Geodesic
Smoothing toroidal crossing spaces
Forum of Mathematics, Pi, Tome 9 (2021)

Voir la notice de l'article provenant de la source Cambridge University Press

We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications. 
@article{10_1017_fmp_2021_8,
     author = {Simon Felten and Matej Filip and Helge Ruddat},
     title = {Smoothing toroidal crossing spaces},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {9},
     year = {2021},
     doi = {10.1017/fmp.2021.8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.8/}
}
TY  - JOUR
AU  - Simon Felten
AU  - Matej Filip
AU  - Helge Ruddat
TI  - Smoothing toroidal crossing spaces
JO  - Forum of Mathematics, Pi
PY  - 2021
VL  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.8/
DO  - 10.1017/fmp.2021.8
LA  - en
ID  - 10_1017_fmp_2021_8
ER  - 
%0 Journal Article
%A Simon Felten
%A Matej Filip
%A Helge Ruddat
%T Smoothing toroidal crossing spaces
%J Forum of Mathematics, Pi
%D 2021
%V 9
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.8/
%R 10.1017/fmp.2021.8
%G en
%F 10_1017_fmp_2021_8
Simon Felten; Matej Filip; Helge Ruddat. Smoothing toroidal crossing spaces. Forum of Mathematics, Pi, Tome 9 (2021). doi: 10.1017/fmp.2021.8

[1] Abouzaid, M., Auroux, D. and Katzarkov, L., ‘Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces’, Publ. Math. Inst. Hautes Études Sci. 123 (2016), 199–282.Google Scholar | DOI

[2] Bănică, C. and Stănăşilă, O., Algebraic Methods in the Global Theory of Complex Spaces (Bucharest; John Wiley & Sons, London, 1976). Translated from Romanian.Google Scholar

[3] Barannikov, S. and Kontsevich, M., ‘Frobenius manifolds and formality of Lie algebras of polyvector fields’, Int. Math. Res. Notices 4 (1998), 201–215.Google Scholar | DOI

[4] Barrott, La. J. and Nabijou, Navid, ‘Tangent curves to degenerating hypersurfaces’ (2020). URL: .Google Scholar | arXiv

[5] Blickle, M., ‘Cartier isomorphism for toric varieties’, J. Algebra 237(1) (2001), 342–357.Google Scholar | DOI

[6] Chan, K., Lau, S.-C. and Conan Leung, N., ‘SYZ mirror symmetry for toric Calabi–Yau manifolds’, J. Differ. Geom. 90(2) (2012), 177–250.Google Scholar | DOI

[7] Chan, K., Leung, N. C. and N. Ma, Z., ‘Geometry of the Maurer–Cartan equation near degenerate Calabi–Yau varieties’ (2015). URL: .Google Scholar | arXiv

[8] Chan, K. and N. Ma, Z., ‘Smoothing pairs over degenerate Calabi–Yau varieties’ (2019). URL: .Google Scholar | arXiv

[9] Chiang, T.-M., Klemm, A., Yau, S.-T. and Zaslow, Eric, ‘Local mirror symmetry: calculations and interpretations’, Adv. Theor. Math. Phys. 3(3) (1999), 495–565.Google Scholar | DOI

[10] Coates, T., Corti, A., Galkin, S., Golyshev, V. and Kasprzyk, A., ‘Mirror symmetry and Fano manifolds’, in European Congress of Mathematics (European Mathematical Society, Zürich, Switzerland, 2013), 285–300.Google Scholar | DOI

[11] Corti, A., Filip, M. and Petracci, A., ‘Mirror symmetry and smoothing Gorenstein toric affine 3-folds’ (2020). URL: .Google Scholar | arXiv

[12] Danilov, V. I., ‘The geometry of toric varieties’, Uspekhi Mat. Nauk. 33(2) (1978), 85–134.Google Scholar

