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  • The homological projective dual of $\operatorname{Sym}^{2}\mathbb{P}(V)$
    Compos. Math. (IF 1.301) Pub Date : 2020-01-17
    Jørgen Vold Rennemo

    We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$ . Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$ . The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

    更新日期:2020-01-17
  • A Witt Nadel vanishing theorem for threefolds
    Compos. Math. (IF 1.301) Pub Date : 2020-01-13
    Yusuke Nakamura; Hiromu Tanaka

    In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_{q}$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$ .

    更新日期:2020-01-13
  • Quantum mirrors of log Calabi–Yau surfaces and higher-genus curve counting
    Compos. Math. (IF 1.301) Pub Date : 2020-01-07
    Pierrick Bousseau

    Gross, Hacking and Keel have constructed mirrors of log Calabi–Yau surfaces in terms of counts of rational curves. Using $q$ -deformed scattering diagrams defined in terms of higher-genus log Gromov–Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding non-commutative algebras of functions.

    更新日期:2020-01-07
  • Loose Engel structures
    Compos. Math. (IF 1.301) Pub Date : 2020-01-06
    Roger Casals; Álvaro del Pino; Francisco Presas

    This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$ -principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

    更新日期:2020-01-06
  • Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations
    Compos. Math. (IF 1.301) Pub Date : 2019-12-09
    Matthew Stoffregen

    We compute the $\text{Pin}(2)$ -equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\unicode[STIX]{x1D6F4}(a_{1},\ldots ,a_{n})$ are not homology cobordant to any $-\unicode[STIX]{x1D6F4}(b_{1},\ldots ,b_{n})$ . We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\text{Pin}(2)$ -equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$ . In particular, we identify an $\mathbb{F}[U]$ -module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

    更新日期:2020-01-04
  • Belyi’s theorem in characteristic two
    Compos. Math. (IF 1.301) Pub Date : 2019-12-18
    Yusuke Sugiyama; Seidai Yasuda

    We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely ramified rational function from the ‘pseudo-tame’ rational function.

    更新日期:2020-01-04
  • Cohomology of generalized configuration spaces
    Compos. Math. (IF 1.301) Pub Date : 2019-12-20
    Dan Petersen

    Let $X$ be a topological space. We consider certain generalized configuration spaces of points on $X$ , obtained from the cartesian product $X^{n}$ by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on $X$ . Suppose that $X$ is a ‘nice’ topological space, $R$ is any commutative ring, $H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$ is the zero map, and that $H_{c}^{\bullet }(X,R)$ is a projective $R$ -module. We prove that the compact support cohomology of any generalized configuration space of points on $X$ depends only on the graded $R$ -module $H_{c}^{\bullet }(X,R)$ . This generalizes a theorem of Arabia.

    更新日期:2020-01-04
  • Hausdorff dimension of divergent trajectories on homogeneous spaces
    Compos. Math. (IF 1.301) Pub Date : 2019-12-19
    Lifan Guan; Ronggang Shi

    For a one-parameter subgroup action on a finite-volume homogeneous space, we consider the set of points admitting divergent-on-average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Cheung [Hausdorff dimension of the set of singular pairs, Ann. of Math. (2) 173 (2011), 127–167].

    更新日期:2020-01-04
  • Vanishing and comparison theorems in rigid analytic geometry
    Compos. Math. (IF 1.301) Pub Date : 2019-12-26
    David Hansen

    We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$ . Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

    更新日期:2020-01-04
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