$$\pi _1$$ of Miranda moduli spaces of elliptic surfaces

We give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over $${\mathbf {P}}^1$$ P 1 which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of $$SL_2{{\mathbb {Z}}}$$ S L 2 Z presented in terms of $$\Bigg (\begin{array}{ll} 1&{}1\\ 0&{}1\end{array} \Bigg )$$ ( 1 1 0 1 ) , $$\Bigg (\begin{array}{ll} 1&{}0\\ {-1}&{}1\end{array} \Bigg )$$ ( 1 0 - 1 1 ) on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other.Our approach exploits the natural $${\mathbb {Z}}_2$$ Z 2 -action on Weierstrass curves and the identification of $${\mathbb {Z}}_2$$ Z 2 -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over $${{\mathbf {P}}}^1$$ P 1 . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition.

Moduli spaces of abstract and embedded Kummer varieties

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack [Formula: see text] of abstract Kummer varieties and the second one is the stack [Formula: see text] of embedded Kummer varieties. We will prove that [Formula: see text] is a Deligne-Mumford stack and its coarse moduli space is isomorphic to [Formula: see text], the coarse moduli space of principally polarized abelian varieties of dimension [Formula: see text]. On the other hand, we give a modular family [Formula: see text] of embedded Kummer varieties embedded in [Formula: see text], meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space [Formula: see text] of embedded Kummer surfaces and prove that it is obtained from [Formula: see text] by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: [Formula: see text] could be obtained from [Formula: see text] via a contraction for all [Formula: see text].

Smooth covers of moduli stacks of Riemann surfaces with symmetry

We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.

Decomposition of degenerate Gromov–Witten invariants

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

Localizing virtual structure sheaves for almost perfect obstruction theories

Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

Uniformizing the Moduli Stacks of Global G-Shtukas

This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants

In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.