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Remarks on quantum ergodicity
Strata of abelian differentials and the Teichmüller dynamics
1. | Department of Mathematics, Boston College, Chestnut Hill, MA 02467, United States |
References:
[1] |
E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, "Geometry of Algebraic Curves," Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267, Springer-Verlag, New York, 1985. |
[2] |
D. Chen, Covers of elliptic curves and the moduli space of stable curves, J. Reine Angew. Math., 649 (2010), 167-205.
doi: 10.1515/CRELLE.2010.092. |
[3] |
D. Chen, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228 (2011), 1135-1162.
doi: 10.1016/j.aim.2011.06.002. |
[4] |
D. Chen and M. Moeller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479. |
[5] |
D. Chen and M. Moeller, Quadratic differentials in low genus: Exceptional and non-varying strata, arXiv:1204.1707, (2012). |
[6] |
D. Chen, M. Moeller and D. Zagier, Siegel-Veech constants and quasimodular forms,, in preparation., ().
|
[7] |
F. Cukierman, Families of Weierstrass points, Duke Math. J., 58 (1989), 317-346.
doi: 10.1215/S0012-7094-89-05815-8. |
[8] |
S. Diaz, Porteous's formula for maps between coherent sheaves, Michigan Math. J., 52 (2004), 507-514.
doi: 10.1307/mmj/1100623410. |
[9] |
A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, preprint, arXiv:1112.5872, (2011). |
[10] |
A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179.
doi: 10.1007/s10240-003-0015-1. |
[11] |
A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL$(2,\mathbb R)$ action on Moduli space, arXiv:1302.3320, (2013). |
[12] |
G. Farkas and A. Verra, The classification of universal Jacobians over the moduli space of curves,, to appear in Comm. Math. Helv., ().
|
[13] |
U. Hamenstädt, Signatures of surface bundles and Milnor Wood inequalities, arXiv:1206.0263, (2012). |
[14] |
J. Harris and I. Morrison, "Moduli of Curves," Graduate Texts in Mathematics, 187, Springer-Verlag, New York, 1998. |
[15] |
J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, With an appendix by William Fulton, Invent. Math., 67 (1982), 23-88.
doi: 10.1007/BF01393371. |
[16] |
David Jensen, Rational fibrations of $\overlineM_{5,1}$ and $\overlineM_{6,1}$, J. Pure Appl. Algebra, 216 (2012), 633-642.
doi: 10.1016/j.jpaa.2011.07.015. |
[17] |
David Jensen, Birational contractions of $\overlineM_{3,1}$ and $\overlineM_{4,1}$, Trans. Amer. Math. Soc., 365 (2013), 2863-2879.
doi: 10.1090/S0002-9947-2012-05581-4. |
[18] |
A. Kokotov, D. Korotkin and P. Zograf, Isomonodromic tau function on the space of admissible covers, Adv. Math., 227 (2011), 586-600.
doi: 10.1016/j.aim.2011.02.005. |
[19] |
M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics" (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, (1997), 318-332. |
[20] |
M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.
doi: 10.1007/s00222-003-0303-x. |
[21] |
D. Korotkin and P. Zograf, Tau function and moduli of differentials, Math. Res. Lett., 18 (2011), 447-458. |
[22] |
R. Lazarsfeld, "Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series," Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 48, Springer-Verlag, Berlin, 2004. |
[23] |
A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math., 125 (2003), 105-138.
doi: 10.1353/ajm.2003.0005. |
[24] |
W. Rulla, "The Birational Geometry of Moduli Space $M(3)$ and Moduli Space $M(2,1)$," Ph.D. Thesis, The University of Texas at Austin, 2001. |
[25] |
B. Thomas, Excess porteous, coherent porteous, and the hyperelliptic locus in $\overline{\mathcal M}_3$, Michigan Math. J., 61 (2012), 359-383.
doi: 10.1307/mmj/1339011531. |
[26] |
G. van der Geer and A. Kouvidakis, The Hodge bundle on Hurwitz spaces, Pure Appl. Math. Q., 7 (2011), 1297-1307. |
[27] |
F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents,, to appear in J. Mod. Dyn., ().
|
[28] |
A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics and Geometry. 1," Springer, Berlin, (2006), 437-583.
