A vanishing theorem on fake projective planes with enough automorphisms

Abstract

Abstract. For every fake projective plane X with automorphism group of order 21, we prove that H i (X, 2L) = 0 for all i and for every ample line bundle L with L 2 = 1. For every fake projective plane with automorphism group of order 9, we prove the same vanishing for every cubic root (and its twist by a 2-torsion) of the canonical bundle K. As an immediate consequence, there are exceptional sequences of length 3 on such fake projective planes.A compact complex surface with the same Betti numbers as the complex projective plane P 2 C is called a fake projective plane if it is not isomorphic to P 2 C . The canonical bundle of a fake projective plane is ample. So a fake projective plane is nothing but a surface of general type with p g = 0 and c 2 1 = 3c 2 = 9. Furthermore, its universal cover is the unit 2-ball in C 2 by [Au] and [Y] and its fundamental group is a co-compact arithmetic subgroup of PU(2, 1) by [Kl].Recently, Prasad and Yeung [PY] classified all possible fundamental groups of fake projective planes. Their proof also shows that the automorphism group of a fake projective plane has order 1, 3, 9, 7, or 21. Then Cartwright and Steger ([CS], [CS2]) have carried out group theoretic enumeration based on computer to obtain more precise result: there are exactly 50 distinct fundamental groups, each corresponding to a pair of fake projective planes, complex conjugate to each other. They also have computed the automorphism groups of all fake projective planes X. In particular Aut(X) ∼ = {1}, C 3 , C 2 3 or 7 : 3, where C n is the cyclic group of order n and 7 : 3 is the unique non-abelian group of order 21. Among the 50 pairs 34 admit a non-trivial group of automorphisms: 3 pairs have Aut ∼ = 7 : 3, 3 pairs have Aut ∼ = C 2 3 and 28 pairs have Aut ∼ = C 3 . For each pair of fake projective planes Cartwright and Steger [CS2] have also computed the torsion group H 1 (X, Z) =Tor(H 2 (X, Z)) =Tor(Pic(X)), which is the abelianization of the fundamental group. According to their computation a fake projective plane with more than 3 automorphisms has no 3-torsion.It can be shown (Lemma 1.5) that if a fake projective plane X has no 3-torsion in H 1 (X, Z), then the canonical class K X is divisible by 3 and has a unique cubic root, i.e., a unique line bundle L 0 , up to isomorphism, such that 3L 0 ∼ = K X . Its isomorphism class [L 0 ] is fixed by Aut(X), since Aut(X) fixes the canonical class.For a fake projective plane X an ample line bundle L is called an ample generator if its isomorphism class [L] generates Pic(X) modulo torsion, or equivalently if the self-intersection number L 2 = 1. Any two ample generators differ by a torsion. We

Exceptional collections and the bicanonical map of Keum’s fake projective planes

“…In this section, we will again assume that the surface X is a fake projective plane, so that P ic(X) is of rank one and K 2 X = 9. The statement below follows from the definition of exceptional collection combined with some complex surface theory, and it has already appeared in the recent literature ([GKMS15], [Keu14], [LY]). We give the details for the reader's convenience.…”

“…More in detail, the subcategory A in the statement is indeed an H-phantom. In order to see this, one can follow Section 4 in [Keu14] and recall that the Hochschild homology of X is isomorphic to its Hodge cohomology. Since X is a fake projective plane, its Hodge cohomology must have dimension equal to 3.…”

Quasiphantom categories on a family of surfaces isogenous to a higher product

“…Quasiphantom categories are surprising new subcategories in the derived categories of algebraic varieties first discovered by Böhning, Bothmer and Sosna in [7]. Their discovery provides new perspectives on the study of derived categories of algebraic varieties and recently many examples of quasiphantom categories were constructed by many authors(see [1,6,7,10,11,15,16,18,19,20,22,23,24] for more details). However their structures are quite mysterious and we do not know whether every surface of general type with p g = q = 0 has a quasiphantom category in its derived category.…”

Exceptional collections on some fake quadrics

“…Motivated by their results now there are lots of studies on derived categories of surfaces of general type with p g = q = 0. See the papers of Böhning, Graf von Bothmer, and Sosna [4], Alexeev and Orlov [1], Galkin and Shinder [12], Böhning, Graf von Bothmer, Katzarkov and Sosna [3], Fakhruddin [10], Galkin, Katzarkov, Mellit and Shinder [11], Coughlan [8], Keum [14] and the first author [15,16] for more details. They constructed categories with vanishing Hochschild homologies as orthogonal complements of exceptional sequences of line bundles of maximal lengths.…”