On the algebraic stringy Euler number
HTML articles powered by AMS MathViewer
- by Victor Batyrev and Giuliano Gagliardi PDF
- Proc. Amer. Math. Soc. 146 (2018), 29-41
Abstract:
We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group $G$ and admitting equivariant desingularizations with only finitely many $G$-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties.References
- Valery Alexeev and Michel Brion, Stable reductive varieties. II. Projective case, Adv. Math. 184 (2004), no. 2, 380–408. MR 2054021, DOI 10.1016/S0001-8708(03)00164-6
- Victor V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 1–32. MR 1672108
- Victor V. Batyrev, Birational Calabi-Yau $n$-folds have equal Betti numbers, New trends in algebraic geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 1–11.
- Victor V. Batyrev, Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 1, 5–33. MR 1677693, DOI 10.1007/PL00011158
- Franziska Bittner, The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140 (2004), no. 4, 1011–1032. MR 2059227, DOI 10.1112/S0010437X03000617
- Michel Brion and Friedrich Knop, Contractions and flips for varieties with group action of small complexity, J. Math. Sci. Univ. Tokyo 1 (1994), no. 3, 641–655. MR 1322696
- Victor Batyrev and Anne Moreau, The arc space of horospherical varieties and motivic integration, Compos. Math. 149 (2013), no. 8, 1327–1352. MR 3103067, DOI 10.1112/S0010437X13007124
- Michel Brion and Emmanuel Peyre, The virtual Poincaré polynomials of homogeneous spaces, Compositio Math. 134 (2002), no. 3, 319–335. MR 1943906, DOI 10.1023/A:1020984924857
- Michel Brion, Variétés sphériques et théorie de Mori, Duke Math. J. 72 (1993), no. 2, 369–404 (French). MR 1248677, DOI 10.1215/S0012-7094-93-07213-4
- Alastair Craw, An introduction to motivic integration, Strings and geometry, Clay Math. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 2004, pp. 203–225. MR 2103724
- Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232. MR 1664700, DOI 10.1007/s002220050284
- V. A. Iskovskikh and V. V. Shokurov, Birational models and flips, Uspekhi Mat. Nauk 60 (2005), no. 1(361), 29–98 (Russian, with Russian summary); English transl., Russian Math. Surveys 60 (2005), no. 1, 27–94. MR 2145659, DOI 10.1070/RM2005v060n01ABEH000807
- Boris Pasquier, An approach of the minimal model program for horospherical varieties via moment polytopes, J. Reine Angew. Math. 708 (2015), 173–212. MR 3420333, DOI 10.1515/crelle-2013-0103
- Boris Pasquier, A survey on the singularities of spherical varieties, arXiv:1510.03995v1.
- Boris Pasquier, KLT singularities of horospherical pairs, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 5, 2157–2167 (English, with English and French summaries). MR 3533280
- Nicolas Perrin, On the geometry of spherical varieties, Transform. Groups 19 (2014), no. 1, 171–223. MR 3177371, DOI 10.1007/s00031-014-9254-0
- Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 395–418. MR 717617
- Jyh-Haur Teh, Motivic integration and projective bundle theorem in morphic cohomology, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 2, 295–321. MR 2525928, DOI 10.1017/S0305004109002588
- Willem Veys, Zeta functions and “Kontsevich invariants” on singular varieties, Canad. J. Math. 53 (2001), no. 4, 834–865. MR 1848509, DOI 10.4153/CJM-2001-034-1
Additional Information
- Victor Batyrev
- Affiliation: Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
- Email: batyrev@math.uni-tuebingen.de
- Giuliano Gagliardi
- Affiliation: Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
- MR Author ID: 1040639
- Email: gagliardi@math.uni-hannover.de
- Received by editor(s): November 28, 2016
- Received by editor(s) in revised form: January 25, 2017
- Published electronically: July 28, 2017
- Communicated by: Lev Borisov
- © Copyright 2017 by the authors
- Journal: Proc. Amer. Math. Soc. 146 (2018), 29-41
- MSC (2010): Primary 14E30; Secondary 14E15, 14E18, 14L30, 14M27
- DOI: https://doi.org/10.1090/proc/13702
- MathSciNet review: 3723118