A survey on strong KT structures - SSMR

A <str<strong>on</strong>g>survey</str<strong>on</strong>g> <strong>on</strong> str<strong>on</strong>g KT structures 1053 Str<strong>on</strong>g KT metrics, currents, blow-ups and resoluti<strong>on</strong>sOn a compact complex manifold the Kähler, balanced and str<strong>on</strong>g KT c<strong>on</strong>diti<strong>on</strong>can be characterized by using c<strong>on</strong>diti<strong>on</strong>s <strong>on</strong> the space of positive currents.We start to review some known facts about positive currents. Let Ω be an openset of C n and let Λ p,q (Ω) (respectively by D p,q (Ω)) be the space of (p,q)-forms(respectively (p,q)-forms with compact support) <strong>on</strong> Ω. On the space D p,q (Ω) <strong>on</strong>ec<strong>on</strong>siders the C ∞ -topology and <strong>on</strong>e defines the space of currents of bi-dimensi<strong>on</strong>(p,q) or of bi-degree (n − p,n − q) as the topological dual D p,q(Ω) ′ of D p,q (Ω). Acurrent of bi-dimensi<strong>on</strong> (p,q) <strong>on</strong> Ω can be viewed as a (n − p,n − q)-form <strong>on</strong> Ωwith coefficients distributi<strong>on</strong>s.A current T ∈ D p,q(Ω) ′ is said to be of order 0 if its coefficients are measuresand is called normal if T and dT are currents of order 0.A current T of bi-dimensi<strong>on</strong> (p,p) is real if T(ϕ) = T(ϕ), for any ϕ ∈ D p,q (Ω).Therefore, if T ∈ D p,p(Ω) ′ is real, then T can be expressed as∑T = σ n−p T IJdz I ∧ dz J ,I,Jwhere σ n−p = i(n−p)22 (n−p) , T IJare distributi<strong>on</strong>s <strong>on</strong> Ω such that T JI= T IJand I, Jare multi-indices of length n − p, I = (i 1 ,...,i n−p ), dz I = dz i1 ∧ · · · ∧ dz in−p .A real current T ∈ D ′ p,p(Ω) is positive ifT(σ p ϕ 1 ∧ · · · ∧ ϕ p ∧ ϕ 1 ∧ · · · ∧ ϕ p ) ≥ 0for any choice of ϕ 1 ,...,ϕ p ∈ D 1,0 (Ω), where σ p = ip22 p . Moreover, T is strictlypositive if ϕ 1 ∧ · · · ∧ ϕ p ≠ 0 implies T(σ p ϕ 1 ∧ · · · ∧ ϕ p ∧ ϕ 1 ∧ · · · ∧ ϕ p ) > 0.If T is a positive current of bi-degree (p,p), then T is of order 0.A real current T of bi-dimensi<strong>on</strong> (p,p) <strong>on</strong> Ω is said to be negative if the current−T is positive and plurisubharm<strong>on</strong>ic if i∂∂T is positive.Given a Hermitian structure (J,g) <strong>on</strong> a complex manifold M, then the fundamental2-form F corresp<strong>on</strong>ds to a real strictly positive current of bi-degree(1,1). In particular, if the Hermitian structure (J,g) is str<strong>on</strong>g KT, then thecorresp<strong>on</strong>ding current is ∂∂-closed.An important type of ∂∂-closed currents is given by the (p,p)-comp<strong>on</strong>ents ofa boundary. We recall that a current T of bi-degree (p,p) is said to be the (p,p)-comp<strong>on</strong>ent of a boundary if there exists a real current S of bi-degree (p,p − 1)such that T = ∂S + ∂S.Harvey and Laws<strong>on</strong> proved that a compact complex manifold admits a Kählermetric if and <strong>on</strong>ly if there is no n<strong>on</strong>-zero positive current of bi-dimensi<strong>on</strong> (1,1)which is the (1,1)-comp<strong>on</strong>ent of a boundary (see [26]). A first generalizati<strong>on</strong> ofthe previous result was obtained by Michels<strong>on</strong> in [40] and it is the following: acompact complex manifold has a balanced metric if and <strong>on</strong>ly if there is no n<strong>on</strong>zeropositive current of bi-dimensi<strong>on</strong> (n − 1,n − 1), which is the (n − 1,n − 1)-comp<strong>on</strong>ent of a boundary. In the case of str<strong>on</strong>g KT metrics, Egidi proved in [14]