2026-09-19T06:10:49Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/166312022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Gang Tian.
Usher, Michael Joseph, 1978-
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2005-05-17T14:44:06Z
2005-05-17T14:44:06Z
2004
2004
http://hdl.handle.net/1721.1/16631
56019436
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
Includes bibliographical references (p. 103-104).
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
This thesis takes up a program initiated by S. Donaldson and I. Smith aimed at using symplectic Lefschetz fibration techniques to obtain information about pseudoholomorphic curves in symplectic 4-manifolds. Donaldson and Smith introduced an invariant DS which counts holomorphic sections of a relative Hilbert scheme constructed from a symplectic Lefschetz fibration, and a number of considerations, including a duality relation for DS proven by Smith, led to the conjecture that DS agrees with the Gromov invariant G[gamma] earlier defined by C. Taubes in his study of Seiberg-Witten theory on symplectic manifolds. Our central result is a proof of this conjecture, which thus makes available new proofs of some results concerning pseudoholomorphic curves which had previously only been accessible via gauge theory. The crucial technical ingredient in the proof is an argument which allows us to work with curves C in the total space of the Lefschetz fibration that are made holomorphic by an almost complex structure which is integrable near C and with respect to which the fibration is a pseudoholomorphic map. We also introduce certain refinements of DS and show that these refinements are equal to Gromov invariants which count pseudoholomorphic subvarieties of symplectic 4-manifolds with a prescribed decomposition into reducible components. We prove a vanishing result for some of these invariants which might bear on the question of the uniqueness of the decomposition of the canonical class of a symplectic 4-manifold into classes with nontrivial Gromov-Witten invariants.
by Michael Joseph Usher.
Ph.D.
104 p.
524163 bytes
523953 bytes
application/pdf
application/pdf
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Relative Hilbert scheme methods in pseudoholomorphic geometry
Thesis
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