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We construct a monomorphism of the De Rham complex of scalar multivaluedmeromorphic forms on the projective line, holomorphic on the complement to afinite set of points, to the chain complex of the Lie algebra of $sl_2$-valuedalgebraic functions on the same complement with coefficients in a tensorproduct of contragradient Verma modules over the affine Lie algebra$\hat{sl}_2$. We show that the existence of singular vectors in the Vermamodules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the newrelations between the cohomology classes of logarithmic differential forms.

Rational differential forms on line and singular vectors in Verma modules over $\widehat {sl}_2$