Hodge theory of degenerations, (II): vanishing cohomology and geometric applications

Abstract

We study the mixed spectrum and vanishing cohomology for several classes of (isolated and nonisolated) hypersurface singularities, and how they contribute to the limiting mixed Hodge structure of a smoothing. Applications are given to several types of singularities arising in KSBA and GIT compactifications and mirror symmetry, including nodes, k-log-canonical singularities, singularities with Calabi-Yau tail, normal-crossing degenerations, slc surface singularities, and the J k,∞ series. ContentsIntroduction Set-up and overview of results Structure of the paper Synopsis of Part I ([KL19]) Acknowledgement 1. Mixed spectra of isolated singularities 2. The quasi-homogeneous case 2.1. Weighted Fermat singularities and deformations 2.2. Nodes on odd-dimensional hypersurfaces 3. Singularities with Calabi-Yau tail 4. Isolated hypersurface singularities: birational invariants 5. Isolated hypersurface singularities: spectral combinatorics 5.1. Toric geometry approach 5.2. Brieskorn lattice approach 6. Sebastiani-Thom formula 6.1. Application to k-log-canonical singularities 7. Non-isolated singularities 7.1. Remarks on rational, du Bois, normal-crossing, and k-lc singularities 7.2. The SSS formula 7.3. Some non-isolated slc surface singularities 7.4. The J k,∞ series 7.5. Clemens-Schmid discrepancies for nodal total spaces References

Deformations of singular Fano and Calabi-Yau varieties

“…Additionally, our definition of higher rational singularities in terms of the vanishing of certain cohomology groups (Definition 3.7) seems new. Under our assumptions, we show it agrees with the previous numerical definition of [KL20], but a priori it has a much wider scope. Section 4 looks at the case of weighted homogeneous singularities, where we have stronger results (e.g.…”

Remarks on eigenspectra of isolated singularities

“…at 0, and its mixed spectrum σ F . These were first computed by Steenbrink in [St], and we briefly review the treatment from [KL2,§2] before passing to eigenspectra.…”

“…for n odd. In the latter case, (C.2) yields no immediate bound on the number of nodes (though one does have results like [KL2,Thm. 2.9+Cor 2.11]).…”

“…Recent work of the third author and R. Laza on the Hodge theory of degenerations [KL1,KL2] re-examined the mixed Hodge theory of the Clemens-Schmid and vanishing-cycle sequences, with an emphasis on applications to limits of period maps and compactifications of moduli. When a degenerating family of varieties has a finite group G acting on its fibers, these become exact sequences in the category of mixed Hodge structures with G × µ k -action, where k is the order of T ss (the semisimple part of monodromy).…”

On k-rational and k-Du Bois local complete intersections

“…Note. The notion of k-rationality for hypersurfaces first appeared in [KL20] precisely as the condition α(Z) > k + 1. Here we are of course using the more natural definition in terms of log resolutions suggested in [FL22].…”