Abstract We prove the S = T $S=T$ upper S equals upper T conjecture proposed by Xiao–Zhu in [XZ17], making use of Scholze’s theory of diamonds and v-stacks and Fargues–Scholze’s geometric Satake equivalence. Following [XZ18], we deduce the Eichler–Shimura relation for Shimura varieties of Hodge type.
Abstract We prove a conjecture due to Sturmfels and Uhler concerning the degree of the projective variety associated to the Gaussian graphical model of the cycle. We involve new methods based on the intersection theory on the space of complete quadrics.
Let k be a field endowed with a complex embedding, and let Gmot $\mathcal {G}_{\mathrm {mot}}$ script upper G Subscript mot be the associated motivic Galois group. The goal of this paper is to show that Gmot $\mathcal {G}_{\mathrm {mot}}$ script upper G Subscript mot admits a natural manifestation in anabelian geometry. Roughly speaking, we prove that Gmot $\mathcal {G}_{\mathrm {mot}}$ script upper G Subscript mot coincides with the automorphism group of the functor X↦π1geo(X) $X\mapsto \pi _1^{\mathrm {geo}}(X)$ upper X right arrow from bar pi 1 Superscript geo Baseline left parenthesis upper X right parenthesis sending a k-variety X to the geometric completion of its topological fundamental group. This can be considered as a motivic version of the Ihara–Matsumoto–Oda Conjecture, greatly generalised and proven by Pop. In a sequel to this paper, we will develop the parallel story for the ℓ $\ell $ script l -adic realisation. For this reason, many of the intermediate results are developed at a greater level of generality than strictly necessary for the main results proven here.
Let R^ ${\hat {R}}_{}$ ModifyingAbove upper R With caret be the I-adic completion of a commutative ring R with respect to a finitely generated ideal I. We give a necessary and sufficient criterion for the category of perfect complexes over R^ ${\hat {R}}_{}$ ModifyingAbove upper R With caret to be equivalent to the subcategory of dualizable objects in the derived category of I-complete complexes of R-modules. Our criterion is always satisfied when R is noetherian. When specialized to R local and noetherian and to I the maximal ideal, our theorem recovers a recent result of Benson, Iyengar, Krause, and Pevtsova.
We explore connections between existence of k $\Bbbk $ double struck k -rational points for Fano varieties defined over k $\Bbbk $ double struck k , a subfield of C $\mathbb {C}$ double struck upper C , and existence of Kähler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over k⊂C $\Bbbk \subset \mathbb {C}$ double struck k subset of double struck upper C admit (orbifold) Kähler–Einstein metrics if they do not have k $\Bbbk $ double struck k -rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit Kähler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus 12 $12$ 12 that admit Kähler–Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over k⊂C $\Bbbk \subset \mathbb {C}$ double struck k subset of double struck upper C whose geometric model is strictly K-semistable.
