AI 中文总结
该研究针对偶维流形的正规光滑化，确定了由微分同胚实现扩展相交形式自同构的条件，并将结果应用于证明复偶维完全交的上同调环自同构均可由微分同胚实现。
AI 中文摘要
设$f: M \rightarrow B$是$2q$维流形$M$在某个$(B,\boldsymbol{\beta})$上的正规$(q-1)$-光滑化，其中$q$为偶数且$B$是单连通的。$M$上覆盖$B$的微分同胚诱导$f$的扩展相交形式（或称“$Q$-形式”）的自同构，该形式由$M$的相交形式与诱导同态$H_q(f)$构成。假设$H_q(B)$是自由的，我们精确确定哪些自同构可由此类微分同胚实现；特别地，若$H_{q-1}(B)$也为自由的，则证明所有自同构均可实现。这些结果通过研究作者前期工作中引入的扩展手术障碍的“双边”版本得到。作为应用，我们证明：对于复$q$维完全交（$q>2$且为偶数），其上同调环的每个自同构均可由微分同胚实现。
英文摘要
Suppose that $f : M \rightarrow B$ is a normal $(q-1)$-smoothing of a $2q$-manifold $M$ over some $(B,ξ)$, where $q$ is even and $B$ is simply-connected. A diffeomorphism of $M$ over $B$ induces an automorphism of the extended intersection form (or ``Q-form") of $f$, which consists of the intersection form of $M$ and the induced homomorphism $H_q(f)$. Assuming that $H_q(B)$ is free, we determine precisely which automorphisms can be realised by such diffeomorphisms. In particular, if $H_{q-1}(B)$ is also free, then we show that every automorphism can be realised. These results are obtained by studying a ``two-sided" version of the extended surgery obstruction, which was introduced in earlier work of the author. As an application, we show that for a complex $q$-dimensional complete intersection, with $q > 2$ even, every automorphism of the cohomology ring is realised by a diffeomorphism.
Comments22 pages
