Fake Projective Planes Overview

• Fake projective planes are smooth compact complex surfaces whose invariants match CP² but differ in complex structure and birational geometry.
• Their classification yields 100 isomorphism classes from 28 arithmetic types, refined by topological and automorphism group properties.
• Recent explicit models use bicanonical embeddings and projective techniques in P⁹ or P⁵ to reveal deep connections between hyperbolic geometry, arithmetic, and derived categories.

Fake projective planes are smooth compact complex surfaces whose Betti numbers agree with those of the complex projective plane but which are not biholomorphic to it. Equivalently, they satisfy

$$b_1=0,\qquad b_2=1,\qquad c_1^2=9,\qquad c_2=3,\qquad p_g=q=0,$$

with ample canonical bundle, so they are surfaces of general type uniformized by the complex $2$-ball and can be written as $B^2/\Gamma$ for a cocompact torsion-free arithmetic lattice $\Gamma<PU(2,1)$ (Borisov et al., 2023, Borisov et al., 2018). Their importance lies in the way they connect complex hyperbolic geometry, arithmetic lattices, explicit projective geometry, and derived-category phenomena. The equality of Betti numbers with $\mathbb P^2$ does not make them close to $\mathbb P^2$ in birational or differential-geometric structure; on the contrary, they are rigid ball quotients of general type.

1. Arithmetic classification and isomorphism types

Prasad–Yeung showed that there are exactly $28$ admissible arithmetic types of ball-quotient surfaces with Euler number $3$, and Cartwright–Steger explicitly enumerated the fake projective planes arising from them. In the resulting classification there are $50$ complex-conjugate pairs, equivalently $100$ isomorphism classes of fake projective planes (Borisov et al., 2023, Stover, 2022). One labeling scheme uses quadruples of the form $2$0, while another packages the arithmetic data as $2$1 (Borisov et al., 2023, Borisov et al., 2 Dec 2025).

This classification has several nonequivalent group-theoretic refinements. By Mostow rigidity, the $2$2 complex-conjugate pairs give $2$3 isomorphism classes of topological fundamental groups. The algebraic fundamental groups, however, are coarser: Stover proved that their profinite completions fall into only $2$4 isomorphism classes, with the only coincidences occurring for the pairs $2$5, $2$6, $2$7, and $2$8 (Stover, 2022). This distinction is central: classification by surface, by lattice, by topological $2$9, and by profinite completion are not identical.

The same arithmetic tables also organize automorphism behavior. The classification data isolate special families with large symmetry, notably the planes with automorphism group $B^2/\Gamma$0, which play a disproportionate role in explicit geometry, vanishing theorems, and equation-finding (Brino et al., 2016, Catanese et al., 2018).

2. Picard groups, positivity, and linear series

A fake projective plane $B^2/\Gamma$1 has Picard number $B^2/\Gamma$2, and the torsion-free part of $B^2/\Gamma$3 is generated by an ample line bundle $B^2/\Gamma$4 with $B^2/\Gamma$5. Every ample line bundle is numerically equivalent to $B^2/\Gamma$6, and the torsion subgroup of $B^2/\Gamma$7 is finite abelian and nontrivial in every case (Cerbo, 2016). In the Cartwright–Steger example

$B^2/\Gamma$8

the Néron–Severi group is described as

$B^2/\Gamma$9

and one may write $\Gamma<PU(2,1)$0 (Borisov et al., 2023).

The local positivity of ample bundles is unusually uniform. Di Cerbo computed the Seshadri constants of all ample line bundles on all fake projective planes and proved

$\Gamma<PU(2,1)$1

for every point $\Gamma<PU(2,1)$2; in particular,

$\Gamma<PU(2,1)$3

The proof uses the Toledo invariant and the fact that fake projective planes contain no totally geodesic curves (Cerbo, 2016). The same analysis gives strong control on low-degree curves: any curve numerically equivalent to $\Gamma<PU(2,1)$4 is smooth of genus $\Gamma<PU(2,1)$5 (Cerbo, 2016).

For the specific plane $\Gamma<PU(2,1)$6, the small linear systems $\Gamma<PU(2,1)$7 with torsion twist $\Gamma<PU(2,1)$8 can be treated explicitly. Riemann–Roch and Kodaira vanishing give

$\Gamma<PU(2,1)$9

except in small cases where vanishing can fail, and the intersection numbers are

$\mathbb P^2$0

Moreover, for all nontrivial torsion $\mathbb P^2$1,

$\mathbb P^2$2

and $\mathbb P^2$3 is base-point-free if and only if $\mathbb P^2$4 (Borisov et al., 2023).

