Toric partial density functions and stability of toric varieties

Let $(L, h)\rightarrow (X, \omega )$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho }_{tk}:X\rightarrow \mathbb {R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along $Y$ , for fixed $t>0$ such that $tk\in \mathbb {N}$ . We prove the existence of a distributional expansion of $\hat{\rho }_{tk}$ as $k\rightarrow \infty $ , including the identification of the coefficient of $k^{n-1}$ as a distribution on $X$ . This expansion is used to give a direct proof that if $\omega $ has constant scalar curvature, then $(X, L)$ must be slope semi-stable with respect to $Y$ (cf. Ross and Thomas in J Differ Geom 72(3): 429–466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.     

Étale cohomology of a DM curve-stack with coefficients in \mathbb G _m

We compute étale cohomology groups $H_{\acute{\mathrm{e}}\mathrm{t}}^r(X, \mathbb G _m)$ in several cases, where $X$ is a connected smooth tame Deligne–Mumford stack of dimension $1$ over an algebraically closed field. We have complete results for orbicurves (and, more generally, for twisted nodal curves) and in the case all stabilizers are cyclic; we give partial results and examples in the general case. In particular, we show that if the stabilizers are abelian then $H_{\acute{\mathrm{e}}\mathrm{t}}^2(X, \mathbb{G }_m)$ does not depend on $X$ but only on the underlying orbicurve $Y$ and on the generic stabilizer. We show with two examples that, in general, the higher cohomology groups $H_{\acute{\mathrm{e}}\mathrm{t}}^r(X, \mathbb{G }_m)$ cannot be computed knowing only the base of the gerbe $X \rightarrow Y$ and the banding group.     

A subelliptic analogue of Aronson–Serrin’s Harnack inequality

We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE $$\begin{aligned} \partial _t u={-}X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu), \end{aligned}$$ in cylinders $\Omega \times (0,T)$ where $\Omega \subset M$ is an open subset of a manifold $M$ endowed with control metric $d$ corresponding to a system of Lipschitz continuous vector fields $X=(X_1,\ldots ,X_m)$ and a measure $d\sigma $ . We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincaré inequality in the metric measure space $(M,d,d\sigma )$ . We also show that such hypothesis hold for a class of Riemannian metrics $g_\epsilon $ collapsing to a sub-Riemannian metric $\lim _{\epsilon \rightarrow 0} g_\epsilon =g_0$ uniformly in the parameter $\epsilon \ge 0$ .     

Singular gradient flow of the distance function and homotopy equivalence

Let $M$ be a Riemannian manifold and let $\varOmega $ be a bounded open subset of $M$ . It is well known that significant information about the geometry of $\varOmega $ is encoded into the properties of the distance, $d_{\partial \varOmega }$ , from the boundary of $\varOmega $ . Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $x_0$ is a singular point of $d_{\partial \varOmega }$ then the generalized characteristic starting at $x_0$ stays singular for all times. As an application, we deduce that the singular set of $d_{\partial \varOmega }$ has the same homotopy type as $\varOmega $ .     

On the pluricanonical maps of varieties of intermediate Kodaira dimension

In this paper we will prove a uniformity result for the Iitaka fibration $f:X\rightarrow Y$ , provided that the generic fiber has a good minimal model and the Prokhorov–Shokurov conjecture holds. In particular, the result holds if the variation of $f$ is zero or $\kappa (X)=\text{ dim}X-1$ .     

Maximal Inequalities for Martingales and Their Differential Subordinates

We introduce a method of proving maximal inequalities for Hilbert- space-valued differentially subordinate local martingales. As an application, we prove that if $X=(X_t)_{t\ge 0},\, Y=(Y_t)_{t\ge 0}$ are local martingales such that $Y$ is differentially subordinate to $X$ , then $$\begin{aligned} ||Y||_1\le \beta ||\sup _{t\ge 0}|X_t|\;||_1, \end{aligned}$$ where $\beta =2.585\ldots $ is the best possible.     

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold $M$ . Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:{\mathbb {S}}^2 \rightarrow M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \le 2$ and that there is some point $\overline{x} \in M$ with scalar curvature $R^M(\overline{x}) > 6$ , we obtain a smooth minimizer $f:{\mathbb {S}}^2 \rightarrow M$ for the functional $\int \frac{1}{4}|H|^2+1$ , where $H$ is the mean curvature.     

On adding a variable to a Frobenius manifold and generalizations

Let $\pi :V\rightarrow M$ be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure $(\circ _{M},e_{M},g_{M})$ and typical fiber has the structure of a Frobenius algebra $(\circ _{V},e_{V},g_{V})$ . Using a connection $D$ on the bundle $\pi : V{\,\rightarrow \,}M$ and a morphism $\alpha :V\rightarrow TM$ , we construct an almost Frobenius structure $(\circ , e_{V},g)$ on the manifold $V$ and we study when it is Frobenius. In particular, we describe all (real) positive definite Frobenius structures on $V$ obtained in this way, when $M$ is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting, we add a real structure $k_{M}$ on $M$ and a real structure $k_{V}$ on the bundle $\pi : V \rightarrow M$ . Using $k_{M}$ , $k_{V}$ and $D$ we define a real structure $k$ on the manifold $V$ . We study when $k$ , together with an almost Frobenius structure $(\circ , e_{V}, g) $ , satisfies the tt*- equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and $tt^{*}$ -geometry.     