doi: 10.1007/978-3-540-31347-2_13. |
[29] |
show all references
References:
[1] |
E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, "Geometry of Algebraic Curves," Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267, Springer-Verlag, New York, 1985. |
[2] |
D. Chen, Covers of elliptic curves and the moduli space of stable curves, J. Reine Angew. Math., 649 (2010), 167-205.
doi: 10.1515/CRELLE.2010.092. |
[3] |
D. Chen, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228 (2011), 1135-1162.
doi: 10.1016/j.aim.2011.06.002. |
[4] |
D. Chen and M. Moeller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479. |
[5] |
D. Chen and M. Moeller, Quadratic differentials in low genus: Exceptional and non-varying strata, arXiv:1204.1707, (2012). |
[6] |
D. Chen, M. Moeller and D. Zagier, Siegel-Veech constants and quasimodular forms,, in preparation., ().
|
[7] |
F. Cukierman, Families of Weierstrass points, Duke Math. J., 58 (1989), 317-346.
doi: 10.1215/S0012-7094-89-05815-8. |
[8] |
S. Diaz, Porteous's formula for maps between coherent sheaves, Michigan Math. J., 52 (2004), 507-514.
doi: 10.1307/mmj/1100623410. |
[9] |
A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, preprint, arXiv:1112.5872, (2011). |
[10] |
A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179.
doi: 10.1007/s10240-003-0015-1. |
[11] |
A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL$(2,\mathbb R)$ action on Moduli space, arXiv:1302.3320, (2013). |
[12] |
G. Farkas and A. Verra, The classification of universal Jacobians over the moduli space of curves,, to appear in Comm. Math. Helv., ().
|
[13] |
U. Hamenstädt, Signatures of surface bundles and Milnor Wood inequalities, arXiv:1206.0263, (2012). |
[14] |
J. Harris and I. Morrison, "Moduli of Curves," Graduate Texts in Mathematics, 187, Springer-Verlag, New York, 1998. |
[15] |
J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, With an appendix by William Fulton, Invent. Math., 67 (1982), 23-88.
doi: 10.1007/BF01393371. |
[16] |
David Jensen, Rational fibrations of $\overlineM_{5,1}$ and $\overlineM_{6,1}$, J. Pure Appl. Algebra, 216 (2012), 633-642.
doi: 10.1016/j.jpaa.2011.07.015. |
[17] |
David Jensen, Birational contractions of $\overlineM_{3,1}$ and $\overlineM_{4,1}$, Trans. Amer. Math. Soc., 365 (2013), 2863-2879.
doi: 10.1090/S0002-9947-2012-05581-4. |
[18] |
A. Kokotov, D. Korotkin and P. Zograf, Isomonodromic tau function on the space of admissible covers, Adv. Math., 227 (2011), 586-600.
doi: 10.1016/j.aim.2011.02.005. |
[19] |
M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics" (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, (1997), 318-332. |
[20] |
M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.
doi: 10.1007/s00222-003-0303-x. |
[21] |
D. Korotkin and P. Zograf, Tau function and moduli of differentials, Math. Res. Lett., 18 (2011), 447-458. |
[22] |
R. Lazarsfeld, "Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series," Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 48, Springer-Verlag, Berlin, 2004. |
[23] |
A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math., 125 (2003), 105-138.
doi: 10.1353/ajm.2003.0005. |
[24] |
W. Rulla, "The Birational Geometry of Moduli Space $M(3)$ and Moduli Space $M(2,1)$," Ph.D. Thesis, The University of Texas at Austin, 2001. |
[25] |
B. Thomas, Excess porteous, coherent porteous, and the hyperelliptic locus in $\overline{\mathcal M}_3$, Michigan Math. J., 61 (2012), 359-383.
doi: 10.1307/mmj/1339011531. |
[26] |
G. van der Geer and A. Kouvidakis, The Hodge bundle on Hurwitz spaces, Pure Appl. Math. Q., 7 (2011), 1297-1307. |
[27] |
F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents,, to appear in J. Mod. Dyn., ().
|
[28] |
A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics and Geometry. 1," Springer, Berlin, (2006), 437-583.
doi: 10.1007/978-3-540-31347-2_13. |
[29] |
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