Hodge theory of degenerations, (I): consequences of the decomposition theorem

We use the Decomposition Theorem to derive several generalizations of the Clemens–Schmid sequence, relating asymptotic Hodge theory of a degeneration to the mixed Hodge theory of its singular fiber(s).

[1]  P. Deligne,et al.  Théorème de Lefschetz et Critères de Dégénérescence de Suites Spectrales , 1968 .

[2]  P. Griffiths,et al.  Some Enumerative Global Properties of Variations of Hodge Structures , 2009 .

[3]  Compactications defined by arrangements II: locally symmetric varieties of type IV , 2002, math/0201218.

[4]  C. Voisin.  Théorème de Torelli pour les cubiques de ℙ5 , 1986 .

[5]  J. Kollár,et al.  Threefolds and deformations of surface singularities , 1988 .

[6]  M. Saito,et al.  Smoothing of rational singularities and Hodge structure , 2019 .

[7]  Sandor J. Kovacs,et al.  Log canonical singularities are Du Bois , 2009, 0902.0648.

[8]  Sandor J. Kovacs,et al.  Families of varieties of general type over compact bases , 2007, 0704.2556.

[9]  S. Grushevsky,et al.  Complete moduli of cubic threefolds and their intermediate Jacobians , 2015, Proceedings of the London Mathematical Society.

[10]  V. Kulikov,et al.  Complex Algebraic Varieties: Periods of Integrals and Hodge Structures , 1998 .

[11]  The Leray spectral sequence is motivic , 2003, math/0301140.

[12]  M. Saito.  Onb-function, spectrum and rational singularity , 1993 .

[13]  M. Kerr,et al.  An Exponential History of Functions with Logarithmic Growth , 2009, 0903.4903.

[14]  Completion of Period Mappings and Ampleness of the Hodge bundle , 2017, 1708.09523.

[15]  D. Abramovich,et al.  WEAK SEMISTABLE REDUCTION IN CHARACTERISTIC 0 , 1997 .

[16]  J. Shah.  A Complete Moduli Space for K3 Surfaces of Degree 2 , 1980 .

[17]  Karim A. Adiprasito,et al.  Semistable reduction in characteristic 0 , 2018, 1810.03131.

[18]  C. Voisin.  Théorème de Torelli pour les cubiques de ℙ5 , 1986 .

[19]  F. Guillén,et al.  Sur le théorème local des cycles invariants , 1990 .

[20]  J. Milnor.  Singular points of complex hypersurfaces , 1968 .

[21]  K. O’Grady,et al.  GIT versus Baily-Borel compactification for quartic K3 surfaces , 2016, 1612.07432.

[22]  L. Migliorini,et al.  Hodge-theoretic aspects of the Decomposition Theorem , 2007, 0710.2708.

[23]  G. E. Bredon.  Introduction to compact transformation groups , 1972 .

[24]  N. M. Katz.  Elliptic convolution, G2, and elliptic surfaces , 2015 .

[25]  J. H. M. Steenbrink,et al.  Mixed Hodge Structure on the Vanishing Cohomology , 1977 .

[26]  Log canonical singularities and complete moduli of stable pairs , 1996, alg-geom/9608013.

[27]  Sandor J. Kovacs,et al.  Singularities of the minimal model program , 2013 .

[28]  A. Ash.  Smooth Compactification of Locally Symmetric Varieties , 1975 .

[29]  C. Voisin,et al.  A hyper-K\"ahler compactification of the Intermediate Jacobian fibration associated to a cubic fourfold , 2016, 1602.05534.

[30]  C. H. Clemens,et al.  Degeneration of Kähler manifolds , 1977 .

[31]  N. A'campo.  La fonction zêta d'une monodromie , 1975 .

[32]  Raymond Wells.  Comparison of de Rham and Dolbeault cohomology for proper surjective mappings. , 1974 .

[33]  G. Pearlstein,et al.  Singularities of admissible normal functions , 2007, 0711.0964.

[34]  C. Schnell.  An overview of Morihiko Saito's theory of mixed Hodge modules , 2014, 1405.3096.

[35]  C. Voisin,et al.  Remarks on degenerations of hyper-K\"ahler manifolds , 2017, 1704.02731.

[36]  Karl Schwede.  A simple characterization of Du Bois singularities , 2007, Compositio Mathematica.

[37]  M. Saito,et al.  Mixed Hodge modules , 1990 .

[38]  M. Kashiwara,et al.  The Poincare´ lemma for variations of polarized Hodge structure , 1987 .

[39]  Weak semistable reduction in characteristic 0 , 1997, alg-geom/9707012.

