(Crepant morphism)
Let \(X\) and \(Y\) be normal varieties whose canonical divisors are \(\QQ\)-Cartier. A proper birational morphism \(f \colon Y \to X\) is crepant if \(K_Y = f^* K_X\) as \(\QQ\)-divisor classes, that is, every \(f\)-exceptional prime divisor has discrepancy \(0\). A crepant resolution is a crepant \(f\) with \(Y\) smooth.
The minimal resolution of an ADE surface singularity is crepant: its exceptional curves \(E\) are smooth rational curves with \(E^2 = -2\), so \(K_Y \cdot E = 0\) for each, and \(K_Y - f^* K_X\), supported on the exceptional curves, is zero because the intersection form on them is negative definite. The blowup \(f \colon Y \to X\) of a smooth point on a surface is not crepant: \(K_Y = f^* K_X + E\). The quotient \(\CC^4 / \{\pm 1\}\) has no crepant resolution, because its singularity is terminal: every exceptional divisor of every resolution has positive discrepancy.
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D-SRFADEADE surface singularitiesrelated-to