References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt145",{id:"formSmash:upper:j_idt145",widgetVar:"widget_formSmash_upper_j_idt145",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt146_j_idt148",{id:"formSmash:upper:j_idt146:j_idt148",widgetVar:"widget_formSmash_upper_j_idt146_j_idt148",target:"formSmash:upper:j_idt146:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

The Bergman Kernel on Toric Kähler ManifoldsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
function selectAll()
{
var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
}
/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
function toggleList(childList, panel)
{
var panelWasOpen = (panel.get(0).style.display == 'none');
// console.log('panel was open ' + panelWasOpen);
for (var c = 0; c < childList.length; c++) {
if (childList[c].classList.contains('authorPanel')) {
clickNode(panelWasOpen, childList[c]);
}
}
}
/*nodes have styleClass ui-corner-top if they are expanded and ui-corner-all if they are collapsed */
function clickNode(collapse, child)
{
if (collapse && child.classList.contains('ui-corner-top')) {
// console.log('collapse');
child.click();
}
if (!collapse && child.classList.contains('ui-corner-all')) {
// console.log('expand');
child.click();
}
}
PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2011 (English)Doctoral thesis, monograph (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

The University of Edinburgh , 2011. , 130 p.
##### National Category

Geometry
##### Identifiers

URN: urn:nbn:se:kth:diva-53788OAI: oai:DiVA.org:kth-53788DiVA: diva2:479144
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt375",{id:"formSmash:j_idt375",widgetVar:"widget_formSmash_j_idt375",multiple:true});
##### Supervisors

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt381",{id:"formSmash:j_idt381",widgetVar:"widget_formSmash_j_idt381",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt387",{id:"formSmash:j_idt387",widgetVar:"widget_formSmash_j_idt387",multiple:true});
##### Note

Sponsorship: Engineering and Physical Sciences Research Council (EPSRC), United KingdomAvailable from: 2012-01-19 Created: 2011-12-30 Last updated: 2012-02-23Bibliographically approved

Let $(L,h)\to (X, \omega)$ be a compact toric polarized Kähler manifold of complex dimension $n$. For each $k\in N$, the fibre-wise Hermitian metric $h^k$ on $L^k$ induces a natural inner product on the vector space $C^{\infty}(X, L^k)$ of smooth global sections of $L^k$ by integration with respect to the volume form $\frac{\omega^n}{n!}$. The orthogonal projection $P_k:C^{\infty}(X, L^k)\to H^0(X, L^k)$ onto the space $H^0(X, L^k)$ of global holomorphic sections of $L^k$ is represented by an integral kernel $B_k$ which is called the Bergman kernel (with parameter $k\in N$). The restriction $\rho_k:X\to R$ of the norm of $B_k$ to the diagonal in $X\times X$ is called the density function of $B_k$.

On a dense subset of $X$, we describe a method for computing the coefficients of the asymptotic expansion of $\rho_k$ as $k\to \infty$ in this toric setting. We also provide a direct proof of a result which illuminates the off-diagonal decay behaviour of toric Bergman kernels.

We fix a parameter $l\in N$ and consider the projection $P_{l,k}$ from $C^{\infty}(X, L^k)$ onto those global holomorphic sections of $L^k$ that vanish to order at least $lk$ along some toric submanifold of $X$. There exists an associated toric partial Bergman kernel $B_{l, k}$ giving rise to a toric partial density function $\rho_{l, k}:X\to R$. For such toric partial density functions, we determine new asymptotic expansions over certain subsets of $X$ as $k\to \infty$. Euler-Maclaurin sums and Laplace's method are utilized as important tools for this. We discuss the case of a polarization of $CP^n$ in detail and also investigate the non-compact Bargmann-Fock model with imposed vanishing at the origin.

We then discuss the relationship between the slope inequality and the asymptotics of Bergman kernels with vanishing and study how a version of Song and Zelditch's toric localization of sums result generalizes to arbitrary polarized Kähler manifolds.

Finally, we construct families of induced metrics on blow-ups of polarized Kähler manifolds. We relate those metrics to partial density functions and study their properties for a specific blow-up of $C^n$ and $CP^n$ in more detail.

References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1080",{id:"formSmash:lower:j_idt1080",widgetVar:"widget_formSmash_lower_j_idt1080",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1081_j_idt1083",{id:"formSmash:lower:j_idt1081:j_idt1083",widgetVar:"widget_formSmash_lower_j_idt1081_j_idt1083",target:"formSmash:lower:j_idt1081:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});