Decomposition into pairs-of-pants for complex algebraic hypersurfaces

By G. Mikhalkin. 2002

185 citations recorded by the source corpus.

Abstract It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to CP n minus n+2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accommodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of hn,o(V) spheres Sn.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/163b2899203217896fe58bf252f089063432cd46

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Brief introduction to tropical geometry (2015, unknown): “9 ([Mik04b]).”
On the homological mirror symmetry conjecture for pairs of pants (2010, background): “This is called the (n-dimensional) pair of pants Pn (see [35]).”
Welschinger invariant and enumeration of real rational curves (2003, background): “weighted complex is called the tropical curve associated with the pair (A, ν); cf. [ 8 ]”
Tropical Geometry and its applications (2006, unknown): “See [17] for a generalization of the pair-of-pants decomposition for the case of higher-dimensional hypersurfaces .”
Morse homology, tropical geometry, and homological mirror symmetry for toric varieties (2006, background): “Rather, following Mikhalkin [33], the graph on the right of Figure 1 is the tropical amoeba of the one-parameter family of Laurent”
Exploded Manifolds (2009, methodology): “An example of a degeneration ‘breaking a manifold into simple pieces’ is given by Mikhalkin’s higher dimensional pair of pants decomposition of projective hypersurfaces from [10].”
Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces (2012, unknown): “, log |xn|) maps Hτ to its amoeba Πτ = Logτ (Hτ ∩ V ); it is known [35, 37] that, for τ → 0, the amoeba Πτ ⊂ R n converges to the tropical hypersurface Π0 ⊂ R n defined by the tropical polynomial”
Enumerative tropical algebraic geometry in R^2 (2003, background): “3 of [17] we can define the J∞-holomorphic curves which happen to be the limits of Jt-holomorphic curves, t→ ∞.”
Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities (2019, unknown): “Definition 2.2 (Mikhalkin, 2004, Definition 3).”
Enumerative tropical algebraic geometry (2003, unknown): “1 =⇒ 3.”
An open string analogue of Viterbo functoriality (2007, background): “However, for the authors of this paper, the main interest lies in potential applications to homological mirror symmetry, using Mikhalkin’s generalized pair-of-pants decompositions [2, 24].”
Homogeneous coordinate rings and mirror symmetry for toric varieties (2005, methodology): “In this situation, we may restate a result of Mikhalkin.”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “More generally, as shown in [35], the pair of pants CP \D is a singular torus fibration over the ‘tropical amoeba of the pair of pants’, which is some space stratified by affine manifolds.”