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Towards a new interpretation of Milnor’s number
Hacia una nueva interpretación del número de Milnor
Palabras clave:
Milnor number, monodromy, right handed Dehn twist, morsification, 2000 Mathematics Subject Classification. 14D05, 14D06, 14B07 (en)Número de Milnor, monodromía, giro de Dehn derecho, morsificación (es)
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Abstract. The Milnor number is a fundamental invariant of the biholomorphism type of the singularity of the germ of a holomorphic function f defined on an open neighborhood W of 0 ∊ ℂn, and such that 0 is the only critical point of f in W. The present article describes a conjecture that would provide an interpretation of this invariant, in the case n = 2, as a sharp lower bound for the number of factors in any factorization in terms of right-handed Dehn twists of the monodromy around the singular fiber of f. Also, towards the end of the paper, an analogue conjecture for proper holomorphic maps f : E → Dr° where E is a complex surface with boundary, Dr° is {z ∊ ℂ : |z | < r}, and f has f - 1(0) as its unique singular fiber and all other fibers are closed and connected 2-manifolds of (necessarily the same) genus g ≥ 0, is briefly described. The latter conjecture has been proved recently by the authors in the case when the regular fiber of f has genus 1 ([3]), and in ([5]), that author provides for each g ≥ 2 an fg : Eg → Dr° having genus g regular fiber and violating this conjecture.
El número de Milnor es un invariante fundamental del tipo de biholomorfismo de un germen de una función holomorfa f definida en una vecindad abierta W of 0 ∊ ℂn, tal que 0 es el único punto crítico de f en W. En este artículo presentamos una conjetura que daría una interpretación de este invariante en el caso n = 2, como una cota inferior exacta para el número de factores de cualquier factorización en términos de giros de Dehn derechos de la monodromía alrededor de la fibra singular de f. Además, hacia el final del artículo, se describe brevemente una conjetura análoga para el caso en que tenemos una función holomorfa propia f : E → Dr° donde E es una superficie compleja con frontera, Dr° is {z ∊ ℂ : |z | < r}, f tiene a f - 1(0) como su única fibra singular y todas las otras fibras son 2-variedades cerradas conexas de género, necesariamente constante, g ≥ 0. Esta última conjetura ha sido demostrada recientemente por los autores en el caso en que el género de la fibra regular es 1 ([3]), y en ([5]), ese autor construye, para cada g ≥ 2, una fibración fg : Eg → Dr° cuya fibra regular tiene género g y que viola esta conjetura.
Referencias
Arnold, V. I. Topological problems in wave propagation theory and topological economy principle in algebraic geometry. In The A moldfest, Proceedings of a Conference in Honour of V.I. Arnold for his Sixtieth Birthday (Providence RI, 1999), E. Bierstone, B. Khesin, A. Khovanskii, and J. E. Marsden, Eds., vol. 24 of Fields Institute Communications, American Mathematical Society.
Birman, J. Braids, Links, and Mapping Class Groups. Princeton University Press, Princeton, 1975.
Cadavid, C., and Vélez, J. On a minimal factorization conjecture. Topology and its Applications 154, 15 (2007), 2786-2794.
Fulton, W. Intersection Theory, second ed. Springer-Verlag, New York, 1984.
Ishizaka, M. One parameter families of Riemann surfaces and presentations of elements of mapping class group by Dehn twists. Journal of the Mathematical Society of Japan 58, 2 (2006), 585-594.
Looijenga, E. Isolated Singular Points on Complete Intersections, vol. 77 of Lecture Notes in Math. Cambridge University Press, Cambridge, 1984.
Mllnor, J. Singular points of Complex Hypersurfaces, vol. 61 of Ann. of Math. Studies. Princeton University Press, Princeton, 1968.
Moishezon, B. Complex Surfaces and Connected Sums of Complex Proyective Planes, vol. 603 of Lecture Notes in Math. Springer-Verlag, Berlin, New York, 1977.
Ozsváth, P., and Szabo, Z. The symplectic Thom conjecture. Annals of Mathematics 151, 1 (2000), 93-124.
Serre, J. P. Algèbre Locale Multiplicités, vol. 11 of Lecture Notes in Math. Springer-Verlag, New York, 1965.
Vasil'ev, V. A. Applied Picard-Lefschetz Theory, vol. 97 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, 2002.
