# 2022-03-28

Refs: ?

## 15:04

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.