Higher syzygetic positivity also admits effective bounds. Ng–Yeung proved that for any ample line bundle $\mathbb P^2$5 and any nef line bundle $\mathbb P^2$6, the bundle

$\mathbb P^2$7

satisfies property $\mathbb P^2$8. They also showed that if $\mathbb P^2$9 is base-point-free, then $\mathbb P^2$0 is projectively normal for $\mathbb P^2$1 and satisfies $\mathbb P^2$2 for $\mathbb P^2$3, while $\mathbb P^2$4 satisfies $\mathbb P^2$5 for $\mathbb P^2$6 (Ng et al., 2019). This suggests a striking regularity in the asymptotic projective geometry of these ball quotients.

3. Automorphisms and quotient geometry

Automorphisms are not uniformly distributed across the classification. Among the $\mathbb P^2$7 complex-conjugate pairs, exactly $\mathbb P^2$8 have nontrivial automorphism group: $\mathbb P^2$9 with $28$0, $28$1 with $28$2, and $28$3 with $28$4 (Catanese et al., 2018). The order-$28$5 cases are especially amenable to explicit analysis.

For the Cartwright–Steger plane $28$6, the full holomorphic automorphism group is

$28$7

with presentation

$28$8

In certain projective realizations, $28$9 acts with seven fixed points, while $3$0 cyclically permutes those fixed points in orbits of length three (Borisov et al., 2023).

The quotient surfaces produced by these automorphisms often have simpler birational geometry than the original plane. Keum’s vanishing arguments rely on the fact that quotients $3$1 or $3$2 are $3$3-homology projective planes with only cyclic quotient singularities, whose minimal resolutions are $3$4-elliptic surfaces of Kodaira dimension $3$5 (Keum, 2014). In one Keum plane studied explicitly, the minimal resolution of the quotient by $3$6 is a Dolgachev surface over $3$7 with one $3$8 fiber, three nodal fibers, and two multiple fibers of multiplicities $3$9 and $50$0; the three singular points of type $50$1 are cyclically permuted by the residual $50$2-action (Borisov et al., 2023).

Keum also established a converse-type geometric criterion: certain $50$3-homology projective planes with prescribed cyclic quotient singularities and resolution data arise as cyclic quotients of fake projective planes, and this leads to a classification of $50$4-homology projective planes with cusp singularities only (Keum, 2010). The quotient perspective is therefore not merely auxiliary; it is one of the principal tools for turning arithmetic ball quotients into concrete algebraic surfaces.

4. Bicanonical geometry, vanishing theorems, and derived categories

Since $50$5, the canonical linear system is empty, but the bicanonical system is substantial. Riemann–Roch gives

$50$6

so the bicanonical map lands in $50$7, and Reider’s theorem shows that $50$8 is base-point-free for fake projective planes (Brino et al., 2016, Catanese et al., 2018). The obstruction to very ampleness is highly constrained: failure can occur only through curves $50$9 with $100$0, $100$1, and $100$2 (Catanese et al., 2018).

For the planes with the largest automorphism group, the picture is stronger. Every fake projective plane with $100$3 has bicanonical map an embedding (Catanese et al., 2018). More generally, several planes with $100$4 also have bicanonical embedding; in the remaining $100$5-cases singled out by Catanese–Keum, the map fails to separate at most two or three points (Catanese et al., 2018). Di Brino–Di Cerbo likewise showed that the bicanonical map of Keum’s fake projective planes is always an embedding (Brino et al., 2016).

Vanishing theorems for low-degree line bundles are equally rigid. Keum proved that for every fake projective plane with automorphism group of order $100$6,

$100$7

and every ample generator $100$8 with $100$9. If $2$00 is the unique cubic root of $2$01, then also

$2$02

for all $2$03. Analogous vanishings hold for the order-$2$04 case under the stated invariance hypothesis on $2$05 (Keum, 2014). In the order-$2$06 family this gives the exceptional sequence

$2$07

in $2$08 (Keum, 2014).

