| 1. |
- Andersson, Mats, 1957, et al.
(författare)
-
Non-pluripolar energy and the complex Monge-Ampere operator
- 2022
-
Ingår i: Journal Fur Die Reine Und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 0075-4102 .- 1435-5345. ; 2022:792, s. 145-188
-
Tidskriftsartikel (refereegranskat)abstract
- Given a domain Omega subset of C-n we introduce a class of plurisubharmonic (psh) functions G(Omega) and Monge-Ampere operators u -> [dd(c)u](p), p <= n, on G(Omega) that extend the Bedford-Taylor-Demailly Monge-Ampere operators. Here [dd(c)u](p) is a closed positive current of bidegree (p, p) that dominates the non-pluripolar Monge-Ampere current < dd(c)u >(p). We prove that [dd(c)u](p) is the limit of Monge-Ampere currents of certain natural regularizations of u. On a compact Kahler manifold (X, omega) we introduce a notion of non-pluripolar energy and a corresponding finite energy class G(X, omega) subset of PSH(X, omega) that is a global version of the class G(Omega). From the local construction we get global Monge-Ampere currents [dd(c)phi + omega](p) for phi is an element of G(X, omega) that only depend on the current dd(c)phi + omega. The limits of Monge-Ampere currents of certain natural regularizations of phi can be expressed in terms of [dd(c)phi + omega](j) for j <= p. We get a mass formula involving the currents [dd(c)phi + omega](p) that describes the loss of mass of the non-pluripolar Monge-Ampere measure < dd(c)phi + omega >(n). The class G(X, omega) includes omega-psh functions with analytic singularities and the class E(X, omega) of omega-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.
|
|
| 2. |
- Andersson, Mats, 1957, et al.
(författare)
-
Segre numbers, a generalized King formula, and local intersections
- 2017
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 1435-5345 .- 0075-4102. ; 728:728, s. 105-136
-
Tidskriftsartikel (refereegranskat)abstract
- Let $\mathcal{J}$ be an ideal sheaf on a reduced analytic space $X$ with zero set $Z$.We show that the Lelong numbers of the restrictions to $Z$ of certain generalized Monge–Ampère products $(dd^c \log |f|^2)^k$, where $f$ is a tuple of generators of $\mathcal{J}$, coincide with theso-called Segre numbers of $\mathcal{J}$, introduced independently by Tworzewski, Achilles–Manaresi,and Gaffney–Gassler. More generally we show that these currents satisfy a generalization ofthe classical King formula that takes into account fixed and moving components of Vogelcycles associated with $\mathcal{J}$. A basic tool is a new calculus for products of positive currents ofBochner–Martinelli type. We also discuss connections to intersection theory.
|
|
| 3. |
- Bergström, Jonas
(författare)
-
Cohomology of moduli spaces of curves of genus three via point counts
- 2008
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - 0075-4102 .- 1435-5345. ; 622, s. 155-187
-
Tidskriftsartikel (refereegranskat)abstract
- In this article we consider the moduli space of smooth n-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make S-n-equivariant counts of its numbers of points defined over finite fields for n <= 7. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the S-n-equivariant Galois and Hodge structure of the (l-adic respectively Betti) cohomology of the moduli space of stable curves of genus 3 for n <= 5 ( to obtain n <= 7 we would need counts of "8-pointed curves of genus 2'').
|
|
| 4. |
- Berman, Robert, 1976, et al.
(författare)
-
Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties
- 2019
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 1435-5345 .- 0075-4102. ; 2019:751, s. 27-89
-
Tidskriftsartikel (refereegranskat)abstract
- We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler-Ricci flow provides weak convergence independently of Perelman's celebrated estimates.
|
|
| 5. |
- Berndtsson, Bo, 1950, et al.
(författare)
-
The volume of Kahler-Einstein varieties and convex bodies
- 2017
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 0075-4102 .- 1435-5345. ; :723, s. 127-152
-
Tidskriftsartikel (refereegranskat)abstract
- We show that the complex projective space Pn has maximal degree (volume) among all n-dimensional Kähler–Einstein Fano manifolds admitting a non-trivial holomorphic C∗-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart’s volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. The case of spherical varieties/multiplicity free symplectic manifolds is also discussed. The proof uses Moser–Trudinger type inequalities for Stein domains and also leads to criticality results for mean field type equations in Cn of independent interest. The paper supersedes our previous preprint [5] concerning the case of toric Fano manifolds.
|
|
| 6. |
- Björklund, Michael, 1981, et al.
(författare)
-
Plünnecke inequalities for countable abelian groups
- 2017
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 1435-5345 .- 0075-4102. ; 730, s. 199-224
-
Tidskriftsartikel (refereegranskat)abstract
- We establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers. We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Plünnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group.
|
|
| 7. |
- Björn, Anders, et al.
(författare)
-
Geometric analysis on Cantor sets and trees
- 2017
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - : WALTER DE GRUYTER GMBH. - 0075-4102 .- 1435-5345. ; 725, s. 63-114
-
Tidskriftsartikel (refereegranskat)abstract
- dUsing uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.
|
|
| 8. |
- Björn, Anders, 1966-, et al.
(författare)
-
The Dirichlet problem for p-harmonic functions on metric spaces
- 2003
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - 0075-4102 .- 1435-5345. ; :556, s. 173-203
-
Tidskriftsartikel (refereegranskat)abstract
- We study the Dirichlet problem for p-harmonic functions (and p-energy minimizers) in bounded domains in proper, pathconnected metric measure spaces equipped with a doubling measure and supporting a PoincarΘ inequality. The Dirichlet problem has previously been solved for Sobolev type boundary data, and we extend this result and solve the problem for all continuous boundary data. We study the regularity of boundary points and prove the Kellogg property, i.e. that the set of irregular boundary points has zero p-capacity. We also construct p-capacitary, p-singular and p-harmonic measures on the boundary. We show that they are all absolutely continuous with respect to the p-capacity. For p = 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide. We give an integral representation for the solution to the Dirichlet problem when p = 2, enabling us to extend the solvability of the problem to L1 boundary data in this case. Moreover, we give a trace result for Newtonian functions when p = 2. Finally, we give an estimate for the Hausdorff dimension of the boundary of a bounded domain in Ahlfors Q-regular spaces.
|
|
| 9. |
- Booker, Andrew, et al.
(författare)
-
Numerical computations with the trace formula and the Selberg eigenvalue conjecture
- 2007
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - 0075-4102 .- 1435-5345. ; 2007:607, s. 113-161
-
Tidskriftsartikel (refereegranskat)abstract
- We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [M. N. Huxley, Introduction to Kloostermania, in: Elementary and analytic theory of numbers, Banach Center Publ. 17, Warsaw (1985), 217–306.]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1/4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois representations of small squarefree conductor.
|
|
| 10. |
- Borcea, Julius
(författare)
-
Classifications of linear operators preserving elliptic, positive and non-negative polynomials
- 2011
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - 0075-4102 .- 1435-5345. ; 650, s. 67-82
-
Tidskriftsartikel (refereegranskat)abstract
- We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer-Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [9] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.
|
|
