Properties of monodromy of a fibration? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:50:16Z http://mathoverflow.net/feeds/question/1912 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1912/properties-of-monodromy-of-a-fibration Properties of monodromy of a fibration? Ilya Nikokoshev 2009-10-22T18:38:57Z 2010-01-15T08:52:41Z <p>Sorry for a loaded question.</p> <p>I'm not an expert on those things, but I do know that a fibration gives rise to the representations of pointed fundamental group of the base on the cohomology of the fiber.</p> <p>What are the properties of this map for different classes of fibrations? I think it's known what the image of this map can be. And the local properties are governed, at least in the complex case, by what type the manifold is.</p> <p>And, most importantly, there is something about uppertriangularity. What exactly is that? </p> http://mathoverflow.net/questions/1912/properties-of-monodromy-of-a-fibration/1915#1915 Answer by Bhargav for Properties of monodromy of a fibration? Bhargav 2009-10-22T18:47:50Z 2009-10-22T19:19:37Z <p>For the last question: at least in the algebro-geometric case, the monodromy is always quasi-unipotent (i.e, some power of is unipotent). There is a beautiful argument due to Grothendieck that proves this by reducing to the p-adic case, and using the action of Frobenius on the (tame) inertia group of a p-adic field. </p> http://mathoverflow.net/questions/1912/properties-of-monodromy-of-a-fibration/2025#2025 Answer by Tony Pantev for Properties of monodromy of a fibration? Tony Pantev 2009-10-23T03:29:27Z 2010-01-15T08:52:41Z <p>A <strong>small clarification</strong> on bhargav's answer: in algebraic geometry we only have quasi-unipotency of the <em>local</em> monodromy in one-parameter families (which is what bhargav is talking about); or in multi-parameter families but only near a normal crossing point of the discriminant. Global monodromies are reductive and local monodromies near bad points of the discriminant can be more general.</p> <p><strong>For concreteness</strong> look at a projective morphism $f : X \to B$, where $X$, $B$ are smooth complex projective varieties. Let $D \subset B$ be the discriminant divisor of $f$, i.e. the divisor where the differential of $f$ is not surjective. The global monodromy of the smooth fibration $f : X - f^{-1}(D) \to B - D$ is always reductive by a theorem of Borel. That is: the Zariski closure of the monodromy in the linear automorphisms of the cohomology of the marked fiber is a complex reductive group. If we take a small analytic ball $U \subset B$ centered at some point of $D$, and if we know that $D\cap U$ is a normal crossings divisor in $U$, then the monodromy of the local fibration $f : f^{-1}(U-D) \to U-D$ is quasi-unipotent as bhargav explained. Note that the normal crossings condition implies that the fundamental group of $U - D$ is abelian, so the quasi-unipotency condition makes sense here. </p> <p>If however $U\cap D$ does not have normal crossings, then $\pi_{1}(U-D)$ need not be abelian and the monodromy of $f : f^{-1}(U-D) \to U-D$ need not be quasi-unipotent. An easy example is to look at a generic projective plane $\mathbb{P}^{2}$ in the $9$ dimensional projective space of cubic curves in $\mathbb{P}^{2}$. This plane parametrizes a family of cubics which degenerates along a discriminant curve $D \subset \mathbb{P}^{2}$ and under the genericity assumption $D$ has only nodes and cusps. The cuspidal points of $D$ correspond to cuspidal cubics, and near a cusp of $D$ the local fundamental group of $U - D$ is the amalgamated product of $\mathbb{Z}/4$ and $\mathbb{Z}/6$ over $\mathbb{Z}/2$ and so is isomorphic to $SL_{2}(\mathbb{Z})$ the local monodromy representation near the cusp, i.e. the representation of $\pi_{1}(U-D,u_{0})$ to the linear automorphisms of the first integral cohomology of the cubic corresponding to $u_{0} \in U - D$, is an inclusion, i.e. has image $SL_{2}(\mathbb{Z})$. In particular it is not quasi-unipotent.</p>