\alpha -Admissibility for Ritt Operators

Let $T:X\rightarrow X$ be a power bounded operator on Banach space. An operator $C:X\rightarrow Y$ is called admissible for $T$ if it satisfies an estimate $\sum _k\Vert CT^k(x)\Vert ^2\,\le M^2\Vert x\Vert ^2$ . Following Harper and Wynn, we study the validity of a certain Weiss conjecture in this discrete setting. We show that when $X$ is reflexive and $T$ is a Ritt operator satisfying a appropriate square function estimate, $C$ is admissible for $T$ if and only if it satisfies a uniform estimate $(1-\vert \omega \vert ^2)^{\frac{1}{2}}\Vert C(I-\omega T)^{-1}\Vert \,\le K\,$ for $\omega \in \mathbb{C }$ , $\vert \omega \vert <1$ . We extend this result to the more general setting of $\alpha $ -admissibility. Then we investigate a natural variant of admissibility involving $R$ -boundedness and provide examples to which our general results apply.     

Rigidity of Einstein manifolds with positive scalar curvature

We prove that if $M^n(n\ge 4)$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $R_0>\sigma _{n}K_{\max }$ , where $\sigma _n\in (\frac{1}{4},1)$ is an explicit positive constant depending only on $n$ , then $M$ must be isometric to a spherical space form. Moreover, we prove that if an $n(\ge {\!\!4})$ -dimensional compact Einstein manifold satisfies $K_{\min }\ge \eta _n R_0,$ where $\eta _n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is locally symmetric. It should be emphasized that the pinching constant $\eta _n$ is optimal when $n$ is even. We then obtain some rigidity theorems for Einstein manifolds under $(n-2)$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $M$ is an $n(\ge {\!\!4})$ -dimensional compact Einstein submanifold in the simply connected space form $F^{N}(c)$ with constant curvature $c\ge 0$ , and the normalized scalar curvature $R_0$ of $M$ satisfies $R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$ where $A_n=n^3-5n^2+8n$ , and $H$ is the mean curvature of $M$ , then $M$ is isometric to a standard $n$ -sphere.     

On the structure of co-Kähler manifolds

By the work of Li, a compact co-Kähler manifold $M$ is a mapping torus $K_\varphi $ , where $K$ is a Kähler manifold and $\varphi $ is a Hermitian isometry. We show here that there is always a finite cyclic cover $\overline{M}$ of the form $\overline{M} \cong K \times S^1$ , where $\cong $ is equivariant diffeomorphism with respect to an action of $S^1$ on $M$ and the action of $S^1$ on $K \times S^1$ by translation on the second factor. Furthermore, the covering transformations act diagonally on $S^1, K$ and are translations on the $S^1$ factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle.     

Strongly embedded subspaces of p-convex Banach function spaces

Let $X(\mu )$ be a p-convex ( $1\le p<\infty $ ) order continuous Banach function space over a positive finite measure  $\mu $ . We characterize the subspaces of  $X(\mu )$ which can be found simultaneously in  $X(\mu )$ and a suitable $L^1(\eta )$ space, where $\eta $ is a positive finite measure related to the representation of  $X(\mu )$ as an $L^p(m)$ space of a vector measure  $m$ . We provide in this way new tools to analyze the strict singularity of the inclusion of  $X(\mu )$ in such an $L^1$ space. No rearrangement invariant type restrictions on  $X(\mu )$ are required.     

Symbolic extensions and dominated splittings for generic C^{1}-diffeomorphisms

Let $\mathrm{Diff }^1(M)$ be the set of all $C^1$ -diffeomorphisms $f:M\rightarrow M$ , where $M$ is a compact boundaryless d-dimensional manifold, $d\ge 2$ . We prove that there is a residual subset $\mathfrak R $ of $\mathrm{Diff }^1(M)$ such that if $f\in \mathfrak R $ and if $H(p)$ is the homoclinic class associated with a hyperbolic periodic point $p$ , then either $H(p)$ admits a dominated splitting of the form $E\oplus F_1\oplus \dots \oplus F_k\oplus G$ , where $F_i$ is not hyperbolic and one-dimensional, or $f|_{H(p)}$ has no symbolic extensions.     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Oka properties of ball complements

Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds $X$ of dimension ${<}n$ to the complement $\mathbb {C}^n{\setminus } L$ of a compact convex set $L\subset \mathbb {C}^n$ satisfy the basic Oka property with approximation and interpolation. If $L$ is polynomially convex then the same holds when $2\dim X \le n$ . We also construct proper holomorphic maps, immersions and embeddings $X\rightarrow \mathbb {C}^n$ with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.     

Nonlinear centralizers in homology

It is shown that every nonlinear centralizer from $L_p$ to $L_q$ is trivial unless $q=p$ . This means that if $q\ne p$ , the only exact sequence of quasi-Banach $L_\infty $ -modules and homomorphisms $0\rightarrow L_q\rightarrow Z\rightarrow L_p\rightarrow 0$ is the trivial one where $Z=L_q\oplus L_p$ . From this it follows that the space of centralizers on $L_p$ is essentially independent on $p\in (0,\infty )$ , which confirms a conjecture by Kalton.     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

CR immersions and Lorentzian geometry

We study the geometry of the second fundamental form of pseudohermitian immersions among nondegenerate CR manifolds. In particular we study existence and uniqueness of pseudohermitian immersions $\phi : M \rightarrow S^{2n+3}$ of a strictly pseudoconvex CR manifold $M$ into an odd dimensional sphere, as determined by the pseudohermitian Gauss and Weingarten equations.