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<head><title>hh -- Hodge numbers of a smooth projective variety</title>
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<div><h1>hh -- Hodge numbers of a smooth projective variety</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>hh^(p,q)(X)</tt></div>
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<li><div class="single">Inputs:<ul><li><span>a pair <tt>(p,q)</tt> of non negative integers</span></li>
<li><span><tt>X</tt>, <span>a <a href="___Projective__Variety.html">projective variety</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div>The command computes the Hodge numbers h^{p,q} of the smooth projective variety X. They are calculated as <tt>HH^q(cotangentSheaf(p,X))</tt><p/>
As an example we compute h^11 of a K3 surface (Fermat quartic in projective threespace:<table class="examples"><tr><td><pre>i1 : X = Proj(QQ[x_0..x_3]/ideal(x_0^4+x_1^4+x_2^4+x_3^4))
o1 = X
o1 : ProjectiveVariety</pre>
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<tr><td><pre>i2 : hh^(1,1)(X)
o2 = 20</pre>
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<div class="single"><h2>Caveat</h2>
<div>There is no check made if the projective variety X is smooth or not.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___H__H^__Z__Z_sp__Sum__Of__Twists.html" title="coherent sheaf cohomology module">HH^ZZ SumOfTwists</a> -- coherent sheaf cohomology module</span></li>
<li><span><a href="___H__H^__Z__Z_sp__Sheaf__Of__Rings.html" title="cohomology of a sheaf of rings on a projective variety">HH^ZZ SheafOfRings</a> -- cohomology of a sheaf of rings on a projective variety</span></li>
<li><span><a href="_euler_lp__Projective__Variety_rp.html" title="topological Euler characteristic of a (smooth) projective variety">euler(ProjectiveVariety)</a> -- topological Euler characteristic of a (smooth) projective variety</span></li>
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<div class="waystouse"><h2>Ways to use <tt>hh</tt> :</h2>
<ul><li>hh(Sequence,ProjectiveVariety)</li>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="_hh.html" title="Hodge numbers of a smooth projective variety">hh</a> is <span>a <a href="___Scripted__Functor.html">scripted functor</a></span>.</p>
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