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Extremal bundles on Calabi-Yau threefolds. (English) Zbl 1328.14068
Motivated by considerations coming from string theory compactifications, the authors study extremal vector bundles on Calabi-Yau threefolds, i.e. stable $$\mathrm{SU}(n)$$ bundles which satisfy the cancellation of the Green-Schwarz anomaly. More explicitly, a stable vector bundle $$V$$ on a Calabi-Yau threefold $$X$$ is called extremal if (1) $$c_1(V)=0$$ and (2) $$c_2(TX)=c_2(V)$$.
Most of the article takes an experimental point of view, using various databases of stable bundles on standard constructions of Calabi-Yau threefolds to determine the rarity of extremal bundles as well as bounds on their $$c_3(V)$$. Section 3, however, is the theoretical core of the paper, where the authors first show that the spectral cover construction cannot produce extremal bundles on elliptic Calabi-Yau threefolds, which serve as the main testing ground for the authors’ investigation. The authors are thus lead to consider a different construction that generalizes the classical Hartshorne-Serre correspondence and gives locally free sheaves of any rank $$n$$. The essential idea is to use semi-continuity and openness of stability to find stable bundles in the same deformation space as $$\mathcal O_X^{n-1}\oplus I_C$$ for some smooth curve $$C$$, but not obtained simply as a non-trivial extension, as these are not always even reflexive sheaves. In rank two, for example, they first consider a coherent sheaf obtained as an extension $0\to\mathcal O_X\to\mathcal E_0\to I_C\to 0,$ where $$C$$ is a smooth curve in $$X$$ with $$[C]=c_2(TX)$$, just as in the classical Serre construction. They consider a seemingly unrelated second extension $0\to\mathcal O_X(-D)\to\mathcal E_1\to \tilde{\mathcal F}\to 0,$ where $$D$$ is an ample divisor not containing $$C$$ and $$\tilde{\mathcal F}$$ is a torsion-free extension in $$\mathrm{Ext}^1(\mathcal O_D,I_C)$$. The crucial observation is that $$\mathcal E_0$$ and $$\mathcal E_1$$ can also be seen as representing linearly independent classes $$v_0,v_1$$ in some other extension space, and by using semi-continuity, openness of stability, and the construction of $$\mathcal E_0,\mathcal E_1$$, the authors are able to show that the generic member of the family of sheaves represented by $$(1-t)v_0+tv_1,t\in\mathbb A^1$$ is a stable bundle. The authors also consider which bundles of monad type on CICY’s (complete intersection Calabi-Yau’s) satisfy the conditions of being extremal.
In Section 4 of the article, the authors use the extremality conditions to translate the DRY conjecture, which gives sufficient conditions on topological invariants for the existence of a stable vector bundle with those invariants on a Calabi-Yau threefold $$X$$, into a purely topological condition on $$X$$. They then determine, for example, that all of the 7890 CICY’s satisfy this topological condition upon using extremal bundles of monad type, except for 37 self-mirror manifolds. The authors proceed similarly in Section 5 by studying the statistics of the bundles they consider with regard to extremality and the DRY conjecture. For extremal bundles, as an example, they plot the “difference” between $$\mathrm{ch}_2(V)$$ and $$\mathrm{ch}_2(TX)$$, appropriately defined as these represent 1-dimensional cycles, versus $$\mathrm{ch}_3(V)$$ as $$X$$ and $$V$$ range over all possibilities in a given construction and rank. In this way, they obtain some interesting general trends for characteristic classes of known bundles.

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
cohomCalg
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