A note on the effective non-vanishing conjecture

Author:
Qihong Xie

Journal:
Proc. Amer. Math. Soc. **137** (2009), 61-63

MSC (2000):
Primary 14E30

Published electronically:
August 4, 2008

MathSciNet review:
2439425

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a reduction of the irregular case for the effective non-vanishing conjecture by virtue of the Fourier-Mukai transform. As a consequence, we reprove that the effective non-vanishing conjecture holds on algebraic surfaces.

**[Am99]**F. Ambro,*Ladders on Fano varieties*, J. Math. Sci. (New York)**94**(1999), no. 1, 1126–1135. Algebraic geometry, 9. MR**1703912**, 10.1007/BF02367253**[CH02]**Jungkai A. Chen and Christopher D. Hacon,*Linear series of irregular varieties*, Algebraic geometry in East Asia (Kyoto, 2001) World Sci. Publ., River Edge, NJ, 2002, pp. 143–153. MR**2030451**, 10.1142/9789812705105_0004**[Ka00]**Yujiro Kawamata,*On effective non-vanishing and base-point-freeness*, Asian J. Math.**4**(2000), no. 1, 173–181. Kodaira’s issue. MR**1802918**, 10.4310/AJM.2000.v4.n1.a11**[KMM87]**Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki,*Introduction to the minimal model problem*, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR**946243****[KM98]**János Kollár and Shigefumi Mori,*Birational geometry of algebraic varieties*, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR**1658959****[Mu81]**Shigeru Mukai,*Duality between 𝐷(𝑋) and 𝐷(𝑋) with its application to Picard sheaves*, Nagoya Math. J.**81**(1981), 153–175. MR**607081****[Mu02]**Mircea Mustaţă,*Vanishing theorems on toric varieties*, Tohoku Math. J. (2)**54**(2002), no. 3, 451–470. MR**1916637****[PP03]**Giuseppe Pareschi and Mihnea Popa,*Regularity on abelian varieties. I*, J. Amer. Math. Soc.**16**(2003), no. 2, 285–302 (electronic). MR**1949161**, 10.1090/S0894-0347-02-00414-9**[PP05]**G. Pareschi, M. Popa, M-regularity and the Fourier-Mukai transform, preprint, math.AG/0512645.**[PP06]**G. Pareschi, M. Popa, GV-sheaves, Fourier-Mukai transform, and generic vanishing, preprint, math.AG/0608127.**[Xie05]**Qihong Xie,*On pseudo-effectivity of the second Chern classes for terminal threefolds*, Asian J. Math.**9**(2005), no. 1, 121–132. MR**2150695****[Xie06]**Qihong Xie,*Effective non-vanishing for algebraic surfaces in positive characteristic*, J. Algebra**305**(2006), no. 2, 1111–1127. MR**2266872**, 10.1016/j.jalgebra.2006.06.008

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
14E30

Retrieve articles in all journals with MSC (2000): 14E30

Additional Information

**Qihong Xie**

Affiliation:
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro, Tokyo 152-8551, Japan

Address at time of publication:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan

Email:
xie_qihong@hotmail.com

DOI:
https://doi.org/10.1090/S0002-9939-08-09539-7

Received by editor(s):
January 17, 2007

Received by editor(s) in revised form:
October 18, 2007, November 16, 2007, and December 27, 2007

Published electronically:
August 4, 2008

Communicated by:
Ted Chinburg

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.