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Orientations for DT Invariants on Quasi-Projective Calabi-Yau 4-Folds

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Orientations for DT Invariants on Quasi-Projective Calabi-Yau 4-Folds Orientations for DT invariants on quasi-projective Calabi{Yau 4-folds Arkadij Bojko∗ Abstract For a Calabi{Yau 4-fold (X; !), where X is quasi-projective and ! is a nowhere vanishing section of its canonical bundle KX , the (derived) moduli stack of compactly supported perfect complexes MX is −2-shifted symplectic and thus has an orientation ! bundle O !MX in the sense of Borisov{Joyce [11] necessary for defining Donaldson{ Thomas type invariants of X. We extend first the orientability result of Cao{Gross{ Joyce [15] to projective spin 4-folds. Then for any smooth projective compactification Y , such that D = Y nX is strictly normal crossing, we define orientation bundles on the stack MY ×MD MY and express these as pullbacks of Z2-bundles in gauge theory, constructed using positive Dirac operators on the double of X. As a result, we relate the ! orientation bundle O !MX to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter, we obtain orientability of MX . We also prove orientability of moduli spaces of stable pairs and Hilbert schemes of proper subschemes. Finally, we consider the compatibility of orientations under direct sums. 1 Introduction 1.1 Background and results Donaldson-Thomas type theory for Calabi{Yau 4-folds has been developed by Borisov{ Joyce [11], Cao{Leung [22] and Oh{Thomas [73] . The construction relies on having Serre duality Ext2(E•;E•) =∼ Ext2(E•;E•)∗ : for a perfect complex E•. This gives a real structure on Ext2(E•;E•), and one can take 2 • • its real subspace Ext (E ;E )R. Borisov{Joyce [11, Definition 3.25] define orientations as a square root of the natural isomorphism arXiv:2008.08441v3 [math.AG] 1 Jul 2021 2 ∼ det(LX ) = OX ∗Address: Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, U.K., E-mail: [email protected] 1 for any −2-shifted symplectic derived scheme X. On moduli stacks of sheaves, this is equiv- 2 • • alent to the orientation of Ext (E ;E )R or in the language of [73] to a choice of a class of positive isotropic subspaces. For the Borisov{Joyce class or Oh{Thomas class to be defined in singular homology, re- −1 spectively Chow homology with Z[2 ] coefficients, one needs to show that these choices can be made continuously. This was proved for compact Calabi{Yau 4-folds in Cao{Gross{Joyce [15]. However, many conjectural formulas have been written down in [17, 18, 19, 20, 21, 23, 25] and [26] for DT4 invariants of non-compact Calabi{Yau 4-folds, where additionally the moduli spaces can be non-compact and one has to use localization (see Oh{Thomas [73, §6] and Cao{Leung [22, §8]). The construction of virtual structure sheaves of Oh{Thomas [73] for non-compact Calabi{Yau 4-folds and its localization also depends on existence of global orientation. Moreover, in string theory the DT4-invariants over Hilbert (respectively Quot- ) schemes of points were studied by Nekrasov [69] and Nekrasov{Piazzalunga [70]. The dependence on signs (orientations) is explained in [70, §2.4.2]. The main goal of this work is to justify these results geometrically by proving ori- entability of the moduli stack MX of compactly supported perfect complexes for any quasi- projective Calabi{Yau 4-fold extending the result of Cao{Gross{Joyce [15]. We also prove orientability of moduli spaces of stable pairs (as in [24, 21]) and of Hilbert schemes of proper subschemes. Our result is also used by Oh{Thomas [73] to extend their theory to quasi- projective Calabi{Yau 4-folds and allows for the construction of global virtual structure sheaves even over quasi-projective moduli spaces when X is also quasi-projective. Of particular importance are general toric Calabi{Yau 4-folds and due to work of 4 Nekrasov and his collaborators especially C which was considered from a mathematical point of view in Cao{Kool [17] and Cao{Kool{Monavari [20]. Recently. Kool{Rennemo 4 [56] described explicitly the orientations on Quot-schemes of points on C by entirely dif- ferent means giving evidence for our more general result. A Calabi{Yau n-fold (X; Ω) here is a smooth quasi-projective variety X of dimension n over C, such that Ω is a nowhere vanishing (algebraic) section of its canonical bundle KX . An important feature of our results is that we are able to express our orientations on MX in terms of differential geometric ones. This allows us to consider compatibility of ! choices of trivializations of the orientation bundle O !MX under direct sums, which is related to constructing vertex algebras and Lie algebras on the homology of MX as done by Joyce in [49] and used for expressing wall-crossing formulae by Gross{Joyce{Tanaka in [43]. As another consequence we obtain that if Mα is a moduli scheme or stack of sheaves 0 with given compactly supported K-theory class α 2 Kcs (X), then there is a natural choice of orientations on it up to a global sign coming from differential geometry independent of how many connected components Mα has after fixing a choice of compactification Y of X by a strictly normal crossing divisor in the sense of Definition 2.21. 