THE FUNDAMENTAL GROUP IS NOT A DERIVED
INVARIANTCHRISTIAN SCHNELL

Abstract. We show that the fundamental group is not invariant under derived equivalence of smooth projective varieties.

1. Acknowledgements. This short paper is based on my talk at the conference
Derived Categories Tokyo 2011, and I am very grateful to Yuujiro Kawamata andYukinobu Toda for inviting me to the conference. I also thank Lev Borisov forsuggesting the example to me in the first place, and Andrei Căldăraru and Sukhendu
Mehrotra for useful discussions.
Some time after giving the talk, I discovered that Anthony Bak [Bak09] alreadyhas a preprint on arXiv in which he obtains the same result. In fact, his proof ismore concrete, and therefore more useful for doing calculations with the derived
equivalence. My apology for nevertheless writing this note is that the proof given
here is different and, by relying on the theorem of Bridgeland and Maciocia [BM02],
a little bit shorter than Bak’s.
This work was supported by the World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan, and by NSF grant DMS-1100606.
2. Introduction. For a smooth complex projective variety X, we denote byDb (X) = Db Coh(X) the bounded derived category of coherent sheaves on X.
Recall that two smooth projective varieties X are Y are said to be derived equivalent if Db (X) ' Db (Y ) as C-linear triangulated categories. We sometimes write
X ∼ Y to indicate that X and Y are derived equivalent.
From the work of Bondal, Orlov, Căldăraru, Kawamata, and others, it is known
that many of the basic invariants of algebraic varieties are preserved under derivedequivalence. These include the dimension, the Kodaira dimension, the canonicalring, and the order of the canonical class. It has also been conjectured that theHodge structure on the cohomology with rational coefficients is a derived invariant,in the sense that if X ∼ Y , then one should have H k (X, Q) ' H k (Y, Q) as rational
Hodge structures, for every k ∈ Z. In particular, it is expected that the Hodgenumbers

hp,q (X) = dimC H p,q (X) = dimC H q X, ΩpX
are invariant under derived equivalences.
In joint work with Mihnea Popa [PS11], we showed that if X and Y are derived
equivalent, then H 1 (X, Q) ' H 1 (Y, Q) as rational Hodge structures; in geometricterms, this means that the two Picard varieties Pic0 (X) and Pic0 (Y ) are isogenous
abelian varieties. Ignoring the choice of basepoint,
H 1 (X, Q) ' HomZ π1 (X), Q ,and so our result naturally leads to the question whether the fundamental group
π1 (X) is itself a derived invariant. The point of this paper is to show that this isnot the case.
Key words and phrases. Derived category, Derived equivalence, Calabi-Yau threefold, Fundamental group, (1,8)-polarized abelian surface.
12CHRISTIAN SCHNELL

