Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Let $$f:S'\longrightarrow S$$ f : S ′ ⟶ S be a cyclic branched covering of smooth projective surfaces over $${\mathbb {C}}$$ C whose branch locus $$\Delta \subset S$$ Δ ⊂ S is a smooth ample divisor. Pick a very ample complete linear system $$|{\mathcal {H}}|$$ | H | on S, such that the polarized surface $$(S, |{\mathcal {H}}|)$$ ( S , | H | ) is not a scroll nor has rational hyperplane sections. For the general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | consider the $$\mu _{n}$$ μ n -equivariant isogeny decomposition of the Prym variety $${{\,\mathrm{Prym}\,}}(C'/C)$$ Prym ( C ′ / C ) of the induced covering $$f:C'{:}{=}f^{-1}(C)\longrightarrow C$$ f : C ′ : = f - 1 ( C ) ⟶ C : $$\begin{aligned} {{\,\mathrm{Prym}\,}}(C'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C). \end{aligned}$$ Prym ( C ′ / C ) ∼ ∏ d | n , d ≠ 1 P d ( C ′ / C ) . We show that for the very general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | the isogeny component $${\mathcal {P}}_{d}(C'/C)$$ P d ( C ′ / C ) is $$\mu _{d}$$ μ d -simple with $${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$$ End μ d ( P d ( C ′ / C ) ) ≅ Z [ ζ d ] . In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map $${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S')$$ P d ( C ′ / C ) ⊂ Jac ( C ′ ) ⟶ Alb ( S ′ ) .

Acylindrical hyperbolicity for Artin groups of dimension 2

In this paper, we show that every irreducible 2-dimensional Artin group $$A_{\Gamma }$$ A Γ of rank at least 3 is acylindrically hyperbolic. We do this by studying the action of $$A_\Gamma $$ A Γ on its modified Deligne complex. Along the way, we prove results of independent interests on the geometry of links of this complex.

A Hurwitz divisor on the moduli of Prym curves

For even genus $$g=2i\ge 4$$ g = 2 i ≥ 4 and the length $$g-1$$ g - 1 partition $$\mu = (4,2,\ldots ,2,-2,\ldots ,-2)$$ μ = ( 4 , 2 , … , 2 , - 2 , … , - 2 ) of 0, we compute the first coefficients of the class of $$\overline{D}(\mu )$$ D ¯ ( μ ) in $$\mathrm {Pic}_{\mathbb {Q}}(\overline{{\mathcal {R}}}_g)$$ Pic Q ( R ¯ g ) , where $$D(\mu )$$ D ( μ ) is the divisor consisting of pairs $$[C,\eta ]\in {\mathcal {R}}_g$$ [ C , η ] ∈ R g with $$\eta \cong {\mathcal {O}}_C(2x_1+x_2+\cdots + x_{i-1}-x_i-\cdots -x_{2i-1})$$ η ≅ O C ( 2 x 1 + x 2 + ⋯ + x i - 1 - x i - ⋯ - x 2 i - 1 ) for some points $$x_1,\ldots , x_{2i-1}$$ x 1 , … , x 2 i - 1 on C. We further provide several enumerative results that will be used for this computation.