TITLE: Semisimple quantum cohomology
QUESTION [7 upvotes]: Dubrovin's conjecture (or Bayer's modified version, if prefer) establishes a condition for the semisimplicity of the quantum cohomology of a manifold X, but, Why is important to know that the quantum cohomology of some manifold X is semisimple?

REPLY [4 votes]: One of the first interesting questions in GW theory asks how much information is needed to determine all GW invariants. The germ of a semisimple Frobenius is determined by a finite amount of data - this follows either from Dubrovin's or Manin's classification data via their structure connections. This implies that all genus-zero GW invariants are determined by this data; combined with Givental's conjecture/Teleman's theorem, this shows that all GW invariants are determined by the same finite amount of data. It also means that you could prove a mirror theorem (an isomorphism of Frobenius manifolds, in this case) by comparing finite amount of data on each side.
The mirror partner of $P^n$ (or any toric variety) is, as Kevin mentioned, constructed from the versal deformation of a function with isolated singularities (essentially this was proved by Givental as the first step of his mirror theorem). The Frobenius manifold associated to such a versal deformation is always generically semisimple. It seems reasonable to ask which other manifolds could have a function with isolated singularities as mirror partner. Clearly, having semisimple quantum cohomology is necessary (and one could motivate Dubrovin's conjecture by suggesting that it is a sufficient condition as well).

While I post here I should as well point out that "Bayer's modified version" was a little to optimistic. A counterexample is given by minimal surfaces of general type which have an exceptional vector bundle. At least one set of examples are the surfaces constructed by Yongnam Lee, Jongil Park in arXiv:math/0609072, on which a construction by Paul Hacking (which is included in arXiv:0808.1550) applies to produce exceptional vector bundles. On the other hand, all genus zero invariants vanish on minimal surfaces of general type by dimension reasons.