The Frölicher spectral sequence can be arbitrarily non degenerate

On the monodromy conjecture for non-degenerate hypersurfaces

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations -- the Denef--Loeser conjecture on topological zeta functions -- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of thr…

Enriques diagrams, arbitrarily near points, and Hilbert schemes

Given a smooth family F/Y of geometrically irreducible surfaces, we study sequences of arbitrarily near T-points of F/Y; they generalize the traditional sequences of infinitely near points of a single smooth surface. We distinguish a special sort of these new sequences, the strict sequences. To each strict sequence, we associate an ordered unweighted Enriques diagram. We prove that the various sequences with a fixed diagram form a functor, and we represent it by a smooth Y-scheme. We equip this Y-scheme with a free action of the automorphism group of the diagram. We equip the diagram with weig…

特征 $p$ 下族的上同调中任意大的跳跃

Arbitrarily large jumps in the de Rham and Hodge cohomology of families in characteristic $p$

本文在特征 $p$ 下构造了光滑射影代数簇族，利用有限平坦群概型的分类栈，使纤维的 de Rham 和 Hodge 上同调群维数发生任意大跳跃，并阐述了 Godeaux--Serre 簇的构造。