We formulate a Beilinson–Bernstein-type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping algebra at a root of 1 with fixed regular Harish-Chandra central character and the category of certain twisted $D$-modules on the corresponding quantized flag manifold. We show that the proof is reduced to a statement about the (derived) global sections of the ring of differential operators on the quantized flag manifold. We also give a reformulation of the conjecture in terms of the (derived) induction functor.

We prove a variant of a formula due to Zhang relating the Beilinson–Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

Let $(X,H)$ be a polarized smooth projective surface satisfying $H^1(X,{\mathcal{O}}_X)=0$, and let $\mathcal{F}$ be either a rank 1 torsion-free sheaf or a rank 2 ${\mu }_H$-stable vector bundle on $X$. Assume that $c_1\left(\mathcal{F}\right)\ne 0$. This article shows that the rank 2—respectively, rank 4—tautological sheaf ${\mathcal{F}}^{\left[2\right]}$ associated with $\mathcal{F}$ on the Hilbert square $X^{\left[2\right]}$ is $\mu$-stable with respect to a certain polarization.

A projective log variety $(X,D)$ is called a log Fano manifold if $X$ is smooth and if $D$ is a reduced simple normal crossing divisor on $X$ with $-(K_X+D)$ ample. The $n$-dimensional log Fano manifolds $(X,D)$ with nonzero $D$ are classified in this article when the log Fano index $r$ of $(X,D)$ satisfies either $r\ge n/2$ with $\rho \left(X\right)\ge 2$ or $r\ge n-2$. This result is a partial generalization of the classification of logarithmic Fano $3$-folds by Maeda.

We introduce a completely integrable system on the Grassmannian of 2-planes in an $n$-space associated with any triangulation of a polygon with $n$ sides, and we compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.

In this paper, the authors characterize pointwise multipliers for Campanato spaces on the Gauss measure space $({\mathbb{R}}^n,|\cdot |,\gamma )$, which includes $BMO\left(\gamma \right)$ as a special case. As applications, several examples of the pointwise multipliers are given. Also, the authors give an example of a nonnegative function in $BMO\left(\gamma \right)$ but not in $BLO\left(\gamma \right)$.

We prove new cases of the direct summand conjecture using fundamental theorems in $p$-adic Hodge theory due to Faltings. The cases tackled include the ones when the ramification locus lies entirely in characteristic $p$.