arXiv math/07032972007-03math.SGmath.DGmath.RT

再论Duistermaat-Heckman测度的对数-对数猜想

The log-concavity conjecture for the Duistermaat-Heckman measure revisited

Yi Lin

备注：: This is the version which is going to appear in Inter. Math. Research Notices. A few references are added. Some minor mistakes are corrected

摘要：

Karshon构造了Duistermaat-Heckman测度的对数-对数猜想的第一个反例：一个Hamilton六流形，其不动点集是两个$T^4$的不交并集。对任意闭辛四流形$N$，若$b+$大于1，证明了存在一个Hamilton六流形$M$，使得它的不动点集是$N$的两个副本的不交并，且其Duistermaat-Heckman函数不是对数凹的.另一方面，证明了若存在复杂度为2的环面作用，使得所有取正则值的辛约化空间满足条件$b+=1$，则其Duistermaat-Heckman函数必为对数凹函数.因此，我们证明了六流形上Hamilton圈作用的log-dummy猜想，使得不动点集没有四维分量，或只有四维片，且有$b+=1$ .

英文摘要：

Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat-Heckman measure: a Hamiltonian six manifold whose fixed points set is the disjoint union of two copies of $T^4$. In this article, for any closed symplectic four manifold $N$ with $b+$ greater than 1, we show that there is a Hamiltonian six manifold $M$ such that its fixed points set is the disjoint union of two copies of $N$ and such that its Duistermaat-Heckman function is not log-concave. On the other hand, we prove that if there is a torus action of complexity two such that all the symplectic reduced spaces taken at regular values satisfy the condition $b+=1$, then its Duistermaat-Heckman function has to be log-concave. As a consequence, we prove the log-concavity conjecture for Hamiltonian circle actions on six manifolds such that the fixed points sets have no four dimensional components, or only have four dimensional pieces with $b+=1$.

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