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<head><title>HH^ZZ CoherentSheaf -- cohomology of a coherent sheaf on a projective variety</title>
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<div><h1>HH^ZZ CoherentSheaf -- cohomology of a coherent sheaf on 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>HH^i F</tt><br/><tt>cohomology(i,F)</tt></div>
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<li><span>Function: <a href="_cohomology.html" title="general cohomology functor">cohomology</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><tt>F</tt>, <span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span>, on a projective variety <tt>X</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, the <tt>i</tt>-th cohomology group of <tt>F</tt> as a vector space over the coefficient field of <tt>X</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_cohomology_lp..._cm_sp__Degree_sp_eq_gt_sp..._rp.html">Degree => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>The command computes the <tt>i</tt>-th cohomology group of <tt>F</tt> as a vector space over the coefficient field of <tt>X</tt>. For i>0 this is currently done via local duality, while for i=0 it is computed as a limmit of Homs. Eventually there will exist an alternative option for computing sheaf cohomology via the Bernstein-Gelfand-Gelfand correspondence<p/>
As examples we compute the Picard numbers, Hodge numbers and dimension of the infinitesimal deformation spaces of various quintic hypersurfaces in projective fourspace (or their Calabi-Yau small resolutions)<p/>
We will make computations for quintics V in the family given by <p align=center><i>x<sub>0</sub><sup>5</sup>+x<sub>1</sub><sup>5</sup>+x<sub>2</sub><sup>5</sup>+x<sub>3</sub><sup>5</sup>+x<sub>4</sub><sup>5</sup>-5λx<sub>0</sub>x<sub>1</sub>x<sub>2</sub>x<sub>3</sub>x<sub>4</sub>=0</i></p> for various values of <i>λ</i>. If <i>λ</i> is general (that is, <i>λ</i> not a 5-th root of unity, 0 or <i>∞</i>), then the quintic <i>V</i> is smooth, so is a Calabi-Yau threefold, and in that case the Hodge numbers are as follows.<p/>
<p align=center><i>h<sup>1,1</sup>(V)=1, h<sup>2,1</sup>(V) = h<sup>1,2</sup>(V) = 101,</i></p><p/>
so the Picard group of V has rank 1 (generated by the hyperplane section) and the moduli space of V (which is unobstructed) has dimension 101:<table class="examples"><tr><td><pre>i1 : Quintic = Proj(QQ[x_0..x_4]/ideal(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-101*x_0*x_1*x_2*x_3*x_4))
o1 = Quintic
o1 : ProjectiveVariety</pre>
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<tr><td><pre>i2 : singularLocus(Quintic)
/QQ[x , x , x , x , x ]\
| 0 1 2 3 4 |
o2 = Proj|----------------------|
\ 1 /
o2 : ProjectiveVariety</pre>
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<tr><td><pre>i3 : omegaQuintic = cotangentSheaf(Quintic);</pre>
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<tr><td><pre>i4 : h11 = rank HH^1(omegaQuintic)
o4 = 1</pre>
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<tr><td><pre>i5 : h12 = rank HH^2(omegaQuintic)
o5 = 101</pre>
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By Hodge duality this is <i>h<sup>2,1</sup></i>. Directly <i>h<sup>2,1</sup></i> could be computed as<table class="examples"><tr><td><pre>i6 : h21 = rank HH^1(cotangentSheaf(2,Quintic))
o6 = 101</pre>
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The Hodge numbers of a (smooth) projective variety can also be computed directly using the <a href="_hh.html" title="Hodge numbers of a smooth projective variety">hh</a> command:<table class="examples"><tr><td><pre>i7 : hh^(2,1)(Quintic)
o7 = 101</pre>
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<tr><td><pre>i8 : hh^(1,1)(Quintic)
o8 = 1</pre>
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Using the Hodge number we compute the topological Euler characteristic of V:<table class="examples"><tr><td><pre>i9 : euler(Quintic)
o9 = -200</pre>
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When <i>λ</i> is a 5th root of unity the quintic V is singular. It has 125 ordinary double points (nodes), namely the orbit of the point <i>(1:λ:λ:λ:λ)</i> under a natural action of <i>ℤ/5<sup>3</sup></i>. Then <i>V</i> has a projective small resolution <i>W</i> which is a Calabi-Yau threefold (since the action of <i>ℤ/5<sup>3</sup></i> is transitive on the sets of nodes of <i>V</i>, or for instance, just by blowing up one of the <i>(1,5)</i> polarized abelian surfaces <i>V</i> contains). Perhaps the most interesting such 3-fold is the one for the value <i>λ=1</i>, which is defined over <i>ℚ</i> and is modular (see Schoen’s work). To compute the Hodge numbers of the small resolution <i>W</i> of <i>V</i> we proceed as follows:<table class="examples"><tr><td><pre>i10 : SchoensQuintic = Proj(QQ[x_0..x_4]/ideal(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-5*x_0*x_1*x_2*x_3*x_4))
o10 = SchoensQuintic
o10 : ProjectiveVariety</pre>
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<tr><td><pre>i11 : Z = singularLocus(SchoensQuintic)
o11 = Z
o11 : ProjectiveVariety</pre>
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<tr><td><pre>i12 : degree Z
o12 = 125</pre>
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<tr><td><pre>i13 : II'Z = sheaf module ideal Z
o13 = image | x_3^4-x_0x_1x_2x_4 x_0x_1x_2x_3-x_4^4 x_2^4-x_0x_1x_3x_4 x_1^4-x_0x_2x_3x_4 x_0^4-x_1x_2x_3x_4 x_2^3x_3^3-x_0^2x_1^2x_4^2 x_1^3x_3^3-x_0^2x_2^2x_4^2 x_0^3x_3^3-x_1^2x_2^2x_4^2 x_1^2x_2^2x_3^2-x_0^3x_4^3 x_0^2x_2^2x_3^2-x_1^3x_4^3 x_0^2x_1^2x_3^2-x_2^3x_4^3 x_1^3x_2^3-x_0^2x_3^2x_4^2 x_0^3x_2^3-x_1^2x_3^2x_4^2 x_0^2x_1^2x_2^2-x_3^3x_4^3 x_0^3x_1^3-x_2^2x_3^2x_4^2 |
1
o13 : coherent sheaf on Proj(QQ[x , x , x , x , x ]), subsheaf of OO
0 1 2 3 4 Proj(QQ[x , x , x , x , x ])
0 1 2 3 4</pre>
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The defect of W (that is, <i>h<sup>1,1</sup>(W)-1</i>) can be computed from the cohomology of the ideal sheaf of the singular locus Z of V twisted by 5 (see Werner’s thesis):<table class="examples"><tr><td><pre>i14 : defect = rank HH^1(II'Z(5))
o14 = 24</pre>
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<tr><td><pre>i15 : h11 = defect + 1
o15 = 25</pre>
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The number <i>h<sup>2,1</sup>(W)</i> (the dimension of the moduli space of W) can be computed (Clemens-Griffiths, Werner) as <i>dim H<sup>0</sup>(<b>I</b><sub>Z</sub>(5))/JacobianIdeal(V)<sub>5</sub></i>.<table class="examples"><tr><td><pre>i16 : quinticsJac = numgens source basis(5,ideal Z)
o16 = 25</pre>
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<tr><td><pre>i17 : h21 = rank HH^0(II'Z(5)) - quinticsJac
o17 = 0</pre>
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In other words W is rigid. It has the following topological Euler characteristic.<table class="examples"><tr><td><pre>i18 : chiW = euler(Quintic)+2*degree(Z)
o18 = 50</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_coherent_spsheaves.html" title="">coherent sheaves</a></span></li>
<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="_hh.html" title="Hodge numbers of a smooth projective variety">hh</a> -- Hodge numbers of a smooth projective variety</span></li>
<li><span><a href="___Coherent__Sheaf.html" title="the class of all coherent sheaves">CoherentSheaf</a> -- the class of all coherent sheaves</span></li>
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