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1 \documentclass[oneside, letterpaper, 11pt]{article}
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40 \begin{document}
41 \leftline{ {\Large Mathematics, Statistics, and Computer Science \quad {\Large\bf @ UIC}} }
42 \vskip 18pt
43 \centerline{{\Large\bf Geometry, Topology and Dynamics Seminar}}
44 {\centering\baselineskip=16pt
45 {\Large\sl Weighted cscK metrics and weighted K-stability}
46
47 }
48 \vskip 6pt
49 \centerline{{\large Abdellah Lahdili (Montreal)}}
50 \vskip 4pt
51 {\bf Abstract:}\enspace
52 {We will introduce a notion of a K\textbackslash{}\textquotesingle\textquotesingle{}ahler metric with constant weighted scalar curvature on a compact K\textbackslash{}\textquotesingle\textquotesingle{}ahler manifold $X$, depending on a fixed real torus $\T$ in the reduced group of automorphisms of $X$, and two smooth (weight) functions defined on the momentum image of $X$. We will also define a notion of weighted Mabuchi energy adapted to our setting, and of a weighted Futaki invariant of a $\T$-compatible smooth K\textbackslash{}\textquotesingle\textquotesingle{}ahler test configuration associated to $(X, \T)$. After that, using the geometric quantization scheme of Donaldson, we will show that if a projective manifold admits in the corresponding Hodge K\textbackslash{}\textquotesingle\textquotesingle{}ahler class a K\textbackslash{}\textquotesingle\textquotesingle{}ahler metric with constant weighted scalar curvature, then this metric minimizes the weighted Mabuchi energy, which implies a suitable notion of weighted K-semistability. As an application, we describe the K\textbackslash{}\textquotesingle\textquotesingle{}ahler classes on a geometrically ruled complex surface of genus greater than 2, which admits conformally K\textbackslash{}\textquotesingle\textquotesingle{}ahler Einstein-Maxwell metrics.}
53 \vskip 10pt minus 8pt
54 {\it }
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59 \end{document}