[13] Deligne, P. and Illusie, L., ‘Relèvements modulo et décomposition du complexe de de Rham’, Invent. Math. 89(2) (1987), 247–270.Google Scholar | DOI

[14] Fantechi, B., Franciosi, M. and Pardini, R., ‘Deformations of semi-smooth varieties’ (2020). URL: .Google Scholar | arXiv

[15] Felten, S., ‘Log smooth deformation theory via Gerstenhaber algebras’, Manuscr. Math. (2020), 35 pages. URL: https://link.springer.com/article/10.1007/s00229-020-01255-6.Google Scholar | DOI

[16] Fiorenza, D., Manetti, M. and Martinengo, E., ‘Cosimplicial DGLAs in deformation theory’, Comm. Algebra, 40(6) (2012), 2243–2260.Google Scholar | DOI

[17] Friedman, R., ‘Global smoothings of varieties with normal crossings’, Ann. Math. 118(1) (1983), 75–114.Google Scholar | DOI

[18] Gross, M., Katzarkov, L. and Ruddat, H., ‘Towards mirror symmetry for varieties of general type’, Adv. Math. 308 (2017), 208–275.Google Scholar | DOI

[19] Gross, M. and Siebert, B., ‘Mirror symmetry via logarithmic degeneration data. I’, J. Differ. Geom. 72(2) (2006), 169–338.Google Scholar | DOI

[20] Gross, M. and Siebert, B., ‘Mirror symmetry via logarithmic degeneration data, II’, J. Algebraic Geom. 19(4) (2010), 679–780.Google Scholar | DOI

[21] Gross, M. and Siebert, B., ‘From real affine geometry to complex geometry’, Ann. Math. 174 (2011), 1301–1428.Google Scholar | DOI

[22] Grothendieck, A., ‘Éléments de géométrie algébrique. I. Le langage des schémas’, Publ. Math. Inst. Hautes Etudes Sci. 4 (1960), 5–228.Google Scholar | DOI

[23] Grothendieck, A., ‘Techniques de construction en géométrie analytique. III. Produits fibrés d’espaces analytiques’, Séminaire Henri Cartan 13(1) (1960–1961), 1–11.Google Scholar

[24] Grothendieck, A., ‘Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II’, Publ. Math. Inst. Hautes Etudes Sci. 24(231) (1965), 5–231.Google Scholar

[25] Grothendieck, A., ‘Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III’, Publ. Math. Inst. Hautes Etudes Sci. 28(255) (1966), 5–255.Google Scholar | DOI

[26] Hashimoto, K. and Sano, T., ‘Examples of non-Kähler Calabi–Yau 3-folds with arbitrarily large b2’ (2019). URL: .Google Scholar | arXiv

[27] Hassett, B. and Kovács, S. J., ‘Reflexive pull-backs and base extension’, J. Algebr. Geom. 13(2) (2004), 233–247.Google Scholar | DOI

[28] Iacono, D. and Manetti, M., ‘An algebraic proof of Bogomolov–Tian–Todorov theorem’, in Deformation Spaces (Vieweg + Teubner, Wiesbaden, Germany, 2010), 113–133.Google Scholar | DOI

[29] Illusie, L., ‘Frobenius and Hodge degeneration. Introduction to Hodge theory’, in SMF/AMS Texts and Monographs, 8 (American Mathematical Society, Providence, RI, 2002). Translated from the 1996 French original by James Lewis and Peters, 96–145.Google Scholar

[30] Kato, F., ‘Log smooth deformation theory’, Tohoku Math. J., Second Series, 48(3) (1996), 317–354.Google Scholar

[31] Kato, F., ‘Log smooth deformation and moduli of log smooth curves’, Int. J. Math. 11(02) (2000), 215–232.Google Scholar | DOI

[32] Kato, K., ‘Logarithmic structures of Fontaine–Illusie’, in Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), (Johns Hopkins University Press, Baltimore, MD, 1989), 191–224.Google Scholar

