Laurent series of characteristic zero carries a natural formal meromorphic connection, which we call the Gauss-Manin connection. We compare it with the Gauss-Manin connection of a proper and smooth variety over a curve, and with the Gauss-Manin connection of the Milnor fibration at an isolated complex hypersurface singularity.

Involutions, obstructions and mirror symmetry

Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya $A_\infty$ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Fl…

奇异概形上微分模复形的完备化

Completions of complexes of differential modules on singular schemes

本文针对域上有限型奇异概形推广Spencer上同调理论，否定了原Vinogradov猜想，借鉴Hartshorne的形式完备化方法在一类奇异概形上构造适用的Spencer复形，并对相应猜想给出更肯定的回答。