Consider a vector bundle with connection on a p$p$-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author’s 2010 book and on subsequent improvements by Baldassarri and by Poineau and Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point.

In Bennett et al. [BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305] it was conjectured that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.

A special linear Grassmann variety \text{SGr}(k,n)$\text{SGr}(k,n)$ is the complement to the zero section of the determinant of the tautological vector bundle over \text{Gr}(k,n)$\text{Gr}(k,n)$. For an SL$SL$-oriented representable ring cohomology theory A^{\ast }(-)$A^{\ast }(-)$ with invertible stable Hopf map {\it\eta}${\it\eta}$, including Witt groups and \text{MSL}_{{\it\eta}}^{\ast ,\ast }$\text{MSL}_{{\it\eta}}^{\ast ,\ast }$, we have A^{\ast }(\text{SGr}(2,2n+1))\cong A^{\ast }(pt)[e]/(e^{2n})$A^{\ast }(\text{SGr}(2,2n+1))\cong A^{\ast }(pt)[e]/(e^{2n})$, and A^{\ast }(\text{SGr}(k,n))$A^{\ast }(\text{SGr}(k,n))$ is a truncated polynomial algebra over A^{\ast }(pt)$A^{\ast }(pt)$ whenever k(n-k)$k(n-k)$ is even. A splitting principle for such theories is established. Using the computations for the special linear Grassmann varieties, we obtain a description of A^{\ast }(\text{BSL}_{n})$A^{\ast }(\text{BSL}_{n})$ in terms of homogeneous power series in certain characteristic classes of tautological bundles.

We prove by the Hilbert–Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves and of Hassett on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solves a question raised by Mumford and Gieseker, to prove the Chow asymptotic stability of stable nodal curves by the Hilbert–Mumford criterion.

We prove a global Torelli theorem for pairs (Y,D)$(Y,D)$ where Y$Y$ is a smooth projective rational surface and D\in |-K_{Y}|$D\in |-K_{Y}|$ is a cycle of rational curves, as conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.

The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton M$M$ with bounded scalar curvature S$S$, it is shown that the curvature operator \text{Rm}$\text{Rm}$ of M$M$ satisfies the estimate |\text{Rm}|\leqslant cS$|\text{Rm}|\leqslant cS$ for some constant c$c$. Moreover, the curvature operator \text{Rm}$\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound \text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, where r$r$ is the distance function to a fixed point in M$M$. As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.

We give explicit formulas for the Hilbert series of residual intersections of a scheme in terms of the Hilbert series of its conormal modules. In a previous paper, we proved that such formulas should exist. We give applications to the number of equations defining projective varieties and to the dimension of secant varieties of surfaces and three-folds.

The Frobenius number F(\boldsymbol{a})$F(\boldsymbol{a})$ of a lattice point \boldsymbol{a}$\boldsymbol{a}$ in \mathbb{R}^{d}$\mathbb{R}^{d}$ with positive coprime coordinates, is the largest integer which can not be expressed as a non-negative integer linear combination of the coordinates of \boldsymbol{a}$\boldsymbol{a}$. Marklof in [The asymptotic distribution of Frobenius numbers, Invent. Math. 181 (2010), 179–207] proved the existence of the limit distribution of the Frobenius numbers, when \boldsymbol{a}$\boldsymbol{a}$ is taken to be random in an enlarging domain in \mathbb{R}^{d}$\mathbb{R}^{d}$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor.

We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated ‘Latin square conjecture’ due to Alon and Tarsi or, more precisely, on the validity of an equivalent ‘column Latin square conjecture’ due to Huang and Rota.

Let G$G$ be a simple algebraic group. A closed subgroup H$H$ of G$G$ is said to be spherical if it has a dense orbit on the flag variety G/B$G/B$ of G$G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

Inspired by symplectic geometry and a microlocal characterizations of perverse (constructible) sheaves we consider an alternative definition of perverse coherent sheaves. We show that a coherent sheaf is perverse if and only if R{\rm\Gamma}_{Z}{\mathcal{F}}$R{\rm\Gamma}_{Z}{\mathcal{F}}$ is concentrated in degree 0$0$ for special subvarieties Z$Z$ of X$X$. These subvarieties Z$Z$ are analogs of Lagrangians in the symplectic case.

