Mauricio D. Garay
We use Kiehl-Verdier's and Houzel's finiteness theorems in the setting of local analytic geometry, and the Whitney-Thom theory of stratified spaces, to prove that fibrewise constructible complex of sheaves have coherent direct images. We review the notions of stratification theory which are needed.
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We show that for quasi-compact quasi-separated schemes of finite dimension, the constructibility condition in real étale cohomology agrees with a notion of constructibility arising naturally from topology. As application we prove that the derived direct image functor preserves constructibility under some assumptions, and compute the Grothendieck group of the constructible rational stable motivic homotopy category for reasonable schemes. We prove the generic base change property for constructible real étale sheaves, and deduce the same property for rational motivic spectra and $b$-sheaves.
We prove a finiteness theorem and a comparison theorem in the theory of étale cohomology of rigid analytic varieties. By a result of Huber, for a quasi-compact separated morphism of rigid analytic varieties with target being of dimension $\le1$, the compactly supported higher direct image preserves quasi-constructibility. Though the analogous statement for morphisms with higher dimensional target fails in general, we prove that, in the algebraizable case, it holds after replacing the target with a modification. We deduce it from a known finiteness result in the theory of nearby cycles over gen…
On finiteness of relative log pluricanonical representations
该研究在复解析框架下证明了相对对数多重典范表示的有限性，建立了复解析空间对数典型翻转的存在性，归约了两类丰度猜想，丰富了代数几何中相关理论。
We prove that the cohomology groups of an etale Q_p-local system on a smooth proper rigid analytic space are finite-dimensional Q_p-vector spaces, provided that the base field is either a finite extension of Q_p or an algebraically closed nonarchimedean field containing Q_p. This result manifests as a special case of a more general finiteness result for the higher direct images of a relative (phi, Gamma)-module along a smooth proper morphism of rigid analytic spaces over a mixed-characterstic nonarchimedean field.
Motivated by the finiteness of the set of automorphisms Aut(X) of a projective manifold X, and by Kobayashi-Ochiai's conjecture that a projective manifold dim(X)-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of Aut(X) for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold X which is (dim(X) -- 1)-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-(dim(X) -- 1)-analytically hyperbol…
Finiteness and duality of cohomology of $(φ,Γ)$-modules and the 6-functor formalism of locally analytic representations
本文通过分析堆和6- functor形式主义，研究了局部解析表示的上同调性质，并推广了Kedlaya-Pottharst-Xiao的结果。
We show that for n $\ge$ 2 the space of closed n-cycles in a strongly (n -- 2)-concave complex space has a natural structure of reduced complex space locally of finite dimension and represents the functor "analytic family of n-cycles" parametrized by Banach analytic sets.
On a complex symplectic manifold we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification (b) in the algebraic case. This extends our previous results in which the symplectic manifold was compact. The main tool is a finiteness theorem for R-constructible sheaves on a real analytic manifold in a non proper situation.
Kolyvagin proved that the Tate-Shafarevich group of an elliptic curve over Q of analytic rank 0 or 1 is finite, and that its algebraic rank is equal to its analytic rank. A program of generalisation of this result to the case of some motives which are quotients of cohomology motives of Shimura varieties is offered. We prove some steps of this program, mainly for quotients of $H^7$ of Siegel sixfolds. For example, we "almost" find analogs of Kolyvagin's trace and reduction relations. Some steps of the present paper are new contribution, because they have no analogs in the case of elliptic curve…
In this paper, the notion of local algebraic fundamental groups of normal complex analytic singularities are generalized to certain profinite groups called $D$-local algebraic fundamental groups which turns out to be useful even for the study of usual local algebraic fundamental groups and the Lefshetz type theorem for $D$-local algebraic fundamental groups is proved under certain conditions. The theorem yields, for example, the finiteness of the local algebraic fundamental groups of a certain class of four dimensional singularities and will be useful for the classification of three dimensiona…
We prove a version of faithfully flat descent in rigid analytic geometry, for almost perfect complexes and without finiteness assumptions on the rings involved. This extends results of Drinfeld for vector bundles.
A ridgeline correspondence criterion: the number of modes of a Gaussian mixture is finite
该研究通过结合脊线理论代数表述与Ax函数超越定理，证明有限多元高斯混合等类混合模型的模态数有限，扩展了同方差高斯混合的相关有限性结果。
Condensed Mathematics and Complex Geometry
本文通过凝聚数学方法重新阐述复几何的经典理论，重点证明紧复流形的有限性、塞雷双对偶、GAGA定理及Grothendieck-Hirzebruch-Riemann-Roch定理。
From Logarithmic Limit Sets to Algebraicity
研究复代数环面中闭解析子簇的对数极限集与代数性的关系，引入有限对数型条件，证明满足该条件且对数极限集为预期维数有限有理球面多面体复形的子簇是代数的，并完全解决一维情形。
Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems
本文在Banach-Colmez空间框架下证明了真p-adic刚性解析空间上pro-étale Q_p-局部系的有限性、庞加莱对偶及最优上同调消失界，并推广至算术上同调。
On the cohomology of p-adic analytic spaces, II: The $C_{\rm st}$-conjecture
本文针对光滑dagger簇提出并证明Fontaine猜想的类比，统一了proper与Stein两种极端情形，核心方法是将相关上同调几何化为拟Banach-Colmez空间并利用其精细性质。
A converse to Cartan's Theorem B: The extension property for real analytic and Nash sets
本文证明Cartan定理B在实解析情形的逆命题，并将刻画推广到Nash集，同时描述不可延拓函数集合，应用于半代数光滑函数与Nash函数一致性的判定。
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