2 1.2 Content of sections and main theorems In §2 we introduce briefly the language of derived algebraic geometry and recall the results of Pantev{To¨en{Vaqui´e{Vezzosi [74] and Brav{Dyckerhoff [13] about existence of shifted symplectic structures. For any smooth (quasi-)projective variety X, we introduce twisted virtual canonical bundles ΛL for any coherent sheaf L on the moduli stack of (compactly supported) perfect complexes MX . If (X; Θ) is a spin manifold as in Definition 2.10, then the virtual canonical bundle is defined by KMX = ΛΘ, where Θ is the corresponding S theta characteristic. In Definition 2.11, we define the orientation bundle O !MX over the moduli stack of perfect complexes on a projective spin 4-fold X and prove that it is trivializable: Theorem 1.1 (see Theorem 2.14). Let (X; Θ) be smooth, projective and spin of complex dimension 4, and S O !MX (1.1) top the orientation bundle from Definition 2.11. Let ΓX :(MX ) !CX be as in Definition 2.13 and apply the topological realization functor (−)top from Blanc [8, Definition 3.1] to 1.1 S top top to obtain a Z2-bundle (O ) ! (MX ) : There is a canonical isomorphism of Z2-bundles = S top ∼ ∗ D+ (O ) = ΓX (OC ) ; D= + where OC !CX is the Z2-bundle from Joyce{Tanaka{Upmeier [51, Definition 2.22] ap- plied to the positive Dirac operator D= + : S+ ! S− as in Cao{Gross{Joyce [15, Theorem S 1.11]. In particular, O !MX is trivializable. We describe examples where this can be used to construct orientations on non-compact Calabi{Yau 4-folds using Corollary 2.16. These include 4 a) C . b) Tot(KV ! V ) for every smooth projective variety V , where KV is the canonical line bundle and dimC(V ) = 3. c) Tot(L1 ⊕ L2 ! S) for any smooth projective surface S, where L1;L2 are line bundles ∼ such that L1 ⊗ L2 = KS. d) Tot(E ! C) for any smooth projective curve C, where E is a rank 3 vector bundle, 3 such that Λ E = KC . This is a simple generalization of the work of Cao{Gross{Joyce [15], . Instead of trying to answer the question of existence of \spin compactifications” of Calabi{Yau 4-folds, we define in Definition 2.24 for any non-compact Calabi{Yau 4-fold X and its smooth projective compactification Y the moduli stack MY ×MD MY of perfect complexes identified at the divisor D = Y nX which we require to be strictly normal crossing ./ (see Definition 2.21). We define an orientation bundle O !MY ×MD MY depending on 3 extension data ./, which contains the information about a choice of order ord of the smooth SN irreducible divisors in D = i=1 Di. Under the inclusion into the first component ζ : MX !MY ×MD MY ./ ! the bundle O pulls back to O !MX . This construction mimics the excision techniques from gauge theory as in Donaldson{Kronheimer [30], Donaldson [29] and Upmeier [95] and is more natural. The analog in topology would be considering vector bundles E; F ! X+ identified on +, where (X+; +) is the one point compactification of X by the point +. 0 This is known to generate Kcs (X) which is the natural topological analog of compactly supported perfect complexes on X. We collect here the orientability results stated in §2 (see Theorem 2.26 and Theorem 2.28): Theorem 1.2 (Theorem 2.26). Let (X; Ω) be a smooth Calabi{Yau 4-fold and Y its smooth projective compactification by a strictly normal crossing divisor D. Choose ord and the extension data ./ as in Definition 2.23, then the Z2-bundle ./ O !MY ×MD MY is trivializable. Let CD; CY be the topological spaces of maps from Definition 2.13. Let C DO !CY ×CD CY be the trivializable Z2-bundle from (4.13). If top Γ:(MY;D) !CY ×CD CY is the natural map from Definition 2.25, then there exists a canonical isomorphism ./ ∗ C ∼ ./ top I :Γ (DO) = (O ) : As a consequence of this we prove: Theorem 1.3 (Theorem 2.28). Let (X; Ω) be a Calabi{Yau 4-fold, then the Z2-bundle ! O !MX (1.2) from Definition 2.9 is trivializable. Moreover, fix a smooth projective compactification Y , such that D = Y nX is a strictly ! top top normal crossing divisor.
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