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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Behrend function</span></span>
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<p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, the <b>Behrend function</b> of a scheme <i>X</i>, introduced by <a href="Kai_Behrend" title="Kai Behrend">Kai Behrend</a>, is a constructible function
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{X}:X\to \mathbb {Z} }">
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<annotation encoding="application/x-tex">{\displaystyle \nu _{X}:X\to \mathbb {Z} }</annotation>
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</math></span><img src="./b26447b75699335e9a7aa284cae56efc76ffc332.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.862ex; height:2.509ex;" alt="{\displaystyle \nu _{X}:X\to \mathbb {Z} }" loading="lazy"></span></dd></dl>
<p>such that if <i>X</i> is a quasi-projective proper moduli scheme carrying a <a href="Symmetric_obstruction_theory" class="mw-redirect" title="Symmetric obstruction theory">symmetric obstruction theory</a>, then the <b>weighted Euler characteristic</b>
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (X,\nu _{X})=\sum _{n\in \mathbb {Z} }n\,\chi (\{\nu _{X}=n\})}">
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<annotation encoding="application/x-tex">{\displaystyle \chi (X,\nu _{X})=\sum _{n\in \mathbb {Z} }n\,\chi (\{\nu _{X}=n\})}</annotation>
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</math></span><img src="./883f014eda3cc13cef9f291115100d4f1a9cf562.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.545ex; height:5.676ex;" alt="{\displaystyle \chi (X,\nu _{X})=\sum _{n\in \mathbb {Z} }n\,\chi (\{\nu _{X}=n\})}" loading="lazy"></span></dd></dl>
<p>is the degree of the <a href="Virtual_fundamental_class" title="Virtual fundamental class">virtual fundamental class</a>
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X]^{\text{vir}}}">
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<p>of <i>X</i>, which is an element of the zeroth <a href="Chow_group" title="Chow group">Chow group</a> of <i>X</i>. Modulo some solvable technical difficulties (e.g., what is the <a href="Chow_group_of_a_stack" title="Chow group of a stack">Chow group of a stack</a>?), the definition extends to moduli stacks such as the moduli stack of stable sheaves (the <a href="Donaldson%E2%80%93Thomas_theory" title="Donaldson–Thomas theory">Donaldson–Thomas theory</a>) or that of <a href="Stable_map" title="Stable map">stable maps</a> (the <a href="Gromov%E2%80%93Witten_theory" class="mw-redirect" title="Gromov–Witten theory">Gromov–Witten theory</a>).
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFBehrend2009" class="citation cs2"><a href="Kai_Behrend" title="Kai Behrend">Behrend, Kai</a> (2009), "Donaldson–Thomas type invariants via microlocal geometry", <i><a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, 2nd Ser., <b>170</b> (3): <span class="nowrap">1307–</span>1338, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0507523">math/0507523</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2009.170.1307">10.4007/annals.2009.170.1307</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2600874">2600874</a></cite>.</li></ul>
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