<div dir="ltr"><p class="gmail-MsoTitle" style="margin:0cm 0cm 0.0001pt;text-align:center;font-size:24pt;font-family:"New York",serif;font-weight:bold;text-decoration-line:underline"><span style="font-size:18pt;font-family:"Times New Roman",serif">SEMINARIO
DI MATEMATICA</span></p>

<p class="gmail-MsoTitle" style="margin:0cm 0cm 0.0001pt;text-align:center;font-size:24pt;font-family:"New York",serif;font-weight:bold;text-decoration-line:underline"><span style="font-size:10pt;font-family:"Times New Roman",serif"><span style="text-decoration-line:none"> </span></span></p>

<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><span style="font-size:16pt;font-family:"Times New Roman",serif">Lunedì 14 ottobre 2019</span></b><b><span style="font-size:16pt;font-family:"Times New Roman",serif"></span></b></p>

<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif">ore 14:00</span></p>

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<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><u><span style="font-size:16pt;font-family:"Times New Roman",serif">Scuola Normale Superiore</span></u></p>

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<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><u><span style="font-size:10pt;font-family:"Times New Roman",serif"><span style="text-decoration-line:none"> </span></span></u></b></p>

<p class="MsoNormal" align="center" style="text-align:center;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><span style="font-size:22pt;font-family:"Times New Roman",serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">Jih-Hsin </span></b><span class="gmail-il"><b><span style="font-size:22pt;font-family:"Times New Roman",serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><span style="text-align:start">Cheng</span></span></b></span><b><span style="font-size:22pt;font-family:"Times New Roman",serif"></span></b></p>

<h2 style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;break-after:avoid;font-size:18pt;font-family:"New York",serif"><i><span style="font-size:12pt;font-family:"Times New Roman",serif;font-weight:normal">(Priceton University</span></i><i><span style="font-size:12pt;font-family:"Times New Roman",serif;font-weight:normal">)</span></i></h2>

<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><i><span style="font-family:"Times New Roman",serif"> </span></i></p>

<p class="MsoNormal" align="center" style="text-align:center;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif">Terrà un
seminario dal titolo:</span></p>

<h1 align="center" style="text-align:center;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;margin:24pt 0cm 0.0001pt;break-after:avoid;font-size:14pt;font-family:Cambria,serif"><span lang="EN-US" style="color:rgb(34,34,34);font-size:18pt;font-family:"Times New Roman",serif">“<span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">Heat kernel and local
index theorem for open complex manifolds</span></span><br style="text-align:start">
<font color="#000000"><span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><span style="text-align:start">with $\mathbb{C}^{\ast }$ action</span></span><span lang="EN-US" style="font-size:18pt;font-family:"Times New Roman",serif">”</span></font><span lang="EN-US" style="font-size:18pt;font-family:"Times New Roman",serif;color:rgb(34,34,34);font-size:18pt"></span></h1>

<p class="MsoNormal" align="center" style="text-align:center;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><i><span lang="EN-US" style="font-size:18pt;font-family:"Times New Roman",serif"> </span></i></b></p>

<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><i><span lang="EN-US" style="font-family:"Times New Roman",serif">Abstract:</span></i></b></p>

<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><i><span lang="EN-US" style="font-size:10pt;font-family:"Times New Roman",serif"> </span></i></b></p>

<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span lang="EN-US" style="font-family:"Times New Roman",serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">For a complex manifold with
$\mathbb{C}^{\ast }$ action, we</span><span lang="EN-US" style="font-family:"Times New Roman",serif"> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">define the $m$-th $\mathbb{C}^{\ast }$
Fourier-Dolbeault cohomology</span></span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">group
and consider the $m$-index. By applying the method of</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">\textit{transversal} heat kernel asymptotics, we
obtain a local index</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">formula for
the $m$-index. We can reinterpret Kawasaki's</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">Hirzebruch-Riemann-Roch formula for a compact complex
orbifold with an</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">orbifold
holomorphic line bundle by our single integral over a</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">(smooth) complex manifold. We generalize
$\mathbb{C}^{\ast }$ action</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">to
complex semisimple Lie group $G$ action on a compact or noncompact</span>
<span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">complex manifold. Among others, we study the
nonextendability of open</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">group
action and the space of all $G$ invariant holomorphic $p$-forms.</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">Finally, in the case of two compatible holomorphic
$\mathbb{C}^{\ast</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">}$ actions, a
mirror-symmetry type isomorphism is found between two</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">linear spaces of holomorphic forms, and the Euler
characteristic</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">associated with
these spaces can be computed by our $\mathbb{C}^{\ast</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">}$ local index formula on the total space. This is
joint work with</span> <span style="background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">Chin-Yu Hsiao and
I-Hsun Tsai.</span></p>

<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span lang="EN-US" style="font-family:Arial,sans-serif"> </span></p>

<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">Tutti gli interessati sono invitati a partecipare.</span></p>

<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"> </span></p>

<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial">Classe di
Scienze</span></p><div><div dir="ltr" class="gmail_signature" data-smartmail="gmail_signature"><div dir="ltr"><div>Valeria Giuliani</div><div>Scuola Normale Superiore</div><div>Servizio alla Didattica e Allievi</div><div>tel. 050 509260</div><div>Piazza dei Cavalieri, 7</div><div>56126 Pisa</div><div>E-mail: <a href="mailto:valeria.giuliani@sns.it" target="_blank">valeria.giuliani@sns.it</a></div><div>E-mail: <a href="mailto:classi@sns.it" target="_blank">classi@sns.it</a></div></div></div></div></div>

<p></p>

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