\documentclass[11pt]{amsart}<br><br><br><br><br>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%<br>\begin{document}<br><br>Dear Editor,<br><br>\bigskip<br>\bigskip<br><br><br>I recommend the paper {\it An update on semisimple quantum<br>
cohomology and F--manifolds} by C. Hertling, Yu. I. Manin and C.<br>Teleman, for publication in Trudy Matematicheskogo Instituta.<br><br><br>The paper deals with two problems in the theory of Frobenius<br>manifolds and closely related \emph{F--manifolds} (introduced by C.<br>
Hertling and Yu.~I.~Manin).  In a purely even version, an<br>$F$--structure on a manifold $M$ endows $\mathcal{T}_M$ with a<br>commutative and associative $\mathcal{O}_M$--bilinear multiplication<br>$\circ$ satisfying the following identity: for all local vector<br>
fields $X,Y,Z,U$<br>$$<br>P_{X\circ Y}(Z,U)=X\circ P_Y(Z,U)+Y\circ P_X(Z,U);<br>$$<br>the existence of an identity vector field $e$ is postulated, too.<br><br>\medskip<br><br><br>\bf Results. \rm The two results presented in the paper are of<br>
foundational character in these related theories. The results answer<br>problems that have been standing for a while and may be of interest<br>to general readership.<br><br>\medskip<br>{ Theorem 1.3.} {\it  If the even quantum cohomology of a projective<br>
algebraic mani\-fold $V$ is generically semi--simple, then $V$ has<br>no odd cohomology and is of Hodge--Tate type.}<br><br><br><br>{Theorem 2.5.} {\it The multiplication $\circ$<br> satisfies the structure identity of $F$--manifolds,<br>
 if and only if the ideal $J(M,\circ )\subset Symm_{\mathcal{O}_M}{\mathcal{T}_M}$<br>generated by $e-1, \; X\circ Y- X \cdot Y$  is stable with respect<br>to the Poisson brackets on $\mathcal{T_M^\ast}$.}<br><br>\bigskip<br>
<br>Possible applications are discussed in the paper, specifically, in<br>mirror symmetry and singularity theory.<br><br><br><br>\bigskip \bf<br>Typos. \rm ... adresses...  p. 1, 1 paragraph from t.<br><br>... cup product... p. 2, 2 par. from b.<br>
<br>... coordinates... p. 4, 1 par. from b.<br><br>\medskip<br><br>\bf Suggestions. \rm A few references to prior work on<br>$F$--manifolds at the beginning of section 2 might be helpful.<br><br>\bigskip<br>\bigskip<br>\bigskip<br>
\bigskip<br><br>Sincerely yours,<br><br><br><br><br><br><br>\end{document}<br><br>
