Defect of projective hypersurfaces with isolated singularities
Seung-Jo Jung, Morihiko Saito
TL;DR
The paper studies defect ${\rm def}(X)=h^{n+1}(X)-h^{n-1}(X)$ for hypersurfaces with isolated singularities, connecting it to cokernels between cohomology and intersection cohomology, vanishing-cycle data, and Milnor fiber cohomology. It develops a framework based on vanishing cycles, mixed Hodge modules, and the pole-order spectral sequence to compute ${\rm def}(X)$, including explicit computations for weighted homogeneous singularities and numerous numerical examples. It proves equalities ${\rm def}(X)=\dim{\rm Coker}\iota^{n-1}=\dim{\rm Coker}\rho=\dim{\rm Im}\sigma=\dim H^n(F_f)_1$ and, under $n=3$ with certain spectral-number conditions, derives relations for graded pieces of IH^3 and H^3 in families. In addition, the paper provides a topological proof of the Park–Popa formula for the $\mathbb{Q}$-factoriality defect, linking Div$ (X)/$CDiv$(X)$ to local Picard data and establishing conditions under which the defect vanishes, thereby connecting singularity theory, Hodge theory, and factoriality in a computable framework.
Abstract
Let $X$ be a hypersurface with isolated singularities defined by $f$ in ${\bf P^{n+1}}$ with $n>1$. The difference ${\rm def}(X):=h^{n+1}(X)-h^{n-1}(X)$ is called the defect of $X$ (for self-duality of the cohomology of $X$). It is known that its vanishing is closely related to ${\bf Q}$-factoriality of $X$ in the rational singularity case with $n=3$. This number coincides with the dimension of the cokernel of the inclusion $H^{n-1}(X)\to{\rm IH}^{n-1}(X)$, the rank of the morphism from the vanishing cohomologies of $X$ to $H^{n+1}(X)$ for a one-parameter smoothing of $X$ with total space smooth, and also with the dimension of the unipotent monodromy part of the Milnor fiber cohomology of $f$ with degree $n$. In the case $X$ has only weighted homogeneous isolated singularities, the defect ${\rm def}(X)$ is then given by the $E_2$-term of the spectral sequence of the double complex with differentials ${\rm d}f\wedge$ and $\rm d$ by the $E_2$-degeneration of the pole order spectral sequence. It can be calculated explicitly using a computer even for analogues of the Hirzebruch quintic threefold with more than one hundred ordinary double points found by B.\ van Geemen and J.\ Werner in a compatible way with their computation. We give also an example with ${\rm def}(X)>0$ and $|{\rm Sing}\,X|=1$ where $n=3$.