Volume 2016, Issue 713

Let S be a smooth affine algebraic curve, and let S˚${\text{{$\mathring{S}$}}}$ be the Riemann surface obtained by removing a point from S . We provide evidence for the congruence subgroup property of mapping class groups by showing that the congruence kernel ker Mod (S˚)^→ Out (π1(S˚)^)$ \ker \Bigl (\widehat{\mathrm {Mod}(\mathring{S})} \rightarrow \mathrm {Out}(\widehat{\pi _1(\mathring{S})})\Bigr ) $ lies in the centralizer of every braid in Mod (S˚)${{\mathrm {Mod}(\text{{$\mathring{S}$}})}}$. As a corollary, we obtain a new proof of Asada's theorem that the congruence subgroup property holds in genus one. We also obtain simple-connectivity of Boggi's procongruence curve complex 𝒞ˇ(S˚)${\check{\mathcal {C}}(\text{{$\mathring{S}$}})}$ when S is affine, and a new proof of Matsumoto's theorem that the congruence kernel depends only on the genus in the affine case.

Let f:X⤏X${f:X\dashrightarrow X}$ be a dominant rational map of a smooth projective variety defined over a characteristic 0 global field K , let δ f be the dynamical degree of f , and let hX:X(K¯)→[1,∞)${h_X:X({\bar{K}})\rightarrow [1,\infty )}$ be a Weil height relative to an ample divisor. We prove that for every ϵ>0${\epsilon >0}$ there is a height bound hX∘fn≪(δf+ϵ)nhX,$ h_X\circ f^n \ll (\delta _f+\epsilon )^n h_X, $ valid for all points whose f -orbit is well-defined, where the implied constant depends only on X , h X , f , and ϵ. An immediate corollary is a fundamental inequality α¯f(P)≤δf${\overline{\alpha }_f(P)\le \delta _f}$ for the upper arithmetic degree. If further f is a morphism and D is a divisor satisfying an algebraic equivalence f*D≡βD${f^*D\equiv \beta D}$ for some β>δf${\beta >\sqrt{\delta _f}}$, we prove that the canonical height h^f,D=limβ-nhD∘fn$ {{\hat{h}}_{f,D}=\lim \beta ^{-n}h_D\circ f^n} $ converges and satisfies h^f,D∘f=βh^f,D${{\hat{h}}_{f,D}\circ f=\beta {\hat{h}}_{f,D}}$ and h^f,D=hD+O(hX)${{\hat{h}}_{f,D}=h_D+O(\sqrt{h_X})}$. We also prove that the arithmetic degree αf(P)${\alpha _f(P)}$, if it exists, gives the main term in the height counting function for the f -orbit of P . We conjecture that α¯f(P)=δf${\overline{\alpha }_f(P)=\delta _f}$ whenever the f -orbit of P is Zariski dense and describe some cases for which we can prove our conjecture.

Using an analogue of Knebusch's generic splitting tower invariant in the theory of non-singular quadratic forms, we study the splitting behaviour of quasilinear (or Fermat-type) forms of degree p over fields of characteristic p > 0. Several new applications of our main results to the theory of quasilinear quadratic forms are provided, including an analogue of a theorem of Vishik relating to the existence of outer excellent connections in the motives of smooth projective quadrics over fields of characteristic different from 2, partial results towards a quasilinear version of Karpenko's theorem on the possible values of the first higher Witt indices of non-singular quadratic forms in characteristic not 2, and a proof of a conjecture of Hoffmann concerning quadratic forms with maximal splitting in the quasilinear case.

Inspired by recent work of Hernandez–Leclerc and Leclerc–Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q . As Leclerc–Plamondon have shown, the points of these varieties can be interpreted as representations of a category, which we call the (singular) Nakajima category 𝒮. We determine the quiver of 𝒮 and the number of minimal relations between any two given vertices. We construct a δ-functor Φ taking each finite-dimensional representation of 𝒮 to an object of the derived category of Q . We show that the functor Φ establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Φ. If the underlying graph of Q is an ADE Dynkin diagram, the image is the whole derived category; otherwise, it is the category of `line bundles over the non-commutative curve given by Q '. We show that the degeneration order between strata corresponds to Jensen–Su–Zimmermann's degeneration order on objects of the derived category. Moreover, if Q is an ADE Dynkin quiver, the singular category 𝒮 is weakly Gorenstein of dimension 1 and its derived category of singularities is equivalent to the derived category of Q .

We apply our integral formula of volumes to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by Witt Nyström to the effect that the sequence of spectral measures for the induced ℂ * -action on the central fiber converges to the canonical measure defined by the associated weak geodesic ray in the space of Kähler metrics. This limit measure coincides with the classical Duistermaat–Heckmann measure if the test configuration is product. As a consequence, we show that the algebraic p -norm of the test configuration is equal to the L p -norm of tangent vectors on the geodesic ray. Using this result, we give a natural energy theoretic explanation for the lower bound estimate on the Calabi functional by Donaldson, extending the statement to any p -norm ( p ≥ 1), and prove an analogous result for Kähler–Einstein metrics.

We discuss static Klein–Gordon–Maxwell–Proca systems in the critical case of 4-dimensional closed manifolds in the continuation of Hebey–Truong [J. reine angew. Math. 667 (2012), 221–248]. We prove phase stability for all possible phases in the nonpositively curved case, and in the positively curved case when the phase lies in some set that we show to be maximal in the case of the round sphere.

We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compact) simply connected solvable Lie groups and prove linear stability of many such metrics. These include an open set of two-step solvsolitons, all two-step nilsolitons, two infinite families of three-step solvable Einstein metrics, all nilsolitons of dimensions six or less, and all solvable Einstein metrics of dimension seven or less with codimension-one nilradical. For each linearly stable metric, dynamical stability follows from a generalization of the techniques of Guenther, Isenberg, and Knopf.

We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain interpolation properties that allow the use of maximal regularity theory and the application of a stability theorem of Simonett. With this, we derive two stability theorems, one for a class of Einstein metrics and one for a class of non-Einstein Ricci solitons. Using linear stability results of Jablonski, Petersen, and the first author, we obtain dynamical stability for many specific Einstein and Ricci soliton metrics on simply connected solvable Lie groups.

We extend our discrete uniformization theorems for planar, m -connected, Jordan domains [J. reine angew. Math. 670 (2012), 65–92] to closed surfaces of genus m ≥ 1.