## Tokyo Journal of Mathematics

### Deformations of a Holomorphic Map and Its Degeneracy Locus

#### Abstract

Let $f: X \to Y$ be a surjective holomorphic map of compact complex manifolds and $\Delta$ the degeneracy locus of $f$. In this paper we shall discuss relationship between infinitesimal deformations of $f$ and the corresponding infinitesimal displacements of $\Delta$ in $Y$. We shall prove that two kinds of Kodaira-Spencer maps are compatible under certain assumptions. As an application of our main theorem, deformations of quadric bundles shall be discussed.

#### Article information

Source
Tokyo J. Math., Volume 35, Number 2 (2012), 253-277.

Dates
First available in Project Euclid: 23 January 2013

https://projecteuclid.org/euclid.tjm/1358951317

Digital Object Identifier
doi:10.3836/tjm/1358951317

Mathematical Reviews number (MathSciNet)
MR3058705

Zentralblatt MATH identifier
1274.14007

#### Citation

EBIHARA, Madoka. Deformations of a Holomorphic Map and Its Degeneracy Locus. Tokyo J. Math. 35 (2012), no. 2, 253--277. doi:10.3836/tjm/1358951317. https://projecteuclid.org/euclid.tjm/1358951317

#### References

• A. Beauville, Variété de Prym ét Jacobiennes intermédiaires, Ann. scient. Éc. Norm. Sup., $4^{\rm e}$ série. t. 10 (1977), 309–391.
• M. Ebihara, Some remarks on infinitesimal deformations of a conic bundle I, Saitama Math. J. 18,(2000), 1–21.
• M. Ebihara, Some remarks on infinitesimal deformations of a conic bundle II, Saitama Math. J. 18,(2000), 23–38.
• M. Ebihara, Some remarks on infinitesimal deformations of a conic bundle III, Saitama Math. J. 24,(2006,$\cdot$,2007), 93–104.
• M. Ebihara, Some remarks on infinitesimal deformations of a conic bundle IV, Saitama Math. J. 28 (2011), 39–54.
• E. Horikawa, On deformations of holomorphic maps I, II, III, J. Math. Soc. Japan 25 (1973), 372–396; 26,(1974), 647–667; Math. Ann. 222 (1976), 275–282.
• K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. 75,(1962), 146–162.
• M. Miyanishi, Algebraic methods in the theory of algebraic threefolds, Algebraic varieties and analytic varieties, eds. S. Iitaka et al., Advanced Studies in Pure Math. 1, Kinokuniya, North-Holland (1983), 69–99.