  We calculate near-horizon solutions for four-dimensional 4-charge and
five-dimensional 3-charge black holes in heterotic string theory from the part
of the ten-dimensional tree-level effective action which is connected to
gravitational Chern-Simons term by supersymmetry. We obtain that the entropies
of large black holes exactly match the alpha'-exact statistical entropies
obtained from microstate counting (D=4) and AdS/CFT correspondence (D=5).
Especially interesting is that we obtain agreement for both BPS and non-BPS
black holes, contrary to the case of R^2-truncated (four-derivative) actions
(D-dimensional N=2 off-shell supersymmetric or Gauss-Bonnet) were used, which
give the entropies agreeing (at best) just for BPS black holes. The key
property of the solutions, which enabled us to tackle the action containing
infinite number of terms, is vanishing of the Riemann tensor \bar{R}_{MNPQ}
obtained from torsional connection defined with \bar{\Gamma} = \Gamma - H/2.
Morover, if every monomial of the remaining part of the effective action would
contain at least two Riemanns \bar{R}_{MNPQ}, it would trivially follow that
our solutions are exact solutions of the full heterotic effective action in
D=10. The above conjecture, which appeared (in this or stronger form) from time
to time in the literature, has controversial status, but is supported by the
most recent calculations of Richards (arXiv:0807.3453 [hep-th]). Agreement of
our results for the entropies with the microscopic ones supports the
conjecture. As for small black holes, our solutions in D=5 still have singular
horizons.