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

Let (G,G^{\prime })$(G,G^{\prime })$ be a type I irreducible reductive dual pair in Sp(W_{\mathbb{R}})$(W_{\mathbb{R}})$. We assume that (G,G^{\prime })$(G,G^{\prime })$ is in the stable range where G$G$ is the smaller member. Let K$K$ and K^{\prime }$K^{\prime }$ be maximal compact subgroups of G$G$ and G^{\prime }$G^{\prime }$ respectively. Let \mathfrak{g}=\mathfrak{k}\bigoplus \mathfrak{p}$\mathfrak{g}=\mathfrak{k}\bigoplus \mathfrak{p}$ and \mathfrak{g}^{\prime }=\mathfrak{k}^{\prime }\bigoplus \mathfrak{p}^{\prime }$\mathfrak{g}^{\prime }=\mathfrak{k}^{\prime }\bigoplus \mathfrak{p}^{\prime }$ be the complexified Cartan decompositions of the Lie algebras of G$G$ and G^{\prime }$G^{\prime }$ respectively. Let \widetilde{K}$\widetilde{K}$ and \widetilde{K}^{\prime }$\widetilde{K}^{\prime }$ be the inverse images of K$K$ and K^{\prime }$K^{\prime }$ in the metaplectic double cover \widetilde{\text{Sp}}(W_{\mathbb{R}})$\widetilde{\text{Sp}}(W_{\mathbb{R}})$ of Sp(W_{\mathbb{R}})$(W_{\mathbb{R}})$. Let {\it\rho}${\it\rho}$ be a genuine irreducible (\mathfrak{g},\widetilde{K})$(\mathfrak{g},\widetilde{K})$-module. Our first main result is that if {\it\rho}${\it\rho}$ is unitarizable, then except for one special case, the full local theta lift {\it\rho}^{\prime }={\rm\Theta}({\it\rho})${\it\rho}^{\prime }={\rm\Theta}({\it\rho})$ is equal to the local theta lift {\it\theta}({\it\rho})${\it\theta}({\it\rho})$. Thus excluding the special case, the full theta lift {\it\rho}^{\prime }${\it\rho}^{\prime }$ is an irreducible and unitarizable (\mathfrak{g}^{\prime },\widetilde{K}^{\prime })$(\mathfrak{g}^{\prime },\widetilde{K}^{\prime })$-module. Our second main result is that the associated variety and the associated cycle of {\it\rho}^{\prime }${\it\rho}^{\prime }$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation {\it\rho}^{\ast }${\it\rho}^{\ast }$ respectively. Finally we obtain some interesting (\mathfrak{g},\widetilde{K})$(\mathfrak{g},\widetilde{K})$-modules whose \widetilde{K}$\widetilde{K}$-spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent K_{\mathbb{C}}$K_{\mathbb{C}}$-orbits in \mathfrak{p}^{\ast }$\mathfrak{p}^{\ast }$.

We prove that the higher direct images of the structure sheaf under a birational and projective morphism between excellent and regular schemes vanish.

We introduce support varieties for rational representations of a linear algebraic group G$G$ of exponential type over an algebraically closed field k$k$ of characteristic p>0$p>0$. These varieties are closed subspaces of the space V(G)$V(G)$ of all 1-parameter subgroups of G$G$. The functor M\mapsto V(G)_{M}$M\mapsto V(G)_{M}$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between V(G)_{M}$V(G)_{M}$ and the family of support varieties V_{r}(G)_{M}$V_{r}(G)_{M}$ obtained by restricting the G$G$ action to Frobenius kernels G_{(r)}\subset G$G_{(r)}\subset G$. These support varieties seem particularly appropriate for the investigation of infinite-dimensional rational G$G$-modules.

The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme S$S$ there exists a motivict$t$-structure for the category \text{DM}_{c}(S)$\text{DM}_{c}(S)$ of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If S$S$ is of finite type over a field, then the heart of this t$t$-structure (the category of mixed motivic sheaves over S$S$) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over S$S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic t$t$-structure is transversal to the Chow weight structure for \text{DM}_{c}(S)$\text{DM}_{c}(S)$ (that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.

We present a new connection between the Hele-Shaw flow, also known as two-dimensional Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this, we prove short-time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

A finitely generated subgroup {\rm\Gamma}${\rm\Gamma}$ of a real Lie group G$G$ is said to be Diophantine if there is {\it\beta}>0${\it\beta}>0$ such that non-trivial elements in the word ball B_{{\rm\Gamma}}(n)$B_{{\rm\Gamma}}(n)$ centered at 1\in {\rm\Gamma}$1\in {\rm\Gamma}$ never approach the identity of G$G$ closer than |B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group G$G$ is said to be Diophantine if for every k\geqslant 1$k\geqslant 1$ a random k$k$-tuple in G$G$ generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5$5$, or derived length at most 2$2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

A technical ingredient in Faltings’ original approach to p$p$-adic comparison theorems involves the construction of K({\it\pi},1)$K({\it\pi},1)$-neighborhoods for a smooth scheme X$X$ over a mixed characteristic discrete valuation ring with a perfect residue field: every point x\in X$x\in X$ has an open neighborhood U$U$ whose generic fiber is a K({\it\pi},1)$K({\it\pi},1)$ scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in p$p$-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether normalization lemma.