## Kodaira dimension and zeros of holomorphic one-forms.(English)Zbl 1297.14011

In this very interesting article, the authors use generic vanishing theorems for Hodge modules on abelian varieties to prove that if $$X$$ is a smooth complex projective variety of general type and $$0\neq \omega \in H^0(\Omega ^1_X)$$ is a global holomorphic 1-form, then the zero locus of $$\omega$$ is non-empty.

### MSC:

 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14K05 Algebraic theory of abelian varieties 32C38 Sheaves of differential operators and their modules, $$D$$-modules 32C05 Real-analytic manifolds, real-analytic spaces
Full Text:

### References:

 [1] D. Arapura, ”Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves,” Bull. Amer. Math. Soc., vol. 26, iss. 2, pp. 310-314, 1992. · Zbl 0759.14016 [2] J. B. Carrell, ”Holomorphic one forms and characteristic numbers,” Topology, vol. 13, pp. 225-228, 1974. · Zbl 0305.32016 [3] H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, Basel: Birkhäuser, 1992, vol. 20. · Zbl 0779.14003 [4] M. Green and R. Lazarsfeld, ”Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville,” Invent. Math., vol. 90, iss. 2, pp. 389-407, 1987. · Zbl 0659.14007 [5] C. D. Hacon and S. J. Kovács, ”Holomorphic one-forms on varieties of general type,” Ann. Sci. École Norm. Sup., vol. 38, iss. 4, pp. 599-607, 2005. · Zbl 1083.14015 [6] Y. Kawamata, ”Characterization of abelian varieties,” Compositio Math., vol. 43, iss. 2, pp. 253-276, 1981. · Zbl 0471.14022 [7] J. Kollár, ”Subadditivity of the Kodaira dimension: fibers of general type,” in Algebraic Geometry, Amsterdam: North-Holland, 1987, vol. 10, pp. 361-398. · Zbl 0659.14024 [8] S. J. Kovács, ”Smooth families over rational and elliptic curves,” J. Algebraic Geom., vol. 5, iss. 2, pp. 369-385, 1996. · Zbl 0861.14033 [9] S. J. Kovács, ”Families over a base with a birationally nef tangent bundle,” Math. Ann., vol. 308, iss. 2, pp. 347-359, 1997. · Zbl 0922.14024 [10] S. J. Kovács, ”Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties,” Compositio Math., vol. 131, iss. 3, pp. 291-317, 2002. · Zbl 1048.14006 [11] T. Luo and Q. Zhang, ”Holomorphic forms on threefolds,” in Recent Progress in Arithmetic and Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2005, vol. 386, pp. 87-94. · Zbl 1216.14013 [12] L. Migliorini, ”A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial,” J. Algebraic Geom., vol. 4, iss. 2, pp. 353-361, 1995. · Zbl 0834.14021 [13] M. Popa and C. Schnell, ”Generic vanishing theory via mixed Hodge modules,” Forum Math. Sigma, vol. 1, p. 1, 2013. · Zbl 1281.14007 [14] M. Saito, ”Modules de Hodge polarisables,” Publ. Res. Inst. Math. Sci., vol. 24, iss. 6, pp. 849-995 (1989), 1988. · Zbl 0691.14007 [15] E. Viehweg and K. Zuo, ”On the isotriviality of families of projective manifolds over curves,” J. Algebraic Geom., vol. 10, iss. 4, pp. 781-799, 2001. · Zbl 1079.14503 [16] Q. Zhang, ”Global holomorphic one-forms on projective manifolds with ample canonical bundles,” J. Algebraic Geom., vol. 6, iss. 4, pp. 777-787, 1997. · Zbl 0922.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.