Finiteness for self-dual classes in integral variations of Hodge structure

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.

[1]  Benjamin Bakker,et al.  Definable structures on flat bundles , 2022, Bulletin of the London Mathematical Society.

[2]  Thomas W. Grimm.  Moduli space holography and the finiteness of flux vacua , 2020, Journal of High Energy Physics.

[3]  Jacob Tsimerman,et al.  Definability of mixed period maps , 2020, Journal of the European Mathematical Society.

[4]  Benjamin Bakker,et al.  The global moduli theory of symplectic varieties , 2018, Journal für die reine und angewandte Mathematik (Crelles Journal).

[5]  B. Klingler.  Tame topology of arithmetic quotients and algebraicity of Hodge loci , 2018, Journal of the American Mathematical Society.

[6]  M. Verbitsky.  Ergodic complex structures on hyperkahler manifolds: an erratum , 2017, 1708.05802.

[7]  M. Orr.  Height bounds and the Siegel property , 2016, 1609.01315.

[8]  M. Verbitsky.  Ergodic complex structures on hyperkähler manifolds , 2013, 1306.1498.

[9]  Zhiqin Lu,et al.  Gauss–Bonnet–Chern theorem on moduli space , 2009, 0902.3839.

[10]  F. Denef.  Les Houches Lectures on Constructing String Vacua , 2008, 0803.1194.

[11]  B. Acharya,et al.  A Finite Landscape , 2006, hep-th/0606212.

[12]  Zhiqin Lu,et al.  On the Geometry of Moduli Space of Polarized Calabi-Yau manifolds , 2006, math/0603414.

[13]  S. Zelditch,et al.  Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics , 2004, math/0406089.

[14]  F. Denef,et al.  Distributions of flux vacua , 2004, hep-th/0404116.

[15]  S. Zelditch,et al.  Critical Points and Supersymmetric Vacua I , 2004, math/0402326.

[16]  M. Douglas,et al.  Counting flux vacua , 2003, hep-th/0307049.

[17]  M. Douglas,et al.  The statistics of string/M theory vacua , 2003, hep-th/0303194.

[18]  Dave Witte Morris,et al.  Introduction to Arithmetic Groups , 2001, math/0106063.

[19]  J. Polchinski,et al.  Hierarchies from fluxes in string compactifications , 2001, hep-th/0105097.

[20]  A. Wilkie.  A theorem of the complement and some new o-minimal structures , 1999 .

[21]  E. Witten,et al.  CFT's from Calabi–Yau four-folds , 1999, hep-th/9906070.

[22]  L. van den Dries,et al.  Tame Topology and O-minimal Structures , 1998 .

[23]  L. Dries,et al.  Geometric categories and o-minimal structures , 1996 .

[24]  C. Vafa.  Evidence for F theory , 1996, hep-th/9602022.

[25]  P. Deligné,et al.  On the locus of Hodge classes , 1994, alg-geom/9402009.

[26]  Helmut Klingen,et al.  Introductory Lectures on Siegel Modular Forms , 1990 .

[27]  A. Todorov.  The Weil-Petersson geometry of the moduli space ofSU(n≧3) (Calabi-Yau) manifolds I , 1989 .

[28]  G. Tian.  Smoothness of the Universal Deformation Space of Compact Calabi-Yau Manifolds and Its Peterson-Weil Metric , 1987 .

[29]  W. Schmid,et al.  DEGENERATION OF HODGE-STRUCTURES , 1986 .

[30]  A. Borel,et al.  Corners and arithmetic groups , 1973 .

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

[32]  Harish-Chandra,et al.  Arithmetic subgroups of algebraic groups , 1961 .

[33]  M. Orr,et al.  CORRECTION TO “HEIGHT BOUNDS AND THE SIEGEL PROPERTY” , 2022 .

[34]  A. Borel,et al.  Compactifications of Locally Symmetric Spaces , 2006 .

[35]  S. Zelditch,et al.  Communications in Mathematical Physics Critical Points and Supersymmetric Vacua , III : String / M Models , 2006 .

[36]  A. Wilkie.  Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function , 1996 .

[37]  Eckart Viehweg,et al.  Quasi-projective moduli for polarized manifolds , 1995, Ergebnisse der Mathematik und ihrer Grenzgebiete.

[38]  E. Cattani,et al.  DEGENERATING VARIATIONS OF HODGE STRUCTURE , 1989 .

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

[40]  H. Hironaka.  Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II , 1964 .