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  "body": "\n## Summary\n\nChern classes are [[characteristic classes|7HACJC0JE2]] of complex vector bundles, with values in cohomology.\nIn algebraic geometry they are defined for algebraic vector bundles, with values in the [[Chow ring|4HV4WTAX9P]].\nA vector bundle $E$ of rank $r$ has classes $c_0(E) = 1, c_1(E), \\ldots, c_r(E)$, with $c_i$ in degree $i$.\nThey measure how far $E$ is from being trivial, and they enter the Riemann–Roch theorems through the [[Todd class|2YDB5089FR]].\nFor a line bundle, $c_1$ is the class of the corresponding divisor.\n\n## Explanations\n\n### L1 human\n\nA vector bundle attaches a vector space to each point of a space, and it may twist as one moves around.\nChern classes are invariants that record this twisting in a way that can be computed and added.\nFor the line bundle $\\mathcal{O}(d)$ on the projective line, the first Chern class is $d$ times the class of a point: its sections vanish at $d$ points.\nFor the tangent bundle of a compact complex curve of genus $g$, the first Chern class has degree $2 - 2g$, the Euler characteristic.\n",
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