---
date: 2022-03-28
tags: [web/quick_notes]
---

# 2022-03-28

Tags:
#geomtop/symplectic-topology 

Refs:
?

## 15:04

> Complex ball quotients, new [symplectic](symplectic.md) 4-manifolds with nonnegative [signature](Unsorted/signature.md), 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:
  - $c_1^2(X) = 2\chi_\Top(X) + 3\sigma$ where $\sigma$ is the signature $\sigma = b_2^+ - b_2^-$.
  - $\chi_{\Hol} = {1\over 4}\qty{\chi_\Top(X) + \sigma}$ the holomorphic Euler characteristic.
- Geography problem: which pairs $(a,b) = (\chi_{\Hol}(X), c_1^2(X)) \in \ZZ_{\geq 0}\cartpower{2}$ can be realized?
  - Famous lines: the BMY line $c_1^2 = 8\chi_{\Hol}$, the signature zero line $c_1^2 = 8 \chi_{\Hol}$ (so $\sigma = 0$), the Noether line $c_1^2 = 2\chi_{\Hol} - 6$.
  - Everything below the $\sigma=0$ line is realized, above is still open.

  <!-- Xournal file: /home/zack/figures/2022-03-28_15-14.xoj -->

  ![](figures/2022-03-28_15-18-41.png)

- 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.

- $\CP^2$ 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 $\CC\dash$surface of general type with $c_1^2(X) = 9 \chi_{\Hol}(X)$, the universal cover of $X$ is biholomorphic to $\tilde X \cong \interior \BB^4 \subseteq \CC^2$.
  In this case $X\cong \interior \BB^4/ G$ for $G = \pi_1 X$ an infinite discrete group.
  - Borel (52), Hirzebruch (57): conversely if $\tilde X \cong \interior \BB^4$ biholomorphically, then $c_1^2(X) = 9\chi_{\Hol}(X)$.

- 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 $\CP^2$s (same Poincare polynomial as $\CP^2$ but not isomorphic)
  - Hirzebruch (83): [branched covers](branched%20covers) of configurations.
  - Ishida (88): smaller constructions
  - Bauer-Catanese (06): new types based on Ishida's construction.
  - Current work: *Galois covers* of $\CP^2$ 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 $\CC(Y) / \CC(X)$.
  Galois iff $G = \Deck(X/Y)$ acts transitively on every fiber.
  For branched covers, Galois iff the unbranched locus yields a Galois cover.

- Take Hasse arrangement in $\CP^2$, 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 $\hat{\CP^2} \da \CP^2 \connectsum 9\bar{\CP}^2 \mapsvia{\pi} \CP^2$.
  Consider $H^1(\hat{\CP^2}\sm D; \ZZ) \cong \ZZ^{?} \surjects G\da C_3^3$, take $G$ as the Galois group of some cover $p: W\to \hat{\CP^2}$.

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


:::{.theorem title="Akhmedov, S, Yeung"}
$\exists W$ a smooth algebraic surface (a ball quotient) on the BMY line, constructed as a $C_3^3$ cover of $\hat{\CP^2}$ branched over the Hesse configuration.
:::

- Next: Cartwright-Steper surfaces and exotic 4-manifolds.
- Fake $\CP^2$s have $c_1^2 = 9\chi_{\Hol}$, 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 $c_1^2 = 9\chi_{\Hol} = 9n$.
  Current work uses the smallest, $M_1$.
  Intersection form is odd and indefinite, Betti numbers are $1,2,5,2,1$ with $\chi_\Top(X) = 3$.

- Lemma: produce $p: \tilde M\to M$ with $G = C_2^2$, find a genus 5 real surface $\Sigma_5 \injects \tilde M \connectsum \bar{\CP^2}$ inducing an isomorphism on $\pi_1$.


:::{.theorem title="?"}
Let $M$ be one of

- $(2n-1)\CP^2 \connectsum (2n-1) \bar{\CP^2}$ for $n\geq 9$ (these come from the $\Sigma_5$ above),
- $(2n-1)\CP^2 \connectsum (2n-2) \bar{\CP^2}$ for $n\geq 9$,
- $(2n-1)\CP^2 \connectsum (2n-3) \bar{\CP^2}$ for $n\geq 10$.

Using various symplectic blowups/surgeries and [symplectic resolution](symplectic%20resolution), 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.

:::