Mathematical Colloquium - On a conjecturte of H. Cartan

00:00 01-11-2010
<p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif">Abstract:  </font></span></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px"><font size="2" face="Arial, Helvetica, sans-serif"></font></span> </p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">In 1928 H. Cartan proved an extension of Montel's normality criterion </span><span style="COLOR: black"></span></font></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">to holomorphic curves in complex projective plane $P^2$. He also conjectured </span><span style="COLOR: black"></span></font></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">that a similar result is true for holomorphic curves in $P^n$ for every $n$. </span><span style="COLOR: black"></span></font></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">I give a counterexample to this conjecture for any $n>2$ and show how to modify Cartan's<br />conjecture so that it becomes true, at least for $n=3$.</span><span style="COLOR: black"></span></font></p> <p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"><span style="COLOR: black"><font face="Arial, Helvetica, sans-serif"><font size="2"> </font></font></span></p>