[40]  Valentine S. Kulikov.  Mixed Hodge Structures and Singularities , 1998 .

[41]  P. Brosnan.  Perverse obstructions to flat regular compactifications , 2016, 1612.01220.

[42]  L. Migliorini,et al.  The perverse filtration and the Lefschetz Hyperplane Theorem , 2008, 0805.4634.

[43]  Notes on Perverse Sheaves and Vanishing Cycles , 1999, math/9908107.

[44]  M. Saito,et al.  Modules de Hodge Polarisables , 1988 .

[45]  W. Schmid.  Variation of hodge structure: The singularities of the period mapping , 1973 .

[46]  K. O’Grady,et al.  Birational geometry of the moduli space of quartic $K3$ surfaces , 2016, Compositio Mathematica.

[47]  M. Saito,et al.  Hodge ideals and spectrum of isolated hypersurface singularities , 2019, Annales de l'Institut Fourier.

[48]  Da Silva,et al.  On the Limiting Behavior of Variations of Hodge Structures , 2016 .

[49]  M. Saito.  Decomposition theorem for proper Kähler morphisms , 1990 .

[50]  L. Migliorini,et al.  The Hodge theory of algebraic maps , 2003, math/0306030.

[51]  J. Shah.  Insignificant limit singularities of surfaces and their mixed Hodge structure , 1979 .

[52]  M. S. Nural,et al.  163 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? , 2005 .

[53]  A. Dimca,et al.  VANISHING CYCLE SHEAVES OF ONE-PARAMETER SMOOTHINGS AND QUASI-SEMISTABLE DEGENERATIONS , 2008, 0810.4896.

[54]  J. Shah.  Degenerations of $K3$ surfaces of degree $4$ , 1981 .

[55]  Sandor J. Kovacs.  Rational, Log Canonical, Du Bois Singularities: On the Conjectures of Kollár and Steenbrink , 1999, Compositio Mathematica.

[56]  M. Verbitsky.  A global Torelli theorem for hyperkahler manifolds , 2009, 0908.4121.

[57]  D. Mumford,et al.  Geometric Invariant Theory , 2011 .

[58]  Otto Forster,et al.  Lectures on Riemann Surfaces , 1999 .

[59]  B. Hassett.  Local stable reduction of plane curve singularities , 2000 .

[60]  W. Schmid,et al.  L2 and intersection cohomologies for a polarizable variation of Hodge structure , 1987 .

[61]  Liang,et al.  Li , 2018, Encyclopedic Dictionary of Archaeology.

[62]  M. Bradley.  Washington University in St , 2006 .

[63]  James D. Lewis,et al.  Specialization of cycles and the $K$-theory elevator , 2017, Communications in Number Theory and Physics.

[64]  M. Saito,et al.  Deformation of rational singularities and Hodge structure , 2019, Algebraic Geometry.

[65]  J. Steenbrink.  Limits of Hodge structures , 1976 .

[66]  Polarized relations on horizontal SL(2)s , 2017, 1705.03117.

[67]  T. Chow,et al.  Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties , 2015, 1511.00773.

[68]  E. Looijenga,et al.  The period map for cubic threefolds , 2006, Compositio Mathematica.

[69]  Motivic Igusa zeta functions , 1998, math/9803040.

[70]  P. Deligne.  Décompositions dans la catégorie dérivée , 1994 .

[71]  R. Laza.  The moduli space of cubic fourfolds via the period map , 2007, 0705.0949.

[72]  P. Deligne,et al.  Equations differentielles à points singuliers reguliers , 1970 .

[73]  K. O’Grady,et al.  GIT versus Baily-Borel compactification for K3's which are double covers of P1×P1 , 2021 .

[74]  John M Conly,et al.  Letter from the authors. , 2010, The Canadian journal of infectious diseases & medical microbiology = Journal canadien des maladies infectieuses et de la microbiologie medicale.

[75]  R. Laza.  The KSBA compactification for the moduli space of degree two K3 pairs , 2012, 1205.3144.

[76]  János Kollár,et al.  Birational Geometry of Algebraic Varieties , 1998 .

[77]  D. Toledo,et al.  The Moduli Space of Cubic Threefolds As a Ball Quotient , 2006, math/0608287.

[78]  A. Dimca,et al.  Some remarks on limit mixed Hodge structures and spectrum , 2012, 1210.3971.

[79]  M. Popa,et al.  Hodge Ideals , 2016, Memoirs of the American Mathematical Society.

[80]  F. Elzein,et al.  Mixed Hodge Structures , 2013, 1302.5811.