Flops and derived categories.

*(English)*Zbl 1085.14017In the present paper, the author gives a new interpretation of smooth threefold flops in terms of Fourier-Mukai transforms, thereby relating birational geometry to the theory of perverse coherent sheaves and their moduli.

More precisely, let \(f: Y\to X\) be a birational morphism of projective varieties with 1-dimensional fibres and the property \({\mathbf R}f_*{\mathcal O}_X = {\mathcal O}_X\). Using techniques from the theory of triangulated categories, the author constructs a \(t\)-structure (à la Beilinson-Bernstein-Deligne) on the derived category \(D(Y)\) of coherent sheaves on \(Y\), which then gives rise to the category \(\text{Per}(Y/X)\) of relative perverse sheaves on \(Y\). An object \(F\) of \(\text{Per}(Y/X)\) is called a perverse point sheaf, iff \(F\) is numerically equivalent to the structure sheaf of a point \(y\in Y\), and the author’s first main result establishes the fact that there is a projective scheme \({\mathcal M}(Y/X)\) representing the functor of equivalence classes of families of perverse point sheaves in \(\text{Per}(Y/X)\). Next it is shown how these fine moduli spaces \({\mathcal M}(Y/X)\) can be used to derive Fourier-Mukai type equivalences of derived categories. This is then applied to complex projective threefolds with terminal singularities, using the observation that crepant resolutions of terminal threefolds are related by a finite chain of flops, on the one hand, and that flops correspond to Fourier-Mukai transforms at the level of the involved derived categories of coherent sheaves, on the other. In this context, the author’s second main theorem concerns birational geometry and reads as follows: If \(X\) is a complex projective threefold with terminal singularities and

\[ \begin{matrix} Y_1 && Y_2\\ f_1\searrow&&\swarrow f_2\\ &X\end{matrix} \] are crepant resolutions, then there is an equivalence of derived categories of coherent sheaves \(D(Y_1)@>\sim>> D(Y_2)\).

In particular, this theorem implies that birationally equivalent Calabi-Yau threefolds have equivalent derived categories of coherent sheaves, thereby confirming a conjecture due to A. I. Bondal and D. O. Orlov [Semiorthogonal decomposition for algebraic varieties, preprint, http://arxiv.org/math.A6/9506012] in full generality.

However, the difficult part in this paper is the construction of the moduli space of perverse point sheaves. This requires several advanced techniques, including geometric invariant theory, stability theory for sheaves, Quot schemes, and the construction of what the author calls the perverse Hilbert scheme \(P\)-Hilb\((Y/X)\). This scheme parametrizes quotients of the structure sheaf of \(Y\) in the category Per\((Y/X)\), and the author’s third main theorem establishes its existence as a projective scheme.

All in all, this paper is a major contribution towards a deeper understanding of the significance of Fourier-Mukai transforms and of perverse sheaves within Mori’s Minimal Model Program (MMP) in higher-dimensional birational geometry.

More precisely, let \(f: Y\to X\) be a birational morphism of projective varieties with 1-dimensional fibres and the property \({\mathbf R}f_*{\mathcal O}_X = {\mathcal O}_X\). Using techniques from the theory of triangulated categories, the author constructs a \(t\)-structure (à la Beilinson-Bernstein-Deligne) on the derived category \(D(Y)\) of coherent sheaves on \(Y\), which then gives rise to the category \(\text{Per}(Y/X)\) of relative perverse sheaves on \(Y\). An object \(F\) of \(\text{Per}(Y/X)\) is called a perverse point sheaf, iff \(F\) is numerically equivalent to the structure sheaf of a point \(y\in Y\), and the author’s first main result establishes the fact that there is a projective scheme \({\mathcal M}(Y/X)\) representing the functor of equivalence classes of families of perverse point sheaves in \(\text{Per}(Y/X)\). Next it is shown how these fine moduli spaces \({\mathcal M}(Y/X)\) can be used to derive Fourier-Mukai type equivalences of derived categories. This is then applied to complex projective threefolds with terminal singularities, using the observation that crepant resolutions of terminal threefolds are related by a finite chain of flops, on the one hand, and that flops correspond to Fourier-Mukai transforms at the level of the involved derived categories of coherent sheaves, on the other. In this context, the author’s second main theorem concerns birational geometry and reads as follows: If \(X\) is a complex projective threefold with terminal singularities and

\[ \begin{matrix} Y_1 && Y_2\\ f_1\searrow&&\swarrow f_2\\ &X\end{matrix} \] are crepant resolutions, then there is an equivalence of derived categories of coherent sheaves \(D(Y_1)@>\sim>> D(Y_2)\).

In particular, this theorem implies that birationally equivalent Calabi-Yau threefolds have equivalent derived categories of coherent sheaves, thereby confirming a conjecture due to A. I. Bondal and D. O. Orlov [Semiorthogonal decomposition for algebraic varieties, preprint, http://arxiv.org/math.A6/9506012] in full generality.

However, the difficult part in this paper is the construction of the moduli space of perverse point sheaves. This requires several advanced techniques, including geometric invariant theory, stability theory for sheaves, Quot schemes, and the construction of what the author calls the perverse Hilbert scheme \(P\)-Hilb\((Y/X)\). This scheme parametrizes quotients of the structure sheaf of \(Y\) in the category Per\((Y/X)\), and the author’s third main theorem establishes its existence as a projective scheme.

All in all, this paper is a major contribution towards a deeper understanding of the significance of Fourier-Mukai transforms and of perverse sheaves within Mori’s Minimal Model Program (MMP) in higher-dimensional birational geometry.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

18E30 | Derived categories, triangulated categories (MSC2010) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14J30 | \(3\)-folds |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |