Mirror principle. II (Q2739542)

For part I see \textit{B. H. Lian}, \textit{K. Lin} and \textit{S.-T. Yau}. NEWLINENEWLINENEWLINEFrom the introduction: The present paper is a sequel to ``Mirror Principle. I'', by the same authors [ibid. 405-454 (1999; see the preceding review Zbl 0999.14010)]. Here, we generalize all the results there to a class of \(T\)-manifolds which we call ballon manifolds.NEWLINENEWLINENEWLINELet \(X\) be a projective \(n\)-fold, and \(d\in H^2(X,\mathbb{Z})\). Let \(M_{0,k}(d,X)\) denote the moduli space of \(k\)-pointed, genus 0, degree \(d\), stable maps \((C,f,x_1,\dots,x_k)\) on \(X\). By the work of \textit{J. Li} and \textit{G. Tian} [J. Am. Math. Soc. 11, No. 1, 119-174 (1998; Zbl 0912.14004)], each non-empty \(M_{0,k}(d,X)\) admits a cycle class \(LT_{0,k}(d,X)\) in the Chow group of degree \(\dim X+\langle c_1(X),d\rangle+n-3\). This cycle plays the role of the fundamental class in topology, hence \(LT_{0,k}(d,X)\) is called the virtual fundamental class.NEWLINENEWLINENEWLINELet \(V\) be a convex vector bundle on \(X\) (i.e. \(H^1 (\mathbb{P}^2,f^*V)=0\) for every holomorphic map \(f:\mathbb{P}^1\to X)\). Then \(V\) induces on each \(M_{0,k}(d,X)\) a vector bundle \(V_d\), with fiber at \((C,f,x_1, \dots, x_k)\) given by the section space \(H^0(C,f^*V)\). Let \(b\) be any multiplicative characteristic class (i.e., if \(0\to E'\to E\to E''\to 0\) is an exact sequence of vector bundles, then \(b(E)=b (E')b(E'').)\) The problem we study here is to compute the characteristic number NEWLINE\[NEWLINEK_d:= \int_{LT_{0,0} (d,X)}b(V_d)NEWLINE\]NEWLINE and their generating function: \(\Phi(t):=\sum K_de^{d\cdot t}\). There is a similar and equally important problem if one starts from a concave vector bundle \(V\) (i.e., \(H^0(\mathbb{P}^1, f^*V)=0\) for every holomorphic map \(f:\mathbb{P}^1\to X)\). More generally, \(V\) can be a direct sum of a convex and a concave bundle. Important progress made on these problems has come from mirror symmetry. All of it seems to point toward the following general phenomenon, which we call the ``mirror principle''. Roughly, it says that the function \(\Phi(t)\) can be computed by a change of variables in terms of certain explicit special functions, loosely called generalized hypergeometric functions.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00007].