448 research outputs found
Logarithmic Gromov-Witten invariants
The goal of this paper is to give a general theory of logarithmic
Gromov-Witten invariants. This gives a vast generalization of the theory of
relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun
Li, and completes a program first proposed by the second named author in 2002.
One considers target spaces X carrying a log structure. Domains of stable log
curves are log smooth curves. Algebraicity of the stack of such stable log maps
is proven, subject only to the hypothesis that the log structure on X is fine,
saturated, and Zariski. A notion of basic stable log map is introduced; all
stable log maps are pull-backs of basic stable log maps via base-change. With
certain additional hypotheses, the stack of basic stable log maps is proven to
be proper. Under these hypotheses and the additional hypothesis that X is log
smooth, one obtains a theory of log Gromov-Witten invariants.Comment: 58 pages, 5 figure
Symplectic Manifolds with Vanishing Action-Maslov Homomorphism
The action--Maslov homomorphism I\co\pi_1(\text{Ham}(X,\omega))\to\R is an
important tool for understanding the topology of the Hamiltonian group of
monotone symplectic manifolds. We explore conditions for the vanishing of this
homomorphism, and show that it is identically zero when the Seidel element has
finite order and the homology satisfies property (a
generalization of having homology generated by divisor classes). We use these
results to show that for products of projective spaces and the
Grassmannian of planes in \C^4.Comment: 21 pages, rewritten to remove unnecessary information and correct
typographical error
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