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  "body": "\n## Definition\n\nLet $X$ be a smooth [[quasi-projective variety|EQSHJPE7H8]] over a field, with rational [[Chow ring|4HV4WTAX9P]] $CH^\\bullet(X)_\\Q = CH^\\bullet(X) \\otimes \\Q$.\nLet $E$ be a [[vector bundle|P4DZM975BV]] of rank $r$ on $X$ with [[Chern classes|PAD37GKDPD]] $c_1, \\ldots, c_r$.\nWrite formally $\\sum_{i=0}^r c_i(E) t^i = \\prod_{j=1}^r (1 + \\alpha_j t)$, where $\\alpha_1, \\ldots, \\alpha_r$ are the *Chern roots* of $E$.\nThe *Todd class* of $E$ is\n$$\\operatorname{td}(E) = \\prod_{j=1}^r \\frac{\\alpha_j}{1 - e^{-\\alpha_j}} \\in CH^\\bullet(X)_\\Q,$$\nwhere each factor is expanded as the power series $\\frac{x}{1 - e^{-x}} = 1 + \\frac{x}{2} + \\frac{x^2}{12} - \\frac{x^4}{720} + \\cdots$.\nThe product is symmetric in the $\\alpha_j$, so it is a polynomial in $c_1, \\ldots, c_r$ in each degree, and only finitely many degrees are nonzero.\nThe Todd class of $X$ is $\\operatorname{td}(X) = \\operatorname{td}(T_X)$, for the tangent bundle $T_X$.\n\n## Remarks\n\nIn low degrees,\n$$\\operatorname{td}(E) = 1 + \\frac{c_1}{2} + \\frac{c_1^2 + c_2}{12} + \\frac{c_1 c_2}{24} + \\cdots.$$\nThe Todd class is multiplicative: $\\operatorname{td}(E) = \\operatorname{td}(E') \\operatorname{td}(E'')$ for an exact sequence $0 \\to E' \\to E \\to E'' \\to 0$.\nExample: for a smooth projective curve $X$ of genus $g$, $\\operatorname{td}(X) = 1 + \\frac{1}{2} c_1(T_X)$, and $\\deg c_1(T_X) = 2 - 2g$, so the degree of the part in $CH^1$ is $1 - g$.\nNon-example: the power series $1 + x$ also gives a multiplicative class, but it is the total Chern class, not the Todd class.\nThe Todd series, with its coefficient $\\frac{1}{12}$ of $x^2$, is the one that makes the Hirzebruch–Riemann–Roch formula hold.\n",
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    "name": "Todd class",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/HD4/HD4EESS2DP.md",
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