Tags: #geomtop/symplectic-topology

Refs: ?


Complex ball quotients, new symplectic 4-manifolds with nonnegative signature, Sümeyra Sakall, UGA topology seminar.

  • Standing assumptions: \((X, \omega)\) a closed, simply connected, symplectic 4-manifold. Minimal: contains no \(-1\) spheres.

  • Important invariants for classification:

    • \(c_1^2(X) = 2\chi_{\mathsf{Top}}(X) + 3\sigma\) where \(\sigma\) is the signature \(\sigma = b_2^+ - b_2^-\).
    • \(\chi_{\mathop{\mathrm{Hol}}} = {1\over 4}\qty{\chi_{\mathsf{Top}}(X) + \sigma}\) the holomorphic Euler characteristic.
  • Geography problem: which pairs \((a,b) = (\chi_{\mathop{\mathrm{Hol}}}(X), c_1^2(X)) \in {\mathbb{Z}}_{\geq 0}{ {}^{ \scriptscriptstyle\times^{2} } }\) can be realized?

    • Famous lines: the BMY line \(c_1^2 = 8\chi_{\mathop{\mathrm{Hol}}}\), the signature zero line \(c_1^2 = 8 \chi_{\mathop{\mathrm{Hol}}}\) (so \(\sigma = 0\)), the Noether line \(c_1^2 = 2\chi_{\mathop{\mathrm{Hol}}} - 6\).
    • Everything below the \(\sigma=0\) line is realized, above is still open.

  • For the \(\sigma \geq 0\) region: Stipsicz (99), Park (03), Niepel (05), all close to BMY. Akhmedov, Park (08, 20). Large \(\chi\)! New work (15): smallest known constructions in this region.

  • \({\mathbb{CP}}^2\) is the first lattice line on the BMY line.

  • Complex ball quotients: 2 big theorems fully characterizing them.

    • Yau (77), Miyaoka (84): for \(X\) a compact \({\mathbb{C}}{\hbox{-}}\)surface of general type with \(c_1^2(X) = 9 \chi_{\mathop{\mathrm{Hol}}}(X)\), the universal cover of \(X\) is biholomorphic to \(\tilde X \cong ^\circ{\mathbb{B}}^4 \subseteq {\mathbb{C}}^2\). In this case \(X\cong ^\circ{\mathbb{B}}^4/ G\) for \(G = \pi_1 X\) an infinite discrete group.
    • Borel (52), Hirzebruch (57): conversely if \(\tilde X \cong ^\circ{\mathbb{B}}^4\) biholomorphically, then \(c_1^2(X) = 9\chi_{\mathop{\mathrm{Hol}}}(X)\).
  • So all Kodaira dimension 2, infinite \(\pi_1\), all on BMY line. How to construct examples? Historical overview of constructions that realize different points on the BMY line:

    • Fake \({\mathbb{CP}}^2\)s (same Poincare polynomial as \({\mathbb{CP}}^2\) but not isomorphic)
    • Hirzebruch (83): branched covers of configurations.
    • Ishida (88): smaller constructions
    • Bauer-Catanese (06): new types based on Ishida’s construction.
    • Current work: Galois covers of \({\mathbb{CP}}^2\) over a Hesse arrangement: 12 lines, each through 3 points, over the 9 points on a \(3\times 3\) grid. Every point has valence 4.
  • Galois coverings: maps of varieties \(h: X\to Y\) for \(X\) normal and \(Y\) smooth, inducing a field extension \({\mathbb{C}}(Y) / {\mathbb{C}}(X)\). Galois iff \(G = \mathop{\mathrm{Deck}}(X/Y)\) acts transitively on every fiber. For branched covers, Galois iff the unbranched locus yields a Galois cover.

  • Take Hasse arrangement in \({\mathbb{CP}}^2\), blow up at the 9 points. Take a divisor \(D\): the sum of the proper transforms of the 12 lines, plus the sum of the exceptional divisors. Yields \(\widehat{{\mathbb{CP}}^2} \coloneqq{\mathbb{CP}}^2 \mathop{ \Large\scalebox{0.8}{\raisebox{0.4ex}{#}}}9\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{CP}}\mkern-1.5mu}\mkern 1.5mu^2 \xrightarrow{\pi} {\mathbb{CP}}^2\). Consider \(H^1(\widehat{{\mathbb{CP}}^2}\setminus D; {\mathbb{Z}}) \cong {\mathbb{Z}}^{?} \twoheadrightarrow G\coloneqq C_3^3\), take \(G\) as the Galois group of some cover \(p: W\to \widehat{{\mathbb{CP}}^2}\).

  • Computing invariants: canonical divisor formula for ramified Galois covers, see Pardini.

\(\exists W\) a smooth algebraic surface (a ball quotient) on the BMY line, constructed as a \(C_3^3\) cover of \(\widehat{{\mathbb{CP}}^2}\) branched over the Hesse configuration.

  • Next: Cartwright-Steper surfaces and exotic 4-manifolds.
  • Fake \({\mathbb{CP}}^2\)s have \(c_1^2 = 9\chi_{\mathop{\mathrm{Hol}}}\), so ball quotients.
    • Constructions due to Mumford (79), Ishida/Kato (98), Keum (06), Prasad/Yeung (07), Cartwright-Steper (10) obtain a full classification.
    • For all \(n\geq 1\), obtain a complex surface of general type \(M_n\) on the BMY line with \(c_1^2 = 9\chi_{\mathop{\mathrm{Hol}}} = 9n\). Current work uses the smallest, \(M_1\). Intersection form is odd and indefinite, Betti numbers are \(1,2,5,2,1\) with \(\chi_{\mathsf{Top}}(X) = 3\).
  • Lemma: produce \(p: \tilde M\to M\) with \(G = C_2^2\), find a genus 5 real surface \(\Sigma_5 \hookrightarrow\tilde M \mathop{ \Large\scalebox{0.8}{\raisebox{0.4ex}{#}}}\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{CP}}^2\mkern-1.5mu}\mkern 1.5mu\) inducing an isomorphism on \(\pi_1\).

Let \(M\) be one of

  • \((2n-1){\mathbb{CP}}^2 \mathop{ \Large\scalebox{0.8}{\raisebox{0.4ex}{#}}}(2n-1) \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{CP}}^2\mkern-1.5mu}\mkern 1.5mu\) for \(n\geq 9\) (these come from the \(\Sigma_5\) above),
  • \((2n-1){\mathbb{CP}}^2 \mathop{ \Large\scalebox{0.8}{\raisebox{0.4ex}{#}}}(2n-2) \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{CP}}^2\mkern-1.5mu}\mkern 1.5mu\) for \(n\geq 9\),
  • \((2n-1){\mathbb{CP}}^2 \mathop{ \Large\scalebox{0.8}{\raisebox{0.4ex}{#}}}(2n-3) \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{CP}}^2\mkern-1.5mu}\mkern 1.5mu\) for \(n\geq 10\).

Using various symplectic blowups/surgeries and symplectic resolution, there are infinitely many

  • minimal simply connected symplectic 4-manifolds
  • minimal simply connected non-symplectic 4-manifolds

which are homeomorphic but not diffeomorphic to \(M\).

Tools used:

  • Preserving homeomorphism types: Freedman.
  • Surgeries used: Gompf, Luttinper.
  • Showing exotic (not diffeomorphic, different smooth structures): Taubes.
  • Minimality: Usher.
  • ?: Finlushel-Stern.
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