| 摘要: | 這個計劃是我上一個計劃(2006量子上同調.和環面幾何)的第二部分, 其主要目的是要去探討比較在 smooth toric flops 中, 它們的 quantum cohomology 的不變性. 我們已經知道對於 ordinary flops, 它們的 Chow motives 是同構的, 這說明它們的整上同調.是同構的, 但從實際計算中知道它們的整上同調環有不同的乘法結構, 我們可以在 simple ordinary flops 之下確切得到它們乘法結構的誤差公式, 並在quantum product 中彌補這個誤差. 我們又知道 smooth toric flops 都是 ordinary flops, 而 smooth toric flops 是比較可進行實際計算的, 因此我們打算先計算 base 是 projective spaces 的 smooth toric flops 的乘法結構的誤差公式, 並期望在此情況之下得到它們的 quantum cohomology 的不變性. 若順利, 我們將探討同樣問題在更一般的 base, 希望能對所有的 smooth toric flops 有充分的瞭解, 進而幫忙研究一般的 ordinary flops. This is a continuation of my 2006 NSC project: Quantum Cohomology and Toric Geometry. The aim of this research is to understand the invariance of quantum cohomology under smooth toric flops. We have shown that for an ordinary flop, the graph closure induces an isomorphism between their Chow motives. The Chow motives certainly imply the universal cohomology theory, so their integral cohomology groups are isomorphic under the graph closure. The problem is that the ring structures of their cohomology are different. For simple ordinary flops, we have computed the defect of the product structure of their cohomology rings and remedy the defect under considering the quantum product. Since we have also shown that any smooth toric flop is ordinary, in this reseach, we try to do the same things for smooth toric flops due to the highly computability of toric varieties, and then to understand the same problem for general ordinary flops. I expect to achieve the following results: 1. To determine the exact formula for the defect of classical product for general smooth toric flops with smooth toric base. 2. To make a detail study of Euler data on toric manifolds to obtain a generalized multiple cover formula. In particular, to obtain the corresponding formula for one point invariants with $\psi$ class at least for the case with $S = \mathbb{P}^m$. 3. To carry out the necessary combinatorial works resulting from the cohomology of base $S = \mathbb{P}^m$. This includes the explicit reduction formula coming from the divisor relation. 研究時間：9608 ~ 9707 |
