[论文解读] Symplectic structures of algebraic surfaces and deformation

Fabrizio Catanese|ArXiv.org|Jul 26, 2002

Geometric and Algebraic Topology参考文献 17被引用 26

一句话总结

本文证明了一般型极小曲面装备有唯一的典范辛结构（在辛同胚意义下唯一），该结构在光滑形变及某些Q-Gorenstein奇点光滑化下保持不变。利用Manetti给出的非形变等价但微分同胚的曲面例子，作者证明这些曲面在典范辛形式下是辛同胚的，并将此结果应用于证明存在具有尖点的平面曲线，它们辛同痕但并非等奇点形变等价。

ABSTRACT

Friedman and Morgan made the "speculation" that deformation equivalence and diffeomorphism should coincide for algebraic surfaces. Counterexamples, for the hitherto open case of surfaces of general type, have been given in the last years by Manetti, by Kharlamov-Kulikov and in my cited article. For the latter much simpler examples, it was shown that there are surfaces $S$ which are not deformation equivalent to their complex conjugate. However, by Seiberg-Witten theory, any (oriented) diffeomorphism of minimal surfaces carries the canonical class K to + K or to - K, and deformation equivalence implies the existence of a diffeomorphism carrying K to +K. In fact, as observed by a referee, the bulk of the proof was to show that our surfaces have no selfhomeomorphism carrying K to - K (the same for the K-K surfaces). In this note we show that Manetti's surfaces provide indeed a counterexample to the reinforced conjecture, since they are symplectomorphic. Our result is that a surface of general type has a canonical symplectic structure (up to symplectomorphism) which is invariant for deformation and for certain degenerations to normal surfaces. Since moreover no simply connected counterexamples to the conjecture are known, we provide explicit families of 1-connected surfaces, which are obtained by glueing together two fixed manifolds with boundary, are not deformation equivalent, but are homeomorphic under a homeomorphism carrying K to +K. We also give as application the existence of symplectically equivalent, but not deformation equivalent cuspidal plane curves.

研究动机与目标

• 建立极小一般型曲面上存在一个在光滑形变下不变的典范辛结构。
• 研究代数曲面在存在奇点时，形变类型与辛同胚类型是否一致。
• 分析商奇点的Q-Gorenstein光滑化对辛不变性与微分同胚类型的影响。
• 将结果应用于尖点平面曲线的同痕问题，证明辛同痕不蕴含等奇点形变等价。

提出的方法

• 通过m次普利坎纳多嵌入到射影空间，将Fubini-Study形式拉回到m ≥ 4时的典范辛形式。
• 利用Veronese嵌入与Moser的辛同痕定理，证明不同m下的辛同胚不变性。
• 通过考虑典范模型并光滑其有理双曲点奇点，将构造推广至非ample典范除子的情形。
• 该构造依赖于Q-Gorenstein光滑化保持辛结构的事实，此点通过变形理论及光滑化局部的不可约性得到证明。
• 该方法适用于中心纤维具有单个光滑化奇点（SSS）的曲面族，确保光滑纤维之间的辛同胚性。
• 通过Milnor纤维粘合与非全纯分支覆盖，将辛不变性推广至通过光滑典范模型得到的曲面。

实验结果

研究问题

• RQ1 极小一般型曲面上的典范辛结构是否在光滑形变及Q-Gorenstein奇点光滑化下保持不变？
• RQ2 两个一般型曲面能否微分同胚且辛同胚但非形变等价？
• RQ3 辛同痕的尖点平面曲线是否必然等奇点形变等价？
• RQ4 不可约光滑化分支在决定光滑纤维的辛同胚类型中起何作用？
• RQ5 商奇点的Q-Gorenstein光滑化如何影响所得光滑曲面的辛类型与微分同胚类型？

主要发现

• 一般型极小曲面装备有唯一的典范辛结构（在辛同胚意义下唯一），其上同调类即为典范类。
• 虽非形变等价，但由同一奇异中心纤维经Q-Gorenstein光滑化得到的曲面是辛同胚的。
• Manetti给出的非形变等价一般型曲面是辛同胚且微分同胚的，且存在保持典范类的微分同胚。
• 即使典范除子非ample，形变允许单个光滑化奇点（SSS）时，辛同胚类型仍保持不变。
• 由非形变等价曲面导出的尖点平面曲线是辛同痕但非等奇点形变等价的。
• 曲面上的辛结构在典范模型构造及有理双曲点光滑化下保持不变，从而确保形变过程中的不变性。