Derived-category refinements were pushed further by Di Brino–Di Cerbo. On the unique Keum plane with

$2$09

they constructed a nonstandard exceptional collection, and the right orthogonal has vanishing Hochschild homology, hence is an $2$10-phantom (Brino et al., 2016). Relatedly, Galkin–Karzhemanov–Shinder proved that on fake projective planes with at least $2$11 automorphisms every cubic root of the canonical bundle is acyclic, and that there are no nonzero automorphic forms of weight $2$12 (Galkin et al., 2016). These results place fake projective planes at the intersection of rigid surface theory and the study of semiorthogonal decompositions and phantom subcategories.

5. Explicit projective models and defining equations

A major recent development is the transition from arithmetic existence theorems to fully explicit projective equations. The first such models were obtained for a conjugate pair with automorphism group $2$13: the fake projective plane is realized in its bicanonical embedding in $2$14 as a smooth degree-$2$15 surface cut out by exactly $2$16 cubic equations with coefficients in $2$17 (Borisov et al., 2017, Borisov et al., 2018). The equations are $2$18-equivariant, and the ambient coordinates can be chosen so that the action of the generators $2$19 and $2$20 is explicit (Borisov et al., 2017).

Model	Ambient space	Defining equations
Order-$2$21 bicanonical model	$2$22	$2$23 cubics over $2$24
$2$25- or $2$26-model in the Cartwright–Steger/Keum families	$2$27	$2$28 sextics over $2$29
$2$30 model	$2$31	$2$32 cubics over $2$33

The order-$2$34 geometry can also be compressed into $2$35. For the plane

$2$36

Borisov–Ji–Li–Mondal showed that

$2$37

with $2$38 the unique nonzero $2$39-invariant $2$40-torsion line bundle gives a projectively normal embedding

$2$41

and the image is cut out by exactly $2$42 sextics with coefficients in $2$43. Galois conjugation $2$44 exchanges these equations with those of the conjugate fake plane. In the same example, $2$45 itself is not very ample: its image in $2$46 has exactly three singular points, namely the images of the three fixed points of $2$47 (Borisov et al., 2023).

A related but distinct $2$48-realization appears in the study of Keum’s plane $2$49. There the map $2$50 yields a degree-$2$51 surface in $2$52, while the bicanonical model in $2$53 is cut out by $2$54 cubic equations. The $2$55 cubics can be simplified substantially by a coordinate change adapted to the $2$56-action and a carefully chosen nonreduced hyperplane section in $2$57 (Borisov et al., 2023).

Explicit equations are no longer confined to the order-$2$58 family. Borisov–Wang constructed a new pair of fake projective planes labeled $2$59 by starting from a commensurable plane $2$60 and passing through a chain of cyclic covers and quotients. The final surface embeds bicanonically in $2$61 and is cut out by exactly $2$62 independent cubic equations over $2$63; a GAP computation shows that its automorphism group is exactly $2$64, generated by the involution corresponding to $2$65 (Borisov et al., 2 Dec 2025). This suggests a broader program in which explicit models are obtained by moving within commensurability classes rather than starting from scratch.

6. Fundamental groups, non-Archimedean uniformization, and broader structure

The arithmetic nature of fake projective planes extends beyond complex ball uniformization. Allcock–Kato constructed a fake projective plane from a lattice in $2$66 with nontrivial torsion, adapting non-Archimedean uniformization to a setting where the lattice is not torsion-free. The resulting surface is commensurable with Mumford’s fake plane but distinct from it and from the other fake planes arising from $2$67-adic uniformization by torsion-free groups (Allcock et al., 2014).

The same construction gives access to genuinely non-Archimedean topological invariants. For the associated Berkovich space $2$68, there is a deformation retraction onto the quotient of the Bruhat–Tits building by the lattice, and that quotient is homotopy-equivalent to a single circle $2$69 with one $2$70-cell attached by a map of degree $2$71. Consequently,

$2$72

(Allcock et al., 2014). This is a different sort of invariant from either the complex topological fundamental group or its profinite completion, and it illustrates how rigid and $2$73-adic analytic viewpoints can separate phenomena that look identical in the classical surface-theoretic language.

Taken together, the classification of topological and algebraic fundamental groups, the cyclic-cover constructions of quotient surfaces, the explicit equations in $2$74 and $2$75, and the vanishing and derived-category results show that fake projective planes are no longer understood only as rare lattice quotients. They now form a domain in which arithmetic classification, explicit algebraic geometry, and homological methods can be compared on a surface-by-surface basis. A plausible implication is that further progress will come from combining these strands: explicit equations make finite covers and automorphism actions computable, while profinite and non-Archimedean methods distinguish planes that are invisible to coarser topological invariants (Stover, 2022).