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.

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Păun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature Θ^{$h$}as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to Θ^{$h$}. Finally we define what it means for$h$to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.

We prove a quantitative bi-Lipschitz non-embedding theorem for the Heisenberg group with its Carnot–Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans–Linial semidefinite relaxation of the sparsest cut problem.

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Based on large N Chern-Simons/topological string duality, in a series of papers [36, 21, 19], Labastida, Marino, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.