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In a previous paper, we described a natural closed subset, M 0 1,k(X,A; J), of the moduli space M1,k(X,A; J) of stable genusone J-holomorphic maps into a symplectic manifold X. In this paper we generalize the definition of the main component to moduli spaces of perturbed, in a restricted way, J-holomorphic maps and conclude that M 0 1,k(X,A; J), just like M1,k(X,A; J), carries a virtual fundamental class, which can be used to define symplectic invariants. These truly genus-one invariants constitute part of the standard genus-one Gromov-Witten invariants, which arise from the entire moduli space M1,k(X,A; J). The new invariants are more geometric and can be used to compute the genus-one GW-invariants of complete intersections, as shown in a separate paper.

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In a symmetric space of noncompact type X = G=K oriented geodesic segments correspond modulo isometries to vectors in the Euclidean Weyl chamber. We can hence assign vector valued lengths to segments. Our main result is a system of homogeneous linear inequalities, which we call the generalized triangle inequalities or stability inequalities, describing the restrictions on the vector valued side lengths of oriented polygons. It is based on the mod 2 Schubert calculus in the real Grassmannians G=P for maximal parabolic subgroups P. The side lengths of polygons in Euclidean buildings are studied in the related paper [KLM2]. Applications of the geomet- ric results in both papers to algebraic group theory are given in[KLM3].

Let X CPN be a smooth subvariety. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X which attempts to deform the given embedding into a balanced one. If L X is an ample line bundle, considering embeddings via H0(Lk) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldsons theorem relating balanced embeddings to metrics with constant scalar curvature [12]. In our proof we combine Donaldson乫s techniques with an asymptotic result of Liu and Ma [17] which, as we explain, describes the asymptotic behavior of the derivative of the map FS . Hilb whose fixed points are balanced metrics.