Deformation of rational singularities and Hodge structure
Deformation of rational singularities and Hodge structure
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DOI：
10.14231/ag-2022-014
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发表时间：
2019-06
影响因子：
1.5
通讯作者：
M. Kerr;R. Laza;M. Saito
中科院分区：
文献类型：
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作者：
M. Kerr;R. Laza;M. Saito
对于n维约化紧致复解析空间的单参数退化，我们证明了边界Hodge数h^{p，q}$（即pq（n{-}p）（n {-}q）=0$）对于纤维的交上同调以及它们的去奇异化的上同调的不变性，假设中心纤维是约化的，投射的，并且只有有理奇点。这可以被证明等价于结构层的上同调维数的不变性（这在可代数化的情况下是已知的），因为我们可以证明所有霍奇数h^{p，q}$的霍奇对称性，以及附近纤维的霍奇到德拉姆谱序列的E_1 $-退化，只假设中心纤维的投射性。为了证明主要定理，我们计算了全空间的交复Hodge模上Hodge滤的第一个非零成员的诱导$V$-滤的分次片，这与去奇异化的对偶层的直接像一致（与Koll\'ar关于光滑簇的对偶层的直接像的猜想有关）。这一计算还意味着，在主定理假设一般纤维进一步光滑的情况下，局部单值性的零阶比一般奇异性情况小2。在某种假设下，我们可以证明主要定理的部分匡威。
For a one-parameter degeneration of reduced compact complex analytic spaces of dimension $n$, we prove the invariance of the frontier Hodge numbers $h^{p,q}$ (that is, with $pq(n{-}p)(n{-}q)=0$) for the intersection cohomology of the fibers and also for the cohomology of their desingularizations, assuming that the central fiber is reduced, projective, and has only rational singularities. This can be shown to be equivalent to the invariance of the dimension of the cohomology of structure sheaf (which is known in the algebraizable case), since we can prove the Hodge symmetry for all the Hodge numbers $h^{p,q}$ together with $E_1$-degeneration of the Hodge-to-de Rham spectral sequence for nearby fibers, assuming only the projectivity of the central fiber. For the proof of the main theorem, we calculate the graded pieces of the induced $V$-filtration for the first non-zero member of the Hodge filtration on the intersection complex Hodge module of the total space, which coincides with the direct image of the dualizing sheaf of a desingularization (related to Koll\'ar's conjecture on the direct images of dualizing sheaves of smooth varieties). This calculation implies also that the order of nilpotence of the local monodromy is smaller than the general singularity case by 2 in the situation of the main theorem assuming further smoothness of general fibers. We can prove a partial converse of the main theorem under some hypothesis.