International Congress of Mathematicians

Three Questions in Gromov-Witten Theory 509(i) the intersection pairing g a t, = J x 7 a U 7&,(ii) the Hodge decomposition 7 a £ H Pa ' Qa (X,C),(iii) the action of the anticanonical class ci(X).The formulas for the operators L k are:Ek =2.^ / A [bg + lrili (C % )gtmdb,m+k-iro=0 j=0+ ^(-l)'»+ 1 [J a -m-l]f(C«)« J Ö 0ira Ö M _ ra „ 1 „ 1 )+ Ç(C k+1 ) ab t a 0t b oà~ko48 JX {(3-r)cr(X)-2c 1(X)c r -i(X)),where the Einstein convention for summing over the repeated indices a,b £ A isfollowed.Several terms require definitions. For each class 7 a , a half integer b a is obtainedfrom the Hodge decomposition,b a =p a + (l-r)/2.The combinatorial factor [x]f is defined by:[x]i = eu+i-i(x,x + l,...,x + k),where e^ is the fcth elementary symmetric function. The matrix C b a is determinedby the action of the anticanonical class,G Q 6 76=ci(X)U7 Q ,The indices of G are lowered and raised by the metric g a t, and its inverse g ab . Theterms t^ and 9 QjTO are defined by:fa l _ 4-a _ x xm — l mu aO"rol)da,m = 9/dt a m,where both are understood to vanish if m < 0.Conjecture 3. For all nonsingular projective varieties X, L\,(Z X ) = 0.The conjecture for varieties X with only (p,p) cohomology was made byEguchi, Hori, and Xiong [4]. The full conjecture involves ideas of Katz. In caseX is a point, the constraints specialize to the known Virasoro formulation of Witten'sconjecture [22], [30] (see also [25]). After the point, the simplest varietiesoccur in two basic families: curves C g of genus g and projective spaces P" of dimensionn. A proof of the Virasoro constraints for target curves C g is presentedin a sequence of papers [26], [27], [28]. Gi ventai has recently proven the Virasoro