Symplectic geometry

$Click here for infomation about last term’s course (Group theory) This term I will be teaching the course “Special Lagrangian fibrations”. The classes take place in the Maths Building, room 611. We will start with symplectic geometry and will follow McDuff and Salamon’s Introduction to Symplectic Topology during the first few lectures. Up to now we have covered sections 2.1 -- 2.4; 3.1 -- 3.3, 6.1 -- 6.2 and 7.1 -- 7.2 of that book as well as parts of sections 8.1 -- 8.3 of Gompf and Stipsicz book. Here is the first exercise sheet for this course. Please, write down solutions for 6 of the boldfaced exercises as well as the extra 4 exercises and hand your solutions by the 2nd of April. Here is the second exercise sheet for this course. Please solve all the questions and hand back before the end of the course. Thursday, May 14, 2009 The course ended with some words about T-duality and the importance of special Lagrangian fibrations for mirror symmetry. Thursday, May 7, 2009 Today we introduced calibrations. Then we showed that, in a Kahler manifold, a surface is calibrated for the symplectic form if and only if it is a complex curve and in general a submanifold calibrated to w^k is a complex submanifold. Then we defined special Lagrangians and went on to consider a elliptic K3 as a double cover of CP^2 # 9 CP^2bar (or equivalently, the fiber connected of CP^2 # 9 CP^2bar with itself). Using the underlying hyperkahler structure, we showed that such a K3 could be made into a special Lagrangian fibration over CP^1. Thursday, April 23, 2009 Today’s topic was pencils and fibrations. We saw how two linearly independent sections s1 and s2 of a complex line bundle over a manifold M give rise to a map from M minus the base locus, i.e., the common zeroes of s1 and s2, to CP^1. Then we saw that if the line bundle is holomorphic and the zeroes of the sections are transverse, one can blow up the base locus to get a well defined map from the blow up to CP^1, obtaining a fibration. We commented that for a Kahler manifold with integral Kahler class, a natural line bundle which can be considered is the one correspoding to the Kahler class and, in this case, the fibers of the pencil are Poincare dual to the Kahler class. We finished explaining the statement of Donaldson’s theorem that the same result obtained above for Kahler manifolds can also be obtained in the symplectic world Thursday, April 16, 2009 We did a few more examples of resolution of singularities, including the resolution of C^2/Z_n and the resolution of the conifold singularity (xy = wz). In the end of the lecture we started our study of pencils and the will continue that next week. Thursday, April 9, 2009 We continued our study of the blow-up. We defined the proper transform of a submanifold and showed that, in the smooth (complex) setting, if tilde{M} is the blow-up of M along a submanifold N and S is another submanifold of M then the proper transform of S is a submanifold of tilde{M} and is equivalent (as complex manifolds) to S blown up along the intersection of S and N. We used the blow up to resolve singularities. This time we focused on how to separate complex curves on the plane through multiple blow-ups. Blow ups are covered in section 7.1 in Mcduff and Salamon and in Griffiths and Harris, page 182 and in several places in Hartshorne (check the index for resolution of singularities). Thursday, April 2, 2009 In this lecture we studied the topology of the blow-up of a point. We saw that, from the differentiable point of view, blowing-up a point is equivalent to doing a connected sum with CP^n with the reverse orientation and that in 4-dimensions the result of a blow up is a in homology is to add a 2-sphere (P^1) with self intersection -1 where before there was only a point. In the complex setting this exceptional divisor is a complex submanifold and in the symplectic setting it is a symplectic submanifold. In both cases whenever such a sphere can be found one can blow it down and the resulting manifold is also complex or symplectic, though in the symplectic case the structure is not canonically determined. Thursday, March 26, 2009 In this class we introduced another method of construction of symplectic manifold: the symplectic blow-up. We studied in detail the blow-up of a point an saw that the same argument also allows us to blow up symplectic submanifolds. This is covered in section 7.1 of McDuff and Salamon and is also covered in Griffiths and Harris, page 182, from the complex point of view. Thursday, March 19, 2009 We moved on to the second method of construction of symplectic manifolds: Gompf’s symplectic connected sums, including symplectic fiber sums. This material is covered in McDuff and Salamon in section 7.2. Thursday, March 12, 2009 In this lecture we proved the Lagrangian neighbourhood theorem. Then we started our study of methods to construct symplectic manifolds. The first method we looked at was Thurston’s symplectic fibrations. This is covered in sections 6.1 and 6.2. Thursday, March 5, 2009 As a warm up, we used Moser’s argument to show that a symplectic structure in a compact, connected, oriented surface is determined its volume. Then we moved to more serious business and proved that a symplectic neighbourhood of a submanifold Q of a symplectic manifold M depends only on the restriction of the symplectic form to TQM. This allowed us to prove Darboux’s Theorem: every point in a symplectic manifold has a neighbourhood symplectomorphic to a neighbourhood of the origin in R^2n endowed with the standard symplectic structure. Then we also proved Weinstein symplectic neighbourhood theorem stating that a neighbourhood of a symplectic submanifold Q is determined by the symplectic structure on Q and the induced symplectic structure on the normal bundle. This material is covered in McDuff and Salamon section 3.3. Thursday, February 26, 2009 In this class we finally introduced symplectic manifolds and studied some of their basic properties. We saw that, in a compact manifold, the existence of a degree 2 cohomology class whose top power is nonzero is a topological requirement for the existence of a symplectic structure, so spheres of dimension more than 2 are not symplectic. Some examples of symplectic manifolds are oriented surfaces, complex projective space and the cotangent bundle of any manifold. We finished the lecture with the first half of Moser’s argument: if a 1-parameter family of symplectic structures wt has exact “time derivative”, then there is a corresponding family of diffeomorphisms ft such that f0 = id and ft*wt = w0. This is covered in McDuff and Salamon in section 3.2. Thursday, February 19, 2009 In this class we studied Cech cohomology, sketched the proof of the Cech to de Rham theorem and used Cech cohomology to understand the (first) Chern class of a complex line bundle and saw that the choice of Cech representatives of that cohomology was equivalent to a choice of connection of the line bundle and the (real) Chern class was represented by the curvature of the chosen connection. We also saw (very quickly) how to define the (higher) Chern classes of Hermitian vector bundles in terms connections and their curvature. You can read more about Chern classes in one of several of Chern’s papers on the subject. Thursday, February 12, 2009 In this class we studied symplectic vector bundles. We showed that every symplectic vector bundle can be endowed with a compatible complex structure and that the space of complex structures compatible with a fix symplectic structure on a vector bundle is contractible. Conversely any complex vector bundle can be endowed with a compatible symplectic structure and the space of such symplectic structures is contractible. Although we presented direct proofs to these facts, one could also see this in terms of the results from previous class (that U(n) is the maximal compact subgroup of Sp(n) as well as GL(n;C)). This material is covered in McDuff and Salamon sections 2.3 and 2.4. Thursday, February 5, 2009 In this class we studied linear symplectic structures. This included symplectic orthogonals, symplectic reduction and Darboux theorem for linear symplectic forms. We also studied the group of linear symplectomorphisms, Sp(n) and got close to the proof that U(n) is a maximal compact subgroup of Sp(n). This material is covered in McDuff and Salamon in sections 2.1 and 2.2.$ shapeimage_2_link_0