More precisely, I will describe an example of a simply connected Calabi-Yau
threefold X, with a nontrivial free action by a finite group G, such that the quotient X/G is derived equivalent to X. Since π1 (X/G) = G, while π1 (X) = {1},this means that neither the fundamental group nor the property of being simply
connected are preserved under derived equivalence.
3. A related problem. Before continuing, I should point out that this result
is connected to a larger question raised by Daniel Huybrechts and Marc NieperWisskirchen, about derived equivalences of varieties with trivial first Chern class.
To set up some notation, suppose that X is a smooth projective variety whose firstChern class c1 (X) is zero as an element of H 2 (X, R). By Yau’s theorem, X admits
a Ricci-flat Kähler metric; by studying the holonomy of this metric, Bogomolov andBeauville [Bea83] have shown that a finite étale cover X 0 → X decomposes into afinite product
YY
X0 ' A ×
Yi ×
Zjijwith A an abelian variety, each Yi a simply connected Calabi-Yau manifold of dimension at least three, and each Zj a holomorphic symplectic manifold. Huybrechts
and Nieper-Wisskirchen ask whether the structure of this decomposition is invariant
under derived equivalences [HNW11, Question 0.2].
In the special case of Calabi-Yau threefolds, the question becomes the following:
Suppose that X is a simply connected Calabi-Yau threefold, and that Y ∼ X.
Because of the example in this paper, we cannot expect Y to be simply connected.
On the other hand, the first Chern class of Y is also trivial, and so a finite étale
cover of Y must be of one of the following three types:(1) A simply connected Calabi-Yau threefold.
(2) An abelian threefold.
(3) The product of an elliptic curve and a K3-surface.
Note that there are examples of finite quotients of abelian threefolds (or productsof an elliptic curve and a K3-surface) with trivial canonical bundle and zero firstBetti number; a partial classification may be found in [OS01]. Nevertheless, it
seems likely that such varieties cannot be derived equivalent to a simply connected
Calabi-Yau threefold. A proof of this would be a useful step towards answering thegeneral question of Huybrechts and Nieper-Wisskirchen.
4. The example. Let us now turn to the description of the example, which wassuggested to me by Lev Borisov. The Calabi-Yau threefold in question is one of aclass of such varieties constructed by Mark Gross and Sorin Popescu [GP98], andhas to do with (1,8)-polarized abelian surfaces.
We shall begin by recalling their construction. Let (A, L) be a (1,8)-polarizedabelian surface. In other words, suppose that A is an abelian surface, and that Lis an ample line bundle on A such that the isogeny
ϕL : A → Pic0 (A),a 7→ t∗a (L) ⊗ L−1 ,has kernel isomorphic to Z8 × Z8 . One can show that L is then automatically very
ample with h0 (L) = 8; by the Riemann-Roch theorem, it follows that we have
8 = χ(L) = L2 /2, which gives L2 = 16. The line bundle therefore embeds A asa surface of degree 16 into P7 , and it is possible to choose the coordinates on theprojective space in such a way that the action of G = Z8 × Z8 on P7 is given by theformulas
σ(x0 : x1 : · · · : x6 : x7 ) = (x1 : x2 : · · · : x7 : x0 )
τ (x0 : x1 : · · · : x6 : x7 ) = (x0 : ζx1 : · · · : ζ 6 x6 : ζ 7 x7 ).

THE FUNDAMENTAL GROUP IS NOT A DERIVED INVARIANT

3Here σ and τ denote the two natural generators of the group G, and ζ is a primitive
eighth root of unity.
The idea of Gross and Popescu is to look at quadrics in P7 that contain theimage of A. Provided that the pair (A, L) is general in moduli, they show thatthe space of such quadrics has dimension four, and that it is generated by the fourquadrics f , σf , σ 2 f , and σ 3 f , wheref = y1 y3 (x20 + x24 ) − y22 (x1 x7 + x3 x5 ) + (y12 + y32 )x2 x6 ,and y ∈ P2 is a general point. The intersection
V8,y = Z(f ) ∩ Z(σf ) ∩ Z(σ 2 )f ∩ Z(σ 3 f )is then a threefold on which the group G acts freely. Conversely, if we assume
that y ∈ P2 is chosen sufficiently general, V8,y will be a complete intersection ofdimension three which is smooth except for 64 ordinary double points, the G-orbitof the point (0 : y1 : y2 : y3 : 0 : −y3 : −y2 : −y1 ). There is always a one-dimensionalfamily of (1, 8)-polarized abelian surfaces contained in V8,y , and every member ofthe family passes through the 64 distinguished points.
1Gross and Popescu discovered that V8,y admits two small resolutions V8,yand
2V8,y , both Calabi-Yau threefolds. The original abelian surface A is a Weil divisor
2on V8,y that is not Cartier; blowing up A produces a small resolution V8,y→ V8,y .
2
Using the Lefschetz theorem and adjunction, one can easily show that V8,yis asimply connected Calabi-Yau threefold. The 64 exceptional curves can be flopped
1
simultaneously to produce another simply connected Calabi-Yau threefold V8,y, andGross and Popescu compute that1
1h1,1 (V8,y) = h1,2 (V8,y) = 2.
1In fact, the Picard group of V8,yis generated (modulo torsion) by the classes of twodivisors: the strict transform A of the original abelian surface, and the preimage
H of a hyperplane section. They satisfy

H 3 = 16,H 2 · A = 16,

H · A2 = 0,

A3 = 0.

