Basics Tools Applications Further Questions References Arrangements with up to 7 Hyperplanes Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF 3-rd February 2016 Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Hyperplane Arrangements Main Definitions A complex hyperplane arrangement is a finite collection A = { H 1 , . . . , H m } of affine hyperplanes in C d . The complement manifold M ( A ) is C d \ � m j = 1 H j . Problem: study the topology of M ( A ) . Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Hyperplane Arrangements The Central Case A complex hyperplane arrangement A = { H 1 , . . . , H m } in C d is central if all the H j ’s contain the origin . Result: to understand M ( A ) we can study the central case. Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Hyperplane Arrangements Milnor Fiber and Fibration For a complex central hyperplane arrangement A = { H 1 , . . . , H m } in C d let α i ∈ ( C d ) ∗ be linear formswith H i = ker α i . The polynomial Q A = � m i = 1 α i is homogeneous of degree m and can → C ∗ that is the projection be considered as a map Q A : M ( A ) − of a fiber bundle called the Milnor fibration of the arrangement. Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Isotopic Hyperplane Arrangements Isotopic Hyperplane Arrangements (Part 1) Theorem ([Ran89]) Let A t be a smooth one-parameter family of central complex hyperplane arrangements in C d . If the underlying matroid M A t does not depend on t, so does the diffeomorphism type of M ( A t ) . Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Isotopic Hyperplane Arrangements Isotopic Hyperplane Arrangements (Part 2) Theorem ([Ran97]) Let A t be a smooth one-parameter family of central complex hyperplane arrangements in C d . If the underlying matroid M A t does not depend on t, so does the isomorphism type of Q A t . Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Small Hyperplane Arrangements Main Results Theorem ([GS16]) Let A = { H 1 , . . . , H m } and B = { K 1 , . . . , K m } be rank d central hyperplane arrangements in C d with same underlying matroid. If 1 � d � m � 7 , then A and B are isotopic arrangements. Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Small Hyperplane Arrangements Some Corollaries (Part 1) Corollary ([GS16]) Let A = { H 1 , . . . , H m } and B = { K 1 , . . . , K m } be rank d central hyperplane arrangements in C d with same underlying matroid. If 1 � d � m � 7 , then the complement manifolds M ( A ) and M ( B ) are diffeomorphic . Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Small Hyperplane Arrangements Some Corollaries (Part 2) Corollary ([GS16]) Let A = { H 1 , . . . , H m } and B = { K 1 , . . . , K m } be rank d central hyperplane arrangements in C d with same underlying matroid. If 1 � d � m � 7 , then the Milnor fibrations Q A and Q B are isomorphic fiber bundles. Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References Further Questions Further Questions Find a non-case-by-case proof of these results. Find more refined techniques to study non-connected matroid realization spaces. Study the Rybnikov matroid realization space. Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Basics Tools Applications Further Questions References References A Small Bibliography Matteo Gallet and Elia Saini, The diffeomorphism type of small hyperplane arrangements is combinatorially determined , [arXiv:1601.05705] (2016). Richard Randell, Lattice-isotopic arrangements are topologically isomorphic , Proc. Amer. Math. Soc. 107 (1989), no. 2, 555–559. , Milnor fibrations of lattice-isotopic arrangements , Proc. Amer. Math. Soc. 125 (1997), no. 10, 3003–3009. Elía Saini Université de Fribourg - Universität Freiburg Swiss National Science Foundation SNSF Arrangements with up to 7 Hyperplanes
Recommend
More recommend
