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<head><title>cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety</title>
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<div><h1>cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a 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>cotangentSheaf(p,X)</tt></div>
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<li><span>Function: <a href="_cotangent__Sheaf.html" title="cotangent sheaf of a projective variety">cotangentSheaf</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>p</tt>, <span>an <a href="___Z__Z.html">integer</a></span></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>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span>, the <tt>p</tt>-th exterior power of the cotangent sheaf</span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Minimize => </tt><span><span>a <a href="___Boolean.html">Boolean value</a></span>, <span>default value true</span>, whether to apply <a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a> to the result before returning it</span></span></li>
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<div class="single"><h2>Description</h2>
<div>This function computes the <tt>p</tt>-th exterior power of the cotangent sheaf of a variety <tt>X</tt>.<p/>
As an example we compute h^11 on a K3 surface (a quartic in projective threespace):<table class="examples"><tr><td><pre>i1 : K3 = Proj(QQ[x_0..x_3]/(x_0^4+x_1^4+x_2^4+x_3^4-11*x_0*x_1*x_2*x_3))
o1 = K3
o1 : ProjectiveVariety</pre>
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<tr><td><pre>i2 : omega1 = cotangentSheaf(1,K3);</pre>
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<tr><td><pre>i3 : HH^1(omega1)
20
o3 = QQ
o3 : QQ-module, free</pre>
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<p/>
As a second example we compute Hodge numbers of the Fermat quintic in projective fourspace:<table class="examples"><tr><td><pre>i4 : FermatQuintic = Proj(QQ[x_0..x_4]/(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5))
o4 = FermatQuintic
o4 : ProjectiveVariety</pre>
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<tr><td><pre>i5 : omega1 = cotangentSheaf(1,FermatQuintic);</pre>
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<tr><td><pre>i6 : HH^1(omega1)
1
o6 = QQ
o6 : QQ-module, free</pre>
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<tr><td><pre>i7 : omega2 = cotangentSheaf(2,FermatQuintic);</pre>
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<tr><td><pre>i8 : HH^1(omega2)
101
o8 = QQ
o8 : QQ-module, free</pre>
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<tr><td><pre>i9 : HH^2(omega1)
101
o9 = QQ
o9 : QQ-module, free</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_tangent__Sheaf.html" title="tangent sheaf of a projective variety">tangentSheaf</a> -- tangent sheaf of a projective variety</span></li>
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