The reader can find a concise summary of all the properties of the Calabi-Yau
1
2threefolds V8,yand V8,yin [GP08]. We shall only list those that are needed below.
(1) The linear system |A| is one-dimensional, and the resulting morphism
1p : V8,y→ P1

is an abelian surface fibration with exactly 64 sections. The images of these
sections are the 64 exceptional curves of the flop.
(2) Every smooth fiber of p is a (1,8)-polarized abelian surface, with polarization induced by the restriction of the line bundle OX (H). The intersection
with the images of the 64 sections is precisely the kernel of the polarization.
(3) There are exactly eight singular fibers, each of them an elliptic translation
scroll. Such a scroll is obtained from an elliptic normal curve E in P7 byfixing a point e ∈ E, and letting Te (E) be the union of all lines through x
and x + e, for x ∈ E. It is not hard to see that Te (E) is singular precisely
along the elliptic curve E, and that E × P1 is a resolution of singularities.
1(4) The group G acts freely on V8,y, and the 64 sections form a single G-orbit.
On each smooth fiber, the action of G is the natural action by the kernelof the polarization; on each singular fiber, the action of G is the naturalaction by the group of 8-torsion points of the elliptic curve.
1
Now let X = V8,y. Since G = Z8 × Z8 acts freely on X, the quotient X/Gis again a smooth projective variety with trivial canonical bundle. The following
theorem is the main result of this paper.

4

CHRISTIAN SCHNELL

Theorem 4.1. The two varieties X and X/G are derived equivalent.
Since X is simply connected, while the quotient X/G has fundamental groupisomorphic to G, it follows that the fundamental group is not invariant underderived equivalences.
Note. For reasons coming from physics, Mark Gross and Simone Pavanelli [GP08]
conjecture that the quotient of X by one of the two Z8 -factors of G should be themirror manifold of X, and that X/G should be the mirror of the mirror. Homological mirror symmetry would therefore predict that Db (X) ' Db (X/G).
5. Proof of the theorem. We now describe one possible proof of Theorem 4.1.
An earlier proof, more concrete but also slightly longer, may be found in the preprint
by Anthony Bak [Bak09].
To explain the basic idea, let us consider one of the smooth fibers A of themorphism p : X → P1 ; it is a (1, 8)-polarized abelian surface, with polarization L
induced by the restriction of H. The group G acts on A, and the 64 points in thekernel of the isogeny
ϕL : A → Pic0 (A)form a single G-orbit. Consequently, ϕL gives rise to an isomorphism A/G 'Pic0 (A). Moreover, once we choose one of the 64 points in the kernel as the unit
element on A, there is a well-defined Poincaré line bundle on A × Pic0 (A), and theassociated Fourier-Mukai transform induces an equivalence Db (A) ' Db (Pic0 (A)).
In our proof, we shall generalize these observations by (1) using the theorem ofBridgeland and Maciocia to prove that X is derived equivalent to the compactified
relative Picard scheme M , and (2) showing that M is actually isomorphic to X/G.
We begin by introducing the space M . Let s0 be one of the 64 sections ofp : X → P1 . The general fiber of p is an abelian surface, and the eight singularfibers are elliptic translation scrolls, and therefore reduced and irreducible. Thus itmakes sense to consider the compactified relative Picard scheme
M = Pic0 (X/P1 )defined by Altman and Kleiman [AK79]. For any smooth fiber A of X → P1 ,the corresponding fiber of M → P1 is also smooth and isomorphic to Pic0 (A). In
general, the compactified relative Picard scheme may be singular, and may fail tobe a fine moduli space because the ambiguity in normalizing the Poincaré bundle
can prevent the existence of a universal sheaf. But in our case, everything works
out nicely.
Lemma 5.1. M is a nonsingular projective Calabi-Yau threefold. Moreover, auniversal sheaf exists on M × X, and induces an equivalence Db (M ) ' Db (X).
Proof. To begin with, Sawon [Saw08, Lemma 8] has shown that M is projective,
because it is an irreducible component of Simpson’s moduli space of stable rank-one
torsion-free sheaves on X. Next, the existence of a section s0 implies that there isa universal sheaf on M ×P1 X ⊆ M × X. Indeed, because X is nonsingular, theimage of s0 has to lie inside the smooth locus of p, and so we can apply [AK79,Theorem 3.4] to obtain the existence of a universal sheaf. In particular, M is a finemoduli space.
The theorem of Bridgeland and Maciocia [BM02, Theorem 1.2] now allows us toconclude that M is also a nonsingular Calabi-Yau threefold, and that the universal
sheaf induces an equivalence between the derived categories of M and X.
The remainder of the proof consists in showing that M is, in fact, isomorphic
to the quotient X/G. Our argument is based on the observation, explained above,

