This is the first of a series of papers about quantization in the context of derived algebraic
geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n …

Derived algebraic geometry Page 1 EMS Surv. Math. Sci. 1 (2014), 153–240 DOI 10.4171/EMSS/4
EMS Surveys in Mathematical Sciences c European Mathematical Society Derived algebraic …

Borisov and Joyce constructed a real virtual cycle on compact moduli spaces of stable
sheaves on Calabi–Yau 4-folds, using derived differential geometry. We construct an …

Enumerative invariants in Algebraic Geometry'count'$\tau $-(semi) stable objects $ E $ with
fixed topological invariants $[E]= a $ in some geometric problem, using a virtual class $[{\cal …

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain
polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a …

Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard
rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter …

This paper is concerned with a non-compact GIT quotient of a vector space, in the presence
of an abelian group action and an equivariant regular function (potential) on the quotient …

We propose a definition of Vafa-Witten invariants counting semistable Higgs pairs on a
polarised surface. We use virtual localisation applied to Mochizuki/Joyce-Song pairs. For …

We prove a Darboux theorem for derived schemes with symplectic forms of degree $ k< 0$,
in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived …

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth
curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure …