Arrangements with up to 7 Hyperplanes Ela Saini Universit de - - PowerPoint PPT Presentation

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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 = {H1, . . . , Hm} of affine hyperplanes in Cd. The complement manifold M(A) is Cd \ m

j=1 Hj.

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 = {H1, . . . , Hm} in Cd is central if all the Hj’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 = {H1, . . . , Hm} in Cd let αi ∈ (Cd)∗ be linear formswith Hi = ker αi. The polynomial QA = m

i=1 αi is homogeneous of degree m and can

be considered as a map QA : M(A) − → C∗ that is the projection

• f 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 At be a smooth one-parameter family of central complex hyperplane arrangements in Cd. If the underlying matroid MAt does not depend on t, so does the diffeomorphism type of M(At).

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 At be a smooth one-parameter family of central complex hyperplane arrangements in Cd. If the underlying matroid MAt does not depend on t, so does the isomorphism type of QAt.

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 = {H1, . . . , Hm} and B = {K1, . . . , Km} be rank d central hyperplane arrangements in Cd 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 = {H1, . . . , Hm} and B = {K1, . . . , Km} be rank d central hyperplane arrangements in Cd 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 = {H1, . . . , Hm} and B = {K1, . . . , Km} be rank d central hyperplane arrangements in Cd with same underlying matroid. If 1 d m 7, then the Milnor fibrations QA and QB 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