THE FUNDAMENTAL GROUP IS NOT A DERIVED INVARIANT

5that the smooth fibers of X/G and M are canonically isomorphic. The main issue
is to extend this isomorphism to the singular fibers.
We begin by constructing a rational map from X/G to M , using the universal
property of the fine moduli space M . The idea is the following: Let (A, L) be apolarized abelian variety, and let PA denote the normalized Poincaré bundle on
A × Pic0 (A). The pullback of PA under the morphismid ×ϕL : A × A → A × Pic0 (A)satisfies(5.2)(id ×ϕL )∗ PA ' m∗ L ⊗ pr ∗1 L−1 ⊗ pr ∗2 L−1 ,where m : A × A → A is the multiplication on A. This allows us to describe themorphism ϕL to the moduli space Pic0 (A) in terms of a line bundle on A × A.
To extend this construction to X, let B ⊆ P1 be the complement of the eight
singular values of p, and set U = p−1 (B). Then p : U → B is smooth, and ourchoice of section s0 makes it into a group scheme over B. We also denote themultiplication morphism by m : U ×B U → U .
Lemma 5.3. There is a morphism f : U/G → M , commuting with the projections
to P1 , whose restriction to any smooth fiber agrees with the natural isomorphism
A/G → Pic0 (A) induced by the (1, 8)-polarization OX (H)|A .
Proof. We begin by constructing a morphism U → M whose restriction to every
smooth fiber A agrees with the natural morphism ϕL : A → Pic0 (A) induced by thepolarization L = OX (H)|A . By the universal property of M , it suffices to constructa line bundle on the product U ×B U whose restriction to A × A is isomorphic tothe pullback of the Poincaré bundle PA . But clearlym∗ OX (H) ⊗ pr ∗1 OX (−H) ⊗ pr ∗2 OX (−H)
is such a line bundle, by virtue of (5.2). The same identity shows that this line
bundle is invariant under the action of G on the second factor of U ×B U , andtherefore descends to a line bundle on U ×B (U/G) whose restriction to A × (A/G)
equals the pullback of PA . The universal property of M now gives us the desired
morphism f : U/G → M .
In particular, f is an isomorphism onto its image, and so X/G and M arebirational. This is already sufficient to conclude that π1 (M ) ' G; but in fact, wecan use the geometry of both varieties to show that they must be isomorphic.
Lemma 5.4. f : U/G → M extends to an isomorphism X/G ' M .
Proof. We consider f as a birational map from X/G to M . Since both are smoothCalabi-Yau threefolds, any birational map between X/G and M is either an isomorphism, or a composition of flops [Kol89, Theorem 4.9]. The second possibility
is easily ruled out: Indeed, since f is an isomorphism over U , the exceptional locus
is contained in the eight singular fibers; moreover, it must be a union of rational
curves [Kol89, Proposition 4.6]. Now each singular fiber is an elliptic translation
scroll, which means that the only rational curves on it are the one-dimensionalfamily of lines on the scroll. Since these lines cover the singular fiber, which is adivisor in X/G, they cannot be flopped. Consequently, the birational map f must
extend to an isomorphism X/G ' M .
Note. Following the argument [GP08, Lemma 1.2], one can show more generallythat X/G does not admit any flops relative to P1 at all.

6CHRISTIAN SCHNELL

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Institute for the Physics and Mathematics of the Universe, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8583, Japan
E-mail address: cschnell@math.uic.edu