Uniruled Symplectic Divisors
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Uniruled Symplectic Divisors Tian-Jun Li · Yongbin Ruan
Received: 2 May 2013 / Accepted: 21 May 2013 / Published online: 9 July 2013 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2013
Abstract In this article, we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry. In particular, we generalize Maulik–Pandharipande’s relative/absolute correspondence to relative-divisor/absolute correspondence. Then, we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold. Keywords Birational symplectic geometry · Gromov–Witten invariants · Symplectic divisor · Uniruled invariant Mathematics Subject Classification (2010) 53D05
1 Introduction The relation between the ambient manifold Gromov–Witten theory and that of a smooth divisor is a basic problem in geometry and physics. It was clear from the history that this problem is a very complicated one and it is unlikely any simple formula can be obtained. However, there are a number of beautiful results computing the divisor invariants in terms of the ambient space invariants (see [2, 4, 5, 12, 25, 32]). In this article, we consider the opposite direction lifting an invariant of divisor to that of ambient manifold. On such a general ground one expects no good solution. Therefore, to have any reasonable answer, we have to choose our invariant T.-J. Li School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA e-mail: [email protected] Y. Ruan () Department of Mathematics, University of Michigan, Ann Arbor, MI, USA e-mail: [email protected]
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T.-J. Li, Y. Ruan
carefully. Our choice is motivated by symplectic birational geometry. In symplectic birational geometry, uniruledness is a central notion defined via GW invariants. More precisely, a symplectic manifold is called uniruled if there is a non-vanishing genus zero GW-invariant involving a point insertion and with a nontrivial class curve. In [9], we proved that uniruledness is invariant under symplectic birational cobordism. The next step of symplectic birational geometry program is to study various surgery operations such as contraction, flip and flop. The main perspective comes from the basic fact in the projective birational program that various birational surgery operations have a common feature: the subset being operated on is necessarily uniruled. Therefore, in our program we also need to understand uniruled symplectic submanifolds. In this article and its sequel, we focus on symplectic uniruled divisors. Our key observation is that, as in the projective birational program, such a divisor admits a dichotomy depending on the positivity of its normal bundle. If the normal bundle is non-negative in certain sense, it will force the ambient manifold to be uniruled. If the normal bundle is negative in certain sense, we can contract it to obtain a simpler symplectic manifold. In this article, we treat the case of
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