[33] Katzarkov, L., Kontsevich, M. and Pantev, T., ‘Hodge theoretic aspects of mirror symmetry, in From Hodge Theory to Integrability and TQFT tt*-Geometry, Vol. 78 of Proc. Sympos. Pure Math (American Mathematical Society, Providence, RI, 2008), 87–174.Google Scholar | DOI

[34] Kawamata, Y. and Namikawa, Y., ‘Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi–Yau varieties’, Invent. Math. 118(3) (1994), 395–409.Google Scholar | DOI

[35] Lee, N.-H., ‘d-Semistable Calabi–Yau threefolds of type III’, Manuscr. Math. 161 (2020), 257–281.Google Scholar | DOI

[36] Lee, N.-H., ‘An example of non-Kähler Calabi–Yau fourfold’ (2021). URL: .Google Scholar | arXiv

[37] Liu, K., Shen, Y. and Chen, X., Applications of the Affine Structures on the Teichmüller Spaces, Vol. 154 of Springer Proc. Math. Stat (Springer, [Tokyo], 2016).Google Scholar

[38] Nakayama, C. and Ogus, A., ‘Relative rounding in toric and logarithmic geometry’, Geom. Topol. 14(4) (2010), 2189–2241.Google Scholar | DOI

[39] Ogus, A., Lectures on Logarithmic Algebraic Geometry , Vol. 178 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge CB2 8BS, UK, 2018).Google Scholar

[40] Olsson, M. C., ‘Universal log structures on semi-stable varieties’, Tohoku Math. J. (2) 55(3) (2003), 397–438.Google Scholar | DOI

[41] Ruddat, H. ‘Log Hodge groups on a toric Calabi–Yau degeneration’, in Mirror Symmetry and Tropical Geometry, No. 527 in Contemporary Mathematics (American Mathematical Society, Providence, RI, 2010), 113–164.Google Scholar | DOI

[42] Ruddat, H., ‘Local uniqueness of approximations and finite determinacy of log morphisms’ (2018).Google Scholar | arXiv

[43] Ruddat, H. and Siebert, B., ‘Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations’, Publ. Math. Inst. Hautes Études Sci. 132 (2020), 1–82.Google Scholar | DOI

[44] Schröer, S. and Siebert, B., ‘Toroidal crossings and logarithmic structures’, Adv. Math. 202(1) (2006), 189–231.Google Scholar | DOI

[45] Steenbrink, J. H.M., ‘Mixed Hodge structure on the vanishing cohomology’, in Real and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) (1977), 525–563.Google Scholar

[46] Steenbrink, J. H.M., ‘Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures’, Math. Ann. 301(1) (1995), 105–118.Google Scholar | DOI

[47] The Stacks Project Authors, Stacks project. URL: http://stacks.math.columbia.edu/.Google Scholar

[48] Tsuji, T., ‘ -Adic étale cohomology and crystalline cohomology in the semi-stable reduction case’, Invent. Math. 137(2) (1999), 233–411.Google Scholar | DOI

[49] Tsuji, T., ‘Poincaré duality for logarithmic crystalline cohomology’, Compositio Math. 118(1) (1999), 11–41.Google Scholar | DOI

[50] Tsuji, T., ‘Saturated morphisms of logarithmic schemes’, Tunis. J. Math. 1(2) (2019), 185–220.Google Scholar | DOI

[51] Tziolas, N. ‘Smoothings of Fano varieties with normal crossing singularities’, Proc. Edinb. Math. Soc. (2) 58(3) (2015), 787–806.Google Scholar | DOI

[52] Yotsutani, N., Global Smoothings of Degenerate K3 Surfaces with Triple Points (2020). URL: .Google Scholar | arXiv

[53] Yotsutani, N., ‘Notes on diffeomorphism classes of the doubling Calabi–Yau threefolds’ (2021). URL: .Google Scholar | arXiv

Cité par Sources :