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<head><title>chern(ZZ,ZZ,AbstractSheaf) -- Get the Chern class of an abstract sheaf</title>
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<div><h1>chern(ZZ,ZZ,AbstractSheaf) -- Get the Chern class of an abstract sheaf</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>c = chern(n,m,A)</tt></div>
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<li><span>Function: <tt>chern</tt> (missing documentation<!-- tag: chern -->)</span></li>
<li><div class="single">Inputs:<ul><li><span><tt>n</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li>
<li><span><tt>m</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li>
<li><span><tt>A</tt>, <span>an <a href="___Abstract__Sheaf.html">abstract sheaf</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>c</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div><div>Chern classes of an abstract sheaf are computed. If called with three arguments as above, a list of the Chern classes <tt>c<sub>n</sub>(A) .. c<sub>m</sub>(A)</tt> are returned. Here <tt>0≤n≤m</tt> are integers. <tt>chern</tt> may also be called without one or both of these integer arguments, in which case just one Chern class, or the total Chern class is returned, respectively. In the following example, we consider two vector bundles <tt>A</tt> and <tt>B</tt> of ranks 2 and 3, respectively, on a variety of which we only know that its dimension is 3.</div>
<table class="examples"><tr><td><pre>i1 : base(3, Bundle => (A,2,a), Bundle => (B,3,b))
o1 = a variety
o1 : an abstract variety of dimension 3</pre>
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<tr><td><pre>i2 : chern(B)
o2 = 1 + b + b + b
1 2 3
o2 : QQ[a , a , b , b , b ]
1 2 1 2 3</pre>
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<tr><td><pre>i3 : chern(-A)
2 3
o3 = 1 - a + (a - a ) + (- a + 2a a )
1 1 2 1 1 2
o3 : QQ[a , a , b , b , b ]
1 2 1 2 3</pre>
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<tr><td><pre>i4 : chern(2,A*B)
2 2
o4 = 3a + 3a + 5a b + b + 2b
1 2 1 1 1 2
o4 : QQ[a , a , b , b , b ]
1 2 1 2 3</pre>
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<tr><td><pre>i5 : chern(2,3,B-A)
2 3 2
o5 = {a - a - a b + b , - a + 2a a + a b - a b - a b + b }
1 2 1 1 2 1 1 2 1 1 2 1 1 2 3
o5 : List</pre>
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<div>The next example gives the Chern classes of the twists of a rank-2 vector bundle on the projective plane</div>
<table class="examples"><tr><td><pre>i6 : pt=base(n,p,q)
o6 = pt
o6 : an abstract variety of dimension 0</pre>
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<tr><td><pre>i7 : P2=projectiveSpace(2,pt)
o7 = P2
o7 : a flag bundle with ranks {2, 1}</pre>
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<tr><td><pre>i8 : E=abstractSheaf(P2,Rank=>2,ChernClass=>1+p*h+q*h^2)
o8 = E
o8 : an abstract sheaf of rank 2 on P2</pre>
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<tr><td><pre>i9 : chern(E*OO(n*h))
2 2
o9 = 1 + (2n + p)h + (n + n*p + q)h
QQ[n, p, q][H , H , h]
1,1 1,2
o9 : -------------------------------
(H + h, H + H h, H h)
1,1 1,2 1,1 1,2</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><tt>segre</tt> (missing documentation<!-- tag: segre